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1004_(GTM170)Sheaf Theory | 0 |
# GraduateTexts inMathematics
Glen E. Bredon
# Sheaf Theory
Second Edition Springer
Editorial Board
S. Axler F.W. Gehring P.R. Halmos Springer-Science+Business Media, LLC
## Graduate Texts in Mathematics
1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 Oxtoby. Measure and Category. 2nd ed.
3 Schaefer. Topological Vector Spaces.
4 Hilton/Stammbach. A Course in Homological Algebra. 2nd ed.
5 MAC LANE. Categories for the Working Mathematician.
6 Hughes/Piper. Projective Planes.
7 Serre. A Course in Arithmetic.
8 TAKEUTI/ZARING. Axiomatic Set Theory.
9 Humphreys. Introduction to Lie Algebras and Representation Theory.
10 Cohen. A Course in Simple Homotopy Theory.
11 Conway. Functions of One Complex Variable I. 2nd ed.
12 BEALS. Advanced Mathematical Analysis.
13 Anderson/Fuller. Rings and Categories of Modules. 2nd ed.
14 Golubitsky/Guillemin. Stable Mappings and Their Singularities.
15 Berberlan. Lectures in Functional Analysis and Operator Theory.
16 WINTER. The Structure of Fields.
17 Rosenblatt. Random Processes. 2nd ed.
18 Halmos. Measure Theory.
19 Halmos. A Hilbert Space Problem Book. 2nd ed.
20 Husemoller. Fibre Bundles. 3rd ed.
21 Humphreys. Linear Algebraic Groups.
22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic.
23 Greub. Linear Algebra. 4th ed.
24 Holmes. Geometric Functional Analysis and Its Applications.
25 Hewitt/Stromberg. Real and Abstract Analysis.
26 Manes. Algebraic Theories.
27 Kelley. General Topology.
28 Zariski/Samuel. Commutative Algebra. Vol.I.
29 Zariski/Samuel. Commutative Algebra. Vol.II.
30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts.
31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra.
32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory.
33 Hirsch. Differential Topology.
34 Spitzer. Principles of Random Walk. 2nd ed.
35 Wermer. Banach Algebras and Several Complex Variables. 2nd ed.
36 Kelley/Namioka et al. Linear Topological Spaces.
37 MONK. Mathematical Logic.
38 Grauert/Fritzsche. Several Complex Variables.
39 Arveson. An Invitation to \( {C}^{ * } \) -Algebras.
40 KEMENY/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed.
41 Apostol. Modular Functions and Dirichlet Series in Number Theory. 2nd ed.
42 Serre. Linear Representations of Finite Groups.
43 Gillman/Jerison. Rings of Continuous Functions.
44 Kendig. Elementary Algebraic Geometry.
45 Loëve. Probability Theory I. 4th ed.
46 Loève. Probability Theory II. 4th ed.
47 Moise. Geometric Topology in
Dimensions 2 and 3.
48 SACHS/WU. General Relativity for Mathematicians.
49 Gruenberg/Weir. Linear Geometry. 2nd ed.
50 Edwards. Fermat's Last Theorem.
51 Klingenberg. A Course in Differential Geometry.
52 Hartshorne. Algebraic Geometry.
53 Manin. A Course in Mathematical Logic.
54 Graver/Watkins. Combinatorics with Emphasis on the Theory of Graphs.
55 Brown/Pearcy. Introduction to Operator Theory I: Elements of Functional Analysis.
56 Massey. Algebraic Topology: An Introduction.
57 Crowell/Fox. Introduction to Knot Theory.
58 KOBLITZ. \( p \) -adic Numbers, \( p \) -adic Analysis, and Zeta-Functions. 2nd ed.
59 LANG. Cyclotomic Fields.
60 Arnold. Mathematical Methods in Classical Mechanics. 2nd ed.
Graduate Texts in Mathematics 1 /
Editorial Board
S. Axler F.W. Gehring P.R. Halmos Springer-Science+Business Media, LLC
## Graduate Texts in Mathematics
1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 Oxtoby. Measure and Category. 2nd ed.
3 Schaefer. Topological Vector Spaces.
4 Hilton/Stammbach. A Course in Homological Algebra.
5 MAC LANE. Categories for the Working Mathematician.
6 Hughes/Piper. Projective Planes.
7 Serre. A Course in Arithmetic.
8 TAKEUTI/ZARING. Axiomatic Set Theory.
9 Humphreys. Introduction to Lie Algebras and Representation Theory.
10 Cohen. A Course in Simple Homotopy Theory.
11 Conway. Functions of One Complex Variable I. 2nd ed.
12 Beals. Advanced Mathematical Analysis.
13 Anderson/Fuller. Rings and Categories of Modules. 2nd ed.
14 Golubitsky/Guillemin. Stable Mappings and Their Singularities.
15 Berberlan. Lectures in Functional Analysis and Operator Theory.
16 Winter. The Structure of Fields.
17 Rosenblatt. Random Processes. 2nd ed.
18 Halmos. Measure Theory.
19 Halmos. A Hilbert Space Problem Book. 2nd ed.
20 Husemoller. Fibre Bundles. 3rd ed.
21 Humphreys. Linear Algebraic Groups.
22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic.
23 Greub. Linear Algebra. 4th ed.
24 Holmes. Geometric Functional Analysis and Its Applications.
25 Hewitt/Stromberg. Real and Abstract Analysis.
26 Manes. Algebraic Theories.
27 Kelley. General Topology.
28 Zariski/Samuel. Commutative Algebra. Vol.I.
29 Zariski/Samuel. Commutative Algebra. Vol.II.
30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts.
31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra.
32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galvis Theory.
33 Hirsch. Differential Topology.
34 Spitzer. Principles of Random Walk. 2nd ed.
35 Wermer. Banach Algebras and Several Complex Variables. 2nd ed.
36 Kelley/Namioka et al. Linear Topological Spaces.
37 Monk. Mathematical Logic.
38 Grauert/Fritzsche. Several Complex Variables.
39 Arveson. An Invitation to \( {C}^{ * } \) -Algebras.
40 KEMENY/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed.
41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed.
42 Serre. Linear Representations of Finite Groups.
43 Gillman/Jerison. Rings of Continuous Functions.
44 KENDIG. Elementary Algebraic Geometry.
45 Loëve. Probability Theory I. 4th ed.
46 Loève. Probability Theory II. 4th ed.
47 MoISE. Geometric Topology in Dimensions 2 and 3.
48 SACHS/WU. General Relativity for Mathematicians.
49 Gruenberg/WEIR. Linear Geometry. 2nd ed.
50 Edwards. Fermat's Last Theorem.
51 KLINGENBERG. A Course in Differential Geometry.
52 Hartshorne. Algebraic Geometry.
53 Manin. A Course in Mathematical Logic.
54 Graver/Watkins. Combinatorics with Emphasis on the Theory of Graphs.
55 Brown/Pearcy. Introduction to Operator Theory I: Elements of Functional Analysis.
56 Massey. Algebraic Topology: An Introduction.
57 Crowell/Fox. Introduction to Knot Theory.
58 KOBLITZ. \( p \) -adic Numbers, \( p \) -adic Analysis, and Zeta-Functions. 2nd ed.
59 LANG. Cyclotomic Fields.
60 Arnol.D. Mathematical Methods in Classical Mechanics. 2nd ed.
Glen E. Bredon
# Sheaf Theory
Second Edition
Glen E. Bredon
Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA
Editorial Board
S. Axler
Department of
Mathematics
Michigan State University
East Lansing, MI 48824
USA
F.W. Gehring
Department of
Mathematics
University of Michigan Ann Arbor, MI 48109 USA
P.R. Halmos
Department of
Mathematics
Santa Clara University Santa Clara, CA 95053 USA
## Mathematics Subject Classification (1991): 18F20, 32L10, 54B40
Library of Congress Cataloging-in-Publication Data
Bredon, Glen E.
Sheaf theory / Glen E. Bredon. - 2nd ed.
p. cm. - (Graduate texts in mathematics ; 170)
Includes bibliographical references and index.
ISBN 978-1-4612-6854-3 ISBN 978-1-4612-0647-7 (eBook)
DOI 10.1007/978-1-4612-0647-7
1. Sheaf theory. I. Title. II. Series.
QA612.36.B74 1997
\( {514}^{\prime }{.224} - \mathrm{{dc}}{20} \) 96-44232
Printed on acid-free paper.
The first edition of this book was published by McGraw Hill Book Co., New York-Toronto, Ont. - London, © 1967.
(C) 1997 Springer Science+Business Media New York
Originally published by Springer-Verlag Berlin Heidelberg New York in 1997
Softcover reprint of the hardcover 2nd edition 1997
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer-Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Production managed by Timothy Taylor; manufacturing supervised by Jacqui Ashri.
Camera-ready copy prepared from the author's LaTeX files.
## Preface
This book is primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems." Sheaves play several roles in this study. For example, they provide a suitable notion of "general coefficient systems." Moreover, they furnish us with a common method of defining various cohomology theories and of comparison between different cohomology theories.
The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory.
Several innovations will be found in this book. Notably, the concept of the "tautness" of a subspace (an adaptation of an analogous notion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book. The fact that sheaf-theoretic cohomology satisfies the homotopy property is proved for general topological spaces. \( {}^{1} \) Also, relative cohomology is introduced into sheaf theory. Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a "single space" theory. This is not without reason, since cohomology relative to a closed subspace |
1004_(GTM170)Sheaf Theory | 1 | ology theories.
The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory.
Several innovations will be found in this book. Notably, the concept of the "tautness" of a subspace (an adaptation of an analogous notion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book. The fact that sheaf-theoretic cohomology satisfies the homotopy property is proved for general topological spaces. \( {}^{1} \) Also, relative cohomology is introduced into sheaf theory. Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a "single space" theory. This is not without reason, since cohomology relative to a closed subspace can be obtained by taking coefficients in a certain type of sheaf, while that relative to an open subspace (or, more generally, to a taut subspace) can be obtained by taking cohomology with respect to a special family of supports. However, even in these cases, it is sometimes of notational advantage to have a relative cohomology theory. For example, in our treatment of characteristic classes in Chapter IV the use of relative cohomology enables us to develop the theory in full generality and with relatively simple notation. Our definition of relative cohomology in sheaf theory is the first fully satisfactory one to be given. It is of interest to note that, unlike absolute cohomology, the relative cohomology groups are not the derived functors of the relative cohomology group in degree zero (but they usually are so in most cases of interest).
The reader should be familiar with elementary homological algebra. Specifically, he should be at home with the concepts of category and functor, with the algebraic theory of chain complexes, and with tensor products and direct limits. A thorough background in algebraic topology is also nec-
---
\( {}^{1} \) This is not even restricted to Hausdorff spaces. This result was previously known only for paracompact spaces. The proof uses the notion of a "relatively Hausdorff subspace" introduced here. Although it might be thought that such generality is of no use, it (or rather its mother theorem II-11.1) is employed to advantage when dealing with the derived functor of the inverse limit functor.
---
essary. In Chapters IV, V and VI it is assumed that the reader is familiar with the theory of spectral sequences and specifically with the spectral sequence of a double complex. In Appendix A we give an outline of this theory for the convenience of the reader and to fix our notation.
In Chapter I we give the basic definitions in sheaf theory, develop some basic properties, and discuss the various methods of constructing new sheaves out of old ones. Chapter II, which is the backbone of the book, develops the sheaf-theoretic cohomology theory and many of its properties.
Chapter III is a short chapter in which we discuss the Alexander-Spanier, singular, de Rham, and Čech cohomology theories. The methods of sheaf theory are used to prove the isomorphisms, under suitable restrictions, of these cohomology theories to sheaf-theoretic cohomology. In particular, the de Rham theorem is discussed at some length. Most of this chapter can be read after Section 9 of Chapter II and all of it can be read after Section 12 of Chapter II.
In Chapter IV the theory of spectral sequences is applied to sheaf cohomology and the spectral sequences of Leray, Borel, Cartan, and Fary are derived. Several applications of these spectral sequences are also discussed. These results, particularly the Leray spectral sequence, are among the most important and useful areas of the theory of sheaves. For example, in the theory of transformation groups the Leray spectral sequence of the map to the orbit space is of great interest, as are the Leray spectral sequences of some related mappings; see [15].
Chapter \( \mathrm{V} \) is an exposition of the homology theory of locally compact spaces with coefficients in a sheaf introduced by A. Borel and J. C. Moore. Several innovations are to be found in this chapter. Notably, we give a definition, in full generality, of the homomorphism induced by a map of spaces, and a theorem of the Vietoris type is proved. Several applications of the homology theory are discussed, notably the generalized Poincaré duality theorem for which this homology theory was developed. Other applications are found in the last few sections of this chapter. Notably, three sections are devoted to a fairly complete discussion of generalized manifolds. Because of the depth of our treatment of Borel-Moore homology, the first two sections of the chapter are devoted to technical development of some general concepts, such as the notion and simple properties of a cosheaf and of the operation of dualization between sheaves and cosheaves. This development is not really needed for the definition of the homology theory in the third section, but is needed in the treatment of the deeper properties of the theory in later sections of the chapter. For this reason, our development of the theory may seem a bit wordy and overcomplicated to the neophyte, in comparison to treatments with minimal depth.
In Chapter VI we investigate the theory of cosheaves (on general spaces) somewhat more deeply than in Chapter V. This is applied to Čech homology, enabling us to obtain some uniqueness results not contained in those of Chaper V.
At the end of each chapter is a list of exercises by which the student may check his understanding of the material. The results of a few of the easier exercises are also used in the text. Solutions to many of the exercises are given in Appendix B. Those exercises having solutions in Appendix B are marked with the symbol \( \text{⑤} \) .
The author owes an obvious debt to the book of Godement [40] and to the article of Grothendieck [41], as well as to numerous other works. The book was born as a private set of lecture notes for a course in the theory of sheaves that the author gave at the University of California in the spring of 1964. Portions of the manuscript for the first edition were read by A. Borel, M. Herrera, and E. Spanier, who made some useful suggestions. Special thanks are owed to Per Holm, who read the entire manuscript of that edition and whose perceptive criticism led to several improvements.
This book was originally published by McGraw-Hill in 1967. For this second edition, it has been substantially rewritten with the addition of over eighty examples and of further explanatory material, and, of course, the correction of the few errors known to the author. Some more recent discoveries have been incorporated, particularly in Sections II-16 and IV- 8 regarding cohomology dimension, in Chapter IV regarding the Oliver transfer and the Conner conjecture, and in Chapter V regarding generalized manifolds. The Appendix B of solutions to selected exercises is also a new feature of this edition, one that should greatly aid the student in learning the theory of sheaves. Exercises were chosen for solution on the basis of their difficulty, or because of an interesting solution, or because of the usage of the result in the main text.
Among the items added for this edition are new sections on Čech cohomology, the Oliver transfer, intersection theory, generalized manifolds, locally homogeneous spaces, homological fibrations and \( p \) -adic transformation groups. Also, Chapter VI on cosheaves and Čech homology is new to this edition. It is based on [12].
Several of the added examples assume some items yet to be proved, such as the acyclicity of a contractible space or that sheaf cohomology and singular cohomology agree on nice spaces. Disallowing such forward references would have impoverished our options for the examples.
As well as the common use of the symbol \( ▱ \) to signal the end, or absence, of a proof, we use the symbol \( \diamond \) to indicate the end of an example, although that is usually obvious.
Throughout the book the word "map" means a morphism in the particular category being discussed. Thus for spaces "map" means "continuous function" and for groups "map" means "homomorphism."
Occasionally we use the equal sign to mean a "canonical" isomorphism, perhaps not, strictly speaking, an equality. The word "canonical" is often used for the concept for which the word "natural" was used before category theory gave that word a precise meaning. That is, "canonical" certainly means natural when the latter has meaning, but it means more: that which might be termed "God-given." We shall make no attempt to define that concept precisely. (Thanks to Dennis Sullivan for a theological discussion in 1969.)
The manuscript for this second edition was prepared using the SCIENTIFIC WORD technical word processing software system published by TCI Software research, Inc. This is a "front end" for Donald Knuth's TEX typesetting system and the LATEX extensions to it developed by Leslie Lamport. Without SCIENTIFIC WORD it is doubtful that the author would have had the energy to complete this project.
North Fork, CA 93643
November 22, 1996
## Contents
Preface
I Sheaves and Presheaves 1
1 Definitions . 1
2 Homomorphisms, subsheaves, and quotient sheaves 8
3 Direct and inverse images 12
4 Cohomomorphisms 14
5 Algebraic constructions 16
6 Supports . 21
7 Classical cohomology theories 24
Exercises 30
II Sheaf Cohomology 33
1 Differential sheaves and resolutions 34
2 The canonical resolution and sheaf cohomology 36
3 Injective sheaves 41
4 Acyclic sheaves 46
5 Flabby sheaves 47
6 Connected sequences of functors 52
7 Axioms for cohomology and the cup product 56
8 Maps of spaces 61
9 Φ-soft and Φ-fin |
1004_(GTM170)Sheaf Theory | 2 | this second edition was prepared using the SCIENTIFIC WORD technical word processing software system published by TCI Software research, Inc. This is a "front end" for Donald Knuth's TEX typesetting system and the LATEX extensions to it developed by Leslie Lamport. Without SCIENTIFIC WORD it is doubtful that the author would have had the energy to complete this project.
North Fork, CA 93643
November 22, 1996
## Contents
Preface
I Sheaves and Presheaves 1
1 Definitions . 1
2 Homomorphisms, subsheaves, and quotient sheaves 8
3 Direct and inverse images 12
4 Cohomomorphisms 14
5 Algebraic constructions 16
6 Supports . 21
7 Classical cohomology theories 24
Exercises 30
II Sheaf Cohomology 33
1 Differential sheaves and resolutions 34
2 The canonical resolution and sheaf cohomology 36
3 Injective sheaves 41
4 Acyclic sheaves 46
5 Flabby sheaves 47
6 Connected sequences of functors 52
7 Axioms for cohomology and the cup product 56
8 Maps of spaces 61
9 Φ-soft and Φ-fine sheaves 65
10 Subspaces 71
11 The Vietoris mapping theorem and homotopy invariance 75
12 Relative cohomology 83
13 Mayer-Vietoris theorems 94
14 Continuity 100
The Künneth and universal coefficient theorems 107
16 Dimension 110
Local connectivity 126
Change of supports; local cohomology groups 134
19 The transfer homomorphism and the Smith sequences 137
Steenrod's cyclic reduced powers 148
The Steenrod operations 162
Exercises 169
III Comparison with Other Cohomology Theories 179
1 Singular cohomology 179
2 Alexander-Spanier cohomology 185
3 de Rham cohomology 187
4 Čech cohomology 189
Exercises 194
IV Applications of Spectral Sequences 197
1 The spectral sequence of a differential sheaf 198
2 The fundamental theorems of sheaves 202
3 Direct image relative to a support family 210
4 The Leray sheaf . 213
5 Extension of a support family by a family on the base space 219
6 The Leray spectral sequence of a map 221
7 Fiber bundles 227
8 Dimension 237
9 The spectral sequences of Borel and Cartan 246
10 Characteristic classes 251
11 The spectral sequence of a filtered differential sheaf . 257
12 The Fary spectral sequence 262
13 Sphere bundles with singularities 264
14 The Oliver transfer and the Conner conjecture 267
Exercises 275
V Borel-Moore Homology 279
1 Cosheaves 281
2 The dual of a differential cosheaf 289
3 Homology theory 292
4 Maps of spaces 299
5 Subspaces and relative homology 303
6 The Vietoris theorem, homotopy, and covering spaces 317
7 The homology sheaf of a map 322
8 The basic spectral sequences . 324
9 Poincaré duality 329
10 The cap product 335
11 Intersection theory 344
12 Uniqueness theorems 349
13 Uniqueness theorems for maps and relative homology 358
14 The Künneth formula 364
15 Change of rings 368
16 Generalized manifolds 373
17 Locally homogeneous spaces 392
18 Homological fibrations and \( p \) -adic transformation groups 394
19 The transfer homomorphism in homology 403
Smith theory in homology 407
Exercises 411
VI Cosheaves and Čech Homology 417
1 Theory of cosheaves 418
2 Local triviality 420
3 Local isomorphisms 421
4 Čech homology 424
5 The reflector 428
6 Spectral sequences 431
7 Coresolutions 432
8 Relative Čech homology 434
9 Locally paracompact spaces 438
10 Borel-Moore homology 439
11 Modified Borel-Moore homology 442
12 Singular homology 443
13 Acyclic coverings 445
14 Applications to maps 446
Exercises 448
A Spectral Sequences 449
1 The spectral sequence of a filtered complex 449
2 Double complexes . 451
3 Products . 453
4 Homomorphisms 454
B Solutions to Selected Exercises 455
Solutions for Chapter I . 455
Solutions for Chapter II 459
Solutions for Chapter III 472
Solutions for Chapter IV 473
Solutions for Chapter V 480
Solutions for Chapter VI 486
Bibliography 487
List of Symbols 491
List of Selected Facts 493
Index 495
## Chapter I Sheaves and Presheaves
In this chapter we shall develop the basic properties of sheaves and pre-sheaves and shall give many of the fundamental definitions to be used throughout the book. In Sections 2 and 5 various algebraic operations on sheaves are introduced. If we are given a map between two topological spaces, then a sheaf on either space induces, in a natural way, a sheaf on the other space, and this is the topic of Section 3. Sheaves on a fixed space form a category whose morphisms are called homomorphisms. In Section 4, this fact is extended to the collection of sheaves on all topological spaces with morphisms now being maps \( f \) of spaces together with so-called \( f \) -cohomomorphisms of sheaves on these spaces. In Section 6 the basic notion of a family of supports is defined and a fundamental theorem is proved concerning the relationship between a certain type of presheaf and the cross-sections of the associated sheaf. This theorem is applied in Section 7 to show how, in certain circumstances, the classical singular, Alexander-Spanier, and de Rham cohomology theories can be described in terms of sheaves.
## 1 Definitions
Of central importance in this book is the notion of a presheaf (of abelian groups) on a topological space \( X \) . A presheaf \( A \) on \( X \) is a function that assigns, to each open set \( U \subset X \), an abelian group \( A\left( U\right) \) and that assigns, to each pair \( U \subset V \) of open sets, a homomorphism (called the restriction)
\[
{r}_{U, V} : A\left( V\right) \rightarrow A\left( U\right)
\]
in such a way that
\[
{r}_{U, U} = 1
\]
and
\[
{r}_{U, V}{r}_{V, W} = {r}_{U, W}\;\text{ when }\;U \subset V \subset W.
\]
Thus, using functorial terminology, we have the following definition:
1.1. Definition. Let \( X \) be a topological space. A "presheaf" \( A \) (of abelian groups) on \( X \) is a contravariant functor from the category of open subsets of \( X \) and inclusions to the category of abelian groups.
In general, one may define a presheaf with values in an arbitrary category. Thus, if each \( A\left( U\right) \) is a ring and the \( {r}_{U, V} \) are ring homomorphisms, then \( A \) is called a presheaf of rings. Similarly, let \( A \) be a presheaf of rings on \( X \) and suppose that \( B \) is a presheaf on \( X \) such that each \( B\left( U\right) \) is an \( A\left( U\right) \) -module and the \( {r}_{U, V} : B\left( V\right) \rightarrow B\left( U\right) \) are module homomorphisms [that is, if \( \alpha \in A\left( V\right) ,\beta \in B\left( V\right) \) then \( \left. {{r}_{U, V}\left( {\alpha \beta }\right) = {r}_{U, V}\left( \alpha \right) {r}_{U, V}\left( \beta \right) }\right\rbrack \) . Then \( B \) is said to be an \( A \) -module.
Occasionally, for reasons to be explained later, we refer to elements of \( A\left( U\right) \) as "sections of \( A \) over \( U \) ." If \( s \in A\left( V\right) \) and \( U \subset V \) then we use the notation \( s \mid U \) for \( {r}_{U, V}\left( s\right) \) and call it the "restriction of \( s \) to \( U \) ."
Examples of presheaves are abundant in mathematics. For instance, if \( M \) is an abelian group, then there is the "constant presheaf" \( A \) with \( A\left( U\right) = M \) for all \( U \) and \( {r}_{U, V} = 1 \) for all \( U \subset V \) . We also have the presheaf \( B \) assigning to \( U \) the group (under pointwise addition) \( B\left( U\right) \) of all functions from \( U \) to \( M \), where \( {r}_{U, V} \) is the canonical restriction. If \( M \) is the group of real numbers, we also have the presheaf \( C \), with \( C\left( U\right) \) being the group of all continuous real-valued functions on \( U \) . Similarly, one has the presheaves of differentiable functions on (open subsets of) a differentiable manifold \( X \) ; of differential \( p \) -forms on \( X \) ; of vector fields on \( X \) ; and so on. In algebraic topology one has, for example, the presheaf of singular \( p \) -cochains of open subsets \( U \subset X \) ; the presheaf assigning to \( U \) its \( p \) th singular cohomology group; the presheaf assigning to \( U \) the \( p \) th singular chain group of \( X \) mod \( X - U \) ; and so on.
It is often the case that a presheaf \( A \) on \( X \) will have a relatively simple structure "locally about a point \( x \in X \) ." To make precise what is meant by this, one introduces the notion of a "germ" of \( A \) at the point \( x \in X \) . Consider the set \( \mathfrak{M} \) of all elements \( s \in A\left( U\right) \) for all open sets \( U \subset X \) with \( x \in U \) . We say that the elements \( s \in A\left( U\right) \) and \( t \in A\left( V\right) \) of \( \mathfrak{M} \) are equivalent if there is a neighborhood \( W \subset U \cap V \) of \( x \) in \( X \) with \( {r}_{W, U}\left( s\right) = {r}_{W, V}\left( t\right) \) . The equivalence classes of \( \mathfrak{M} \) under this equivalence relation are called the germs of \( A \) at \( x \) . The equivalence class containing \( s \in A\left( U\right) \) is called the germ of \( s \) at \( x \in U \) . Thus, for example, one has the notion of the germ of a continuous real-valued function \( f \) at any point of the domain of \( f \) .
Of course, the set \( {\mathcal{A}}_{x} \) of germs of \( A \) at \( x \) that we have constructed is none other than the direct limit
\[
{\mathcal{A}}_{x} = \mathop{\lim }\limits_{ \rightarrow }A\left( U\right)
\]
where \( U \) ranges over the open neighborhoods of \( x \) in \( X \) . The set \( {\mathcal{A}}_{x} \) inherits a canonical group structure from the groups \( A\left( U\right) \) . The disjoint union \( \mathcal{A} \) of the \( {\mathcal{A}}_{x} \) for \( x \in X \) provides information about the local structure of \( A \), but most global structure has been lost, since we have discarded all relationships between the \( {\mathcal{A}}_{x} \) for \( x \) varying. In order to retrieve some global structure, a topology is introduced into the set \( \mathcal{A} \) of germs of \( A \), as follows. Fix an element |
1004_(GTM170)Sheaf Theory | 3 | m of \( s \) at \( x \in U \) . Thus, for example, one has the notion of the germ of a continuous real-valued function \( f \) at any point of the domain of \( f \) .
Of course, the set \( {\mathcal{A}}_{x} \) of germs of \( A \) at \( x \) that we have constructed is none other than the direct limit
\[
{\mathcal{A}}_{x} = \mathop{\lim }\limits_{ \rightarrow }A\left( U\right)
\]
where \( U \) ranges over the open neighborhoods of \( x \) in \( X \) . The set \( {\mathcal{A}}_{x} \) inherits a canonical group structure from the groups \( A\left( U\right) \) . The disjoint union \( \mathcal{A} \) of the \( {\mathcal{A}}_{x} \) for \( x \in X \) provides information about the local structure of \( A \), but most global structure has been lost, since we have discarded all relationships between the \( {\mathcal{A}}_{x} \) for \( x \) varying. In order to retrieve some global structure, a topology is introduced into the set \( \mathcal{A} \) of germs of \( A \), as follows. Fix an element \( s \in A\left( U\right) \) . Then for each \( x \in U \) we have the germ \( {s}_{x} \) of \( s \) at \( x \) . For \( s \) fixed, the set of all germs \( {s}_{x} \in {\mathcal{A}}_{x} \) for \( x \in U \) is taken
to be an open set in \( \mathcal{A} \) . The topology of \( \mathcal{A} \) is taken to be the topology generated by these open sets. (We shall describe this more precisely later in this section.) With this topology, \( \mathcal{A} \) is called "the sheaf generated by the presheaf \( A \) " or "the sheaf of germs of \( A \) ," and we denote this by
\[
\mathcal{A} = \text{ Sheaf }\left( A\right) \;\text{ or }\;\mathcal{A} = \text{ Sheaf }\left( {U \mapsto A\left( U\right) }\right) .
\]
In general, the topology of \( \mathcal{A} \) is highly non-Hausdorff.
There is a natural map \( \pi : \mathcal{A} \rightarrow X \) taking \( {\mathcal{A}}_{x} \) into the point \( x \) . It will be verified later in this section that \( \pi \) is a local homeomorphism. That is, each point \( t \in \mathcal{A} \) has a neighborhood \( N \) such that the restriction \( \pi \mid N \) is a homeomorphism onto a neighborhood of \( \pi \left( t\right) \) . (The set \( \left\{ {{s}_{x} \mid x \in U}\right\} \) for \( s \in A\left( U\right) \) is such a set \( N \) .) Also it is the case that in a certain natural sense, the group operations in \( {\mathcal{A}}_{x} \), for \( x \) varying, are continuous in \( x \) . These facts lead us to the basic definition of a sheaf on \( X \) :
1.2. Definition. A "sheaf" (of abelian groups) on \( X \) is a pair \( \left( {\mathcal{A},\pi }\right) \) where:
(i) \( \mathcal{A} \) is a topological space (not Hausdorff in general);
(ii) \( \pi : \mathcal{A} \rightarrow X \) is a local homeomorphism onto \( X \) ;
(iii) each \( {\mathcal{A}}_{x} = {\pi }^{-1}\left( x\right) \), for \( x \in X \), is an abelian group (and is called the "stalk" of \( \mathcal{A} \) at \( x \) );
(iv) the group operations are continuous.
(In practice, we always regard the map \( \pi \) as being understood and we speak of the sheaf \( \mathcal{A} \) .) The meaning of (iv) is as follows: Let \( \mathcal{A}\mathcal{A} \) be the subspace of \( \mathcal{A} \times \mathcal{A} \) consisting of those pairs \( \langle \alpha ,\beta \rangle \) with \( \pi \left( \alpha \right) = \) \( \pi \left( \beta \right) \) . Then the function \( \mathcal{A}\Delta \mathcal{A} \rightarrow \mathcal{A} \) taking \( \langle \alpha ,\beta \rangle \mapsto \alpha - \beta \) is continuous. [Equivalently, \( \alpha \mapsto - \alpha \) of \( \mathcal{A} \rightarrow \mathcal{A} \) is continuous and \( \langle \alpha ,\beta \rangle \mapsto \alpha + \beta \) of \( \mathcal{A}\Delta \mathcal{A} \rightarrow \mathcal{A} \) is continuous.]
Similarly one may define, for example, a sheaf of rings or a module (sheaf of modules) over a sheaf of rings.
Thus, for a sheaf \( \mathcal{R} \) of abelian groups to be a sheaf of rings, each stalk is assumed to have the (given) structure of a ring, and the map \( \langle \alpha ,\beta \rangle \mapsto {\alpha \beta } \) of \( \mathcal{R}\bigtriangleup \mathcal{R} \rightarrow \mathcal{R} \) is assumed to be continuous (in addition to (iv)). By a sheaf of rings with unit we mean a sheaf of rings in which each stalk has a unit and the assignment to each \( x \in X \) of the unit \( {1}_{x} \in {\mathcal{R}}_{x} \) is continuous. \( {}^{1} \)
If \( \mathcal{R} \) is a sheaf of rings and if \( \mathcal{A} \) is a sheaf in which each stalk \( {\mathcal{A}}_{x} \) has a given \( {\mathcal{R}}_{x} \) -module structure, then \( \mathcal{A} \) is called an \( \mathcal{R} \) -module (or a module over \( \mathcal{R} \) ) if the map \( \mathcal{R}\bigtriangleup \mathcal{A} \rightarrow \mathcal{A} \) given by \( \langle \rho ,\alpha \rangle \mapsto {\rho \alpha } \) is continuous, where, of course, \( \mathcal{R}\bigtriangleup \mathcal{A} = \{ \langle \rho ,\alpha \rangle \in \mathcal{R} \times \mathcal{A} \mid \pi \left( \rho \right) = \pi \left( \alpha \right) \} \) .
\( {}^{1} \) Example 1.13 shows that this latter condition is not superfluous.
For example, the sheaf \( {\Omega }^{0} \) of germs of smooth real-valued functions on a differentiable manifold \( {M}^{n} \) is a sheaf of rings with unit, and the sheaf \( {\Omega }^{p} \) of germs of differential \( p \) -forms on \( {M}^{n} \) is an \( {\Omega }^{0} \) -module; see Section 7.
If \( \mathcal{A} \) is a sheaf on \( X \) with projection \( \pi : \mathcal{A} \rightarrow X \) and if \( Y \subset X \), then the restriction \( \mathcal{A} \mid Y \) of \( \mathcal{A} \) is defined to be
\[
\mathcal{A} \mid Y = {\pi }^{-1}\left( Y\right)
\]
which is a sheaf on \( Y \) .
If \( \mathcal{A} \) is a sheaf on \( X \) and if \( Y \subset X \), then a section (or cross section) of \( \mathcal{A} \) over \( Y \) is a map \( s : Y \rightarrow \mathcal{A} \) such that \( \pi \circ s \) is the identity. Clearly the pointwise sum or difference (or product in a sheaf of rings, and so on) of two sections over \( Y \) is a section over \( Y \) .
Every point \( x \in Y \) admits a section \( s \) over some neighborhood \( U \) of \( x \) by (ii). It follows that \( s - s \) is a section over \( U \) taking the value 0 in each stalk. This shows that the zero section \( 0 : X \rightarrow \mathcal{A} \), is indeed a section. It follows that for any \( Y \subset X \), the set \( \mathcal{A}\left( Y\right) \) of sections over \( Y \) forms an abelian group. Similarly, \( \mathcal{R}\left( Y\right) \) is a ring (with unit) if \( \mathcal{R} \) is a sheaf of rings (with unit), and moreover, \( \mathcal{A}\left( Y\right) \) is an \( \mathcal{R}\left( Y\right) \) -module if \( \mathcal{A} \) is an \( \mathcal{R} \) -module.
Clearly, the restriction \( \mathcal{A}\left( Y\right) \rightarrow \mathcal{A}\left( {Y}^{\prime }\right) \), for \( {Y}^{\prime } \subset Y \), is a homomorphism. Thus, in particular, the assignment \( U \mapsto \mathcal{A}\left( U\right) \), for open sets \( U \subset X \), defines a presheaf on \( X \) . This presheaf is called the presheaf of sections of \( \mathcal{A} \) .
Another common notation for the group of all sections of \( \mathcal{A} \) is
\[
\Gamma \left( \mathcal{A}\right) = \mathcal{A}\left( X\right)
\]
See Section 6 for an elaboration on this notation.
We shall now list some elementary consequences of Definition 1.2. The reader may supply any needed argument.
(a) \( \pi \) is an open map.
(b) Any section of \( \mathcal{A} \) over an open set is an open map.
(c) Any element of \( \mathcal{A} \) is in the range of some section over some open set.
(d) The set of all images of sections over open sets is a base for the topology of \( \mathcal{A} \) .
(e) For any two sections \( s \in \mathcal{A}\left( U\right) \) and \( t \in \mathcal{A}\left( V\right), U \) and \( V \) open, the set \( W \) of points \( x \in U \cap V \) such that \( s\left( x\right) = t\left( x\right) \) is open.
Note that if \( \mathcal{A} \) were Hausdorff then the set \( W \) of (e) would also be closed in \( U \cap V \) . That is generally false for sheaves. Thus (e) indicates the "strangeness" of the topology of \( \mathcal{A} \) . It is a consequence of part (ii) of Definition 1.2.
1.3. Example. A simple example of a non-Hausdorff sheaf is the sheaf on the real line that has zero stalk everywhere but at 0, and has stalk \( {\mathbb{Z}}_{2} \) at 0 . There is only one topology consistent with Definition 1.2, and the two points in the stalk at 0 cannot be separated by open sets (sections over open sets in \( \mathbb{R} \) ). As a topological space, this is the standard example of a non-Hausdorff 1-manifold.
1.4. Example. Perhaps a more illuminating and more important example of a non-Hausdorff sheaf is the sheaf \( \mathcal{C} \) of germs of continuous real-valued functions on \( \mathbb{R} \) . The function \( f\left( x\right) = x \) for \( x \geq 0 \) and \( f\left( x\right) = 0 \) for \( x \leq 0 \) has a germ \( {f}_{0} \) at \( 0 \in \mathbb{R} \) that does not equal the germ \( {0}_{0} \) of the zero function, but a section through \( {f}_{0} \) takes value 0 in the stalk at \( x \) for all \( x < 0 \) sufficiently near 0 . Thus \( {f}_{0} \) and \( {0}_{0} \) cannot be separated by open sets in \( \mathcal{C} \) . The sheaf of germs of differentiable functions gives a similar example, but the sheaf of germs of real analytic functions is Hausdorff.
1.5. We now describe more precisely the construction of the sheaf generated by a given presheaf.
Let \( A \) be a presheaf on \( X \) . For each open set \( U \subset X \) consider the space \( U \times A\left( U\right) \), where \( U \) has the subspace topology and \( A\left( U\right) \) has the discrete topology. Form the topological sum
\[
E = \mathop{+}\limits_{{U \subset X}}\left( {U \times A\left( U\right) }\right)
\]
Consider the following equivalence relation \( R \) on \( E \) : If \( \langle x, s\rangle \in U \ |
1004_(GTM170)Sheaf Theory | 4 | eq 0 \) has a germ \( {f}_{0} \) at \( 0 \in \mathbb{R} \) that does not equal the germ \( {0}_{0} \) of the zero function, but a section through \( {f}_{0} \) takes value 0 in the stalk at \( x \) for all \( x < 0 \) sufficiently near 0 . Thus \( {f}_{0} \) and \( {0}_{0} \) cannot be separated by open sets in \( \mathcal{C} \) . The sheaf of germs of differentiable functions gives a similar example, but the sheaf of germs of real analytic functions is Hausdorff.
1.5. We now describe more precisely the construction of the sheaf generated by a given presheaf.
Let \( A \) be a presheaf on \( X \) . For each open set \( U \subset X \) consider the space \( U \times A\left( U\right) \), where \( U \) has the subspace topology and \( A\left( U\right) \) has the discrete topology. Form the topological sum
\[
E = \mathop{+}\limits_{{U \subset X}}\left( {U \times A\left( U\right) }\right)
\]
Consider the following equivalence relation \( R \) on \( E \) : If \( \langle x, s\rangle \in U \times A\left( U\right) \) and \( \langle y, t\rangle \in V \times A\left( V\right) \) then \( \langle x, s\rangle R\langle y, t\rangle \Leftrightarrow (x = y \) and there exists an open neighborhood \( W \) of \( x \) with \( W \subset U \cap V \) and \( s\left| {W = t}\right| W \) ).
Let \( \mathcal{A} \) be the quotient space \( E/R \) and let \( \pi : \mathcal{A} \rightarrow X \) be the projection induced by the map \( p : E \rightarrow X \) taking \( \langle x, s\rangle \mapsto x \) . We have the commutative diagram
Recall that the topology of \( \mathcal{A} = E/R \) is defined by: \( Y \subset \mathcal{A} \) is open \( \Leftrightarrow {q}^{-1}\left( Y\right) \) is open in \( E \) . Note also that for any open subset \( {E}^{\prime } \) of \( E \), the saturation \( R\left( {E}^{\prime }\right) = {q}^{-1}q\left( {E}^{\prime }\right) \) of \( {E}^{\prime } \) is open. Thus \( q \) is an open map. Now \( \pi \) is continuous, since \( p \) is open and \( q \) is continuous; \( \pi \) is locally one-to-one, since \( p \) is locally one-to-one and \( q \) is onto. Thus \( \pi \) is a local homeomorphism.
Clearly \( {\mathcal{A}}_{x} = {\pi }^{-1}\left( x\right) \) is the direct limit of \( A\left( U\right) \) for \( U \) ranging over the open neighborhoods of \( x \) . Thus the stalk \( {\mathcal{A}}_{x} \) has a canonical group structure. It is easy to see that the group operations in \( \mathcal{A} \) are continuous since they are so in \( E \) . Therefore \( \mathcal{A} \) is a sheaf.
\( \mathcal{A} \) is called the sheaf generated by the presheaf \( A \) . As we have noted, this is denoted by \( \mathcal{A} = \) Sheaf \( \left( A\right) \) or \( \mathcal{A} = \) Sheaf \( \left( {U \mapsto A\left( U\right) }\right) \) .
1.6. Let \( {\mathcal{A}}_{0} \) be a sheaf, \( A \) the presheaf of sections of \( {\mathcal{A}}_{0} \), and \( \mathcal{A} = \) Then \( f\left( A\right) \) . Any element of \( {\mathcal{A}}_{0} \) lying over \( x \in X \) has a local section about it, and this determines an element of \( \mathcal{A} \) over \( x \) . This gives a canonical function \( \lambda : {\mathcal{A}}_{0} \rightarrow \mathcal{A} \) . By the definition of the topology of \( \mathcal{A},\lambda \) is open and continuous. It is also bijective on each stalk, and hence globally. Therefore \( \lambda \) is a homeomorphism. It also preserves group operations. Thus \( {\mathcal{A}}_{0} \) and \( \mathcal{A} \) are essentially the same. For this reason we shall usually not distinguish between a sheaf and its presheaf of sections and shall denote them by the same symbol.
1.7. Let \( A \) be a presheaf and \( \mathcal{A} \) the sheaf that it generates. For any open set \( U \subset X \) there is a natural map \( {\theta }_{U} : A\left( U\right) \rightarrow \mathcal{A}\left( U\right) \) (recall the construction of \( \mathcal{A} \) ) that is a homomorphism and commutes with restrictions (which is the meaning of "natural"). When is \( {\theta }_{U} \) an isomorphism for all \( U \) ? Recalling that \( {\mathcal{A}}_{x} = \mathop{\lim }\limits_{{x \in U}}A\left( U\right) \), it follows that an element \( s \in A\left( U\right) \) is in \( \operatorname{Ker}{\theta }_{U} \Leftrightarrow s \) is "locally trivial" (that is, for every \( x \in U \) there is a neighborhood \( V \) of \( x \) such that \( s \mid V = 0 \) ).
Thus \( {\theta }_{U} \) is a monomorphism for all \( U \subset X \Leftrightarrow \) the following condition holds:
(S1) If \( U = \mathop{\bigcup }\limits_{\alpha }{U}_{\alpha } \), with \( {U}_{\alpha } \) open in \( X \), and \( s, t \in A\left( U\right) \) are such that
\[
s\left| {{U}_{\alpha } = t}\right| {U}_{\alpha }\text{for all}\alpha \text{, then}s = t{.}^{2}
\]
A presheaf satisfying condition (S1) is called a monopresheaf.
Similarly, let \( t \in \mathcal{A}\left( U\right) \) . For each \( x \in U \) there is a neighborhood \( {U}_{x} \) of \( x \) and an element \( {s}_{x} \in A\left( {U}_{x}\right) \) with \( {\theta }_{{U}_{x}}\left( {s}_{x}\right) \left( x\right) = t\left( x\right) \) . Since \( \pi : \mathcal{A} \rightarrow X \) is a local homeomorphism, \( \theta \left( {s}_{x}\right) \) and \( t \) coincide in some neighborhood \( {V}_{x} \) of \( x \) . We may assume that \( {V}_{x} = {U}_{x} \) . Now \( \theta \left( {{s}_{x} \mid {U}_{x} \cap {U}_{y}}\right) = \theta \left( {{s}_{y} \mid {U}_{x} \cap {U}_{y}}\right) \) so that if (S1) holds, we obtain that \( {s}_{x}\left| {{U}_{x} \cap {U}_{y} = {s}_{y}}\right| {U}_{x} \cap {U}_{y} \) . If \( A \) were a presheaf of sections (of any map), then this condition would imply that the \( {s}_{x} \) are restrictions to \( {U}_{x} \) of a section \( s \in A\left( U\right) \) . Conversely, if there is an element \( s \in A\left( U\right) \) with \( s \mid {U}_{x} = {s}_{x} \) for all \( x \), then \( \theta \left( s\right) = t \) .
We have shown that if (S1) holds, then \( {\theta }_{U} \) is surjective for all \( U \) (and hence is an isomorphism) \( \Leftrightarrow \) the following condition is satisfied:
(S2) Let \( \left\{ {U}_{\alpha }\right\} \) be a collection of open sets in \( X \) and let \( U = \bigcup {U}_{\alpha } \) . If \( {s}_{\alpha } \in A\left( {U}_{\alpha }\right) \) are given such that \( {s}_{\alpha }\left| {{U}_{\alpha } \cap {U}_{\beta } = {s}_{\beta }}\right| {U}_{\alpha } \cap {U}_{\beta } \) for all \( \alpha ,\beta \) , then there exists an element \( s \in A\left( U\right) \) with \( s \mid {U}_{\alpha } = {s}_{\alpha } \) for all \( \alpha \) .
A presheaf satisfying (S2) is called conjunctive. If it only satisfies (S2) for a particular collection \( \left\{ {U}_{\alpha }\right\} \), then it is said to be conjunctive for \( \left\{ {U}_{\alpha }\right\} \) .
Thus, sheaves are in one-to-one correspondence with presheaves satisfying (S1) and (S2), that is, with conjunctive monopresheaves. For this
\( {}^{2} \) Clearly, we could take \( t = 0 \) here, i.e., replace \( \left( {s, t}\right) \) with \( \left( {s - t,0}\right) \) . However, the condition is phrased so that it applies to presheaves of sets. reason it is common practice not to distinguish between sheaves and conjunctive monopresheaves. \( {}^{3} \)
Note that with the notation \( {U}_{\alpha ,\beta } = {U}_{\alpha } \cap {U}_{\beta } \) ,(S1) and (S2) are equivalent to the hypothesis that the sequence
\[
0 \rightarrow A\left( U\right) \overset{f}{ \rightarrow }\mathop{\prod }\limits_{\alpha }A\left( {U}_{\alpha }\right) \overset{g}{ \rightarrow }\mathop{\prod }\limits_{{\langle \alpha ,\beta \rangle }}A\left( {U}_{\alpha ,\beta }\right)
\]
is exact, where \( f\left( s\right) = \mathop{\prod }\limits_{\alpha }\left( {s \mid {U}_{\alpha }}\right) \) and
\[
g\left( {\mathop{\prod }\limits_{\alpha }{s}_{\alpha }}\right) = \mathop{\prod }\limits_{{\langle \alpha ,\beta \rangle }}\left( {{s}_{\alpha }\left| {{U}_{\alpha ,\beta } - {s}_{\beta }}\right| {U}_{\alpha ,\beta }}\right)
\]
where \( \langle \alpha ,\beta \rangle \) denotes ordered pairs of indices.
1.8. Definition. Let \( \mathcal{A} \) be a sheaf on \( X \) and let \( Y \subset X \) . Then \( \mathcal{A} \mid Y = \) \( {\pi }^{-1}\left( Y\right) \) is a sheaf on \( Y \) called the "restriction" of \( \mathcal{A} \) to \( Y \) .
1.9. Definition. Let \( G \) be an abelian group. The "constant" sheaf on \( X \) with stalk \( G \) is the sheaf \( X \times G \) (giving \( G \) the discrete topology). It is also denoted by \( G \) when the context indicates this as a sheaf. A sheaf \( \mathcal{A} \) on \( X \) is said to be "locally constant" if every point of \( X \) has a neighborhood \( U \) such that \( \mathcal{A} \mid U \) is constant.
1.10. Definition. If \( \mathcal{A} \) is a sheaf on \( X \) and \( s \in \mathcal{A}\left( X\right) \) is a section, then the "support" of \( s \) is defined to be the closed set \( \left| s\right| = \{ x \in X \mid s\left( x\right) \neq 0\} \) .
The set \( \left| s\right| \) is closed since its complement is the set of points at which \( s \) coincides with the zero section, and that is open by item (e) on page 4 .
1.11. Example. An important example of a sheaf is the orientation sheaf on an \( n \) -manifold \( {M}^{n} \) . Using singular homology, this can be defined as the sheaf \( {\mathcal{O}}_{n} = {\mathcal{H}}_{\text{heaf }}\left( {U \mapsto {H}_{n}\left( {{M}^{n},{M}^{n} - U;\mathbb{Z}}\right) }\right) \) . It is easy to see that this is a locally constant sheaf with stalks \( \mathbb{Z} \) . It is constant if \( {M}^{n} \) is orientable. If \( {M}^{n} \) has a boundary then \( {\mathcal{O}}_{n} \) is no longer locally constant since its stalks are zero over points of the boundary.
More generally, for any space \( X \) and index \( p \) there is the sheaf \( {\mathcal{H}}_{p}\left( X\right) = \) Then \( \ell \left( {U \mapsto {H}_{p}\left( {X, X - U;\mathbb{Z}} |
1004_(GTM170)Sheaf Theory | 5 | ight| = \{ x \in X \mid s\left( x\right) \neq 0\} \) .
The set \( \left| s\right| \) is closed since its complement is the set of points at which \( s \) coincides with the zero section, and that is open by item (e) on page 4 .
1.11. Example. An important example of a sheaf is the orientation sheaf on an \( n \) -manifold \( {M}^{n} \) . Using singular homology, this can be defined as the sheaf \( {\mathcal{O}}_{n} = {\mathcal{H}}_{\text{heaf }}\left( {U \mapsto {H}_{n}\left( {{M}^{n},{M}^{n} - U;\mathbb{Z}}\right) }\right) \) . It is easy to see that this is a locally constant sheaf with stalks \( \mathbb{Z} \) . It is constant if \( {M}^{n} \) is orientable. If \( {M}^{n} \) has a boundary then \( {\mathcal{O}}_{n} \) is no longer locally constant since its stalks are zero over points of the boundary.
More generally, for any space \( X \) and index \( p \) there is the sheaf \( {\mathcal{H}}_{p}\left( X\right) = \) Then \( \ell \left( {U \mapsto {H}_{p}\left( {X, X - U;\mathbb{Z}}\right) }\right) \), which is called the " \( p \) -th local homology sheaf" of \( X \) . Generally, it has a rather complicated structure. The reader would benefit by studying it for some simple spaces. For example, the sheaf \( {\mathcal{H}}_{1}\left( \bot \right) \) has stalk \( \mathbb{Z} \oplus \mathbb{Z} \) at the triple point, stalks 0 at the three end points, and stalks \( \mathbb{Z} \) elsewhere. How do these stalks fit together?
---
\( {}^{3} \) Indeed, in certain generalizations of the theory, Definition 1.2 is not available and the other notion is used. This will not be of concern to us in this book.
---
1.12. Example. Consider the presheaf \( P \) on the real line \( \mathbb{R} \) that assigns to an open set \( U \subset \mathbb{R} \), the group \( P\left( U\right) \) of all real-valued polynomial functions on \( U \) . Then \( P \) is a monopresheaf that is conjunctive for coverings of \( \mathbb{R} \) , but it is not conjunctive for arbitrary collections of open sets. For example, the element \( 1 \in P\left( \left( {0,1}\right) \right) \) (the constant function with value 1) and the element \( x \in P\left( \left( {2,3}\right) \right) \) do not come from any single polynomial on \( \left( {0,1}\right) \cup \) \( \left( {2,3}\right) \) . The sheaf \( \mathcal{P} = \mathcal{{Meaf}}\left( P\right) \) has for \( \mathcal{P}\left( U\right) \) the functions that are "locally polynomials"; e.g.,1 and \( x \), as before, do combine to give an element of \( \mathcal{P}\left( {\left( {0,1}\right) \cup \left( {2,3}\right) }\right) \) . Important examples of this type of behavior are given in Section 7 and Exercise 12.
1.13. Example. Consider the presheaf \( A \) on \( X = \left\lbrack {0,1}\right\rbrack \) with \( A\left( U\right) = \mathbb{Z} \) for all \( U \neq \varnothing \) and with \( {r}_{V, U} : A\left( U\right) \rightarrow A\left( V\right) \) equal to the identity if \( 0 \in V \) or if \( 0 \notin U \) but \( {r}_{V, U} = 0 \) if \( 0 \in U - V \) . Let \( \mathcal{A} = \mathcal{R} \) heaf \( \left( A\right) \) . Then \( {\mathcal{A}}_{x} \approx \mathbb{Z} \) for all \( x \) . However, any section over \( \lbrack 0,\varepsilon ) \) takes the value \( 0 \in {\mathcal{A}}_{x} \) for \( x \neq 0 \) , but can be arbitrary in \( {\mathcal{A}}_{0} \approx \mathbb{Z} \) for \( x = 0 \) . The restriction \( \mathcal{A}|(0,1\rbrack \) is constant. Thus \( \mathcal{A} \) is a sheaf of rings but not a sheaf of rings with unit. (In the notation of 2.6 and Section 5, \( \mathcal{A} \approx {\mathbb{Z}}_{\{ 0\} } \oplus {\mathbb{Z}}_{(0,1\rbrack } \) .)
1.14. Example. A sheaf can also be described as being generated by a "presheaf" defined only on a basis of open sets. For example, on the circle \( {\mathbb{S}}^{1} \), consider the basis \( \mathcal{B} \) consisting of open arcs \( U \) of \( {\mathbb{S}}^{1} \) . For \( U \in \mathcal{B} \) and for \( x, y \in U \) we write \( x > y \) if \( y \) is taken into \( x \) through \( U \) by a counterclockwise rotation. Fix a point \( {x}_{0} \in {\mathbb{S}}^{1} \) . For \( U \in \mathcal{B} \) let \( A\left( U\right) = \mathbb{Z} \) and for \( U, V \in \mathcal{B} \) with \( V \subset U \) let \( {r}_{V, U} = 1 \) if \( {x}_{0} \in V \) or if \( {x}_{0} \notin U \) (i.e., if \( {x}_{0} \) is in both \( U \) and \( V \) or in neither). If \( {x}_{0} \in U - V \) then let \( {r}_{V, U} = 1 \) if \( {x}_{0} > y \) for all \( y \in V \) , and \( {r}_{V, U} = n \) (multiplication by the integer \( n \) ) if \( {x}_{0} < y \) for all \( y \in V \) . This generates an interesting sheaf \( {\mathcal{A}}_{n} \) on \( {\mathbb{S}}^{1} \) . It can be described directly (and more easily) as the quotient space \( \left( {-1,1}\right) \times \mathbb{Z} \) modulo the identification \( \langle t, k\rangle \sim \langle t - 1,{nk}\rangle \) for \( 0 < t < 1 \), and with the projection \( \left\lbrack {\langle t, k\rangle }\right\rbrack \mapsto \left\lbrack t\right\rbrack \) to \( {\mathbb{S}}^{1} = \left( {-1,1}\right) /\{ t \sim \left( {t - 1}\right) \} \) . Note, in particular, the cases \( n = 0, - 1 \) . The sheaf \( {\mathcal{A}}_{n} \) is Hausdorff for \( n \neq 0 \) but not for \( n = 0 \) .
## 2 Homomorphisms, subsheaves, and quotient sheaves
In this section we fix the base space \( X \) . A homomorphism of presheaves \( h \) : \( A \rightarrow B \) is a collection of homomorphisms \( {h}_{U} : A\left( U\right) \rightarrow B\left( U\right) \) commuting with restrictions. That is, \( h \) is a natural transformation of functors.
A homomorphism of sheaves \( h : \mathcal{A} \rightarrow \mathcal{B} \) is a map such that \( h\left( {\mathcal{A}}_{x}\right) \subset \) \( {\mathcal{B}}_{x} \) for all \( x \in X \) and the restriction \( {h}_{x} : {\mathcal{A}}_{x} \rightarrow {\mathcal{B}}_{x} \) of \( h \) to stalks is a homomorphism for all \( x \) .
A homomorphism of sheaves induces a homomorphism of the presheaves of sections in the obvious way. Conversely, let \( h : A \rightarrow B \) be a homomorphism of presheaves (not necessarily satisfying (S1) and (S2)). For each
\( x \in X, h \) induces a homomorphism \( {h}_{x} : {\mathcal{A}}_{x} = \mathop{\lim }\limits_{{x \in U}}A\left( U\right) \rightarrow \mathop{\lim }\limits_{{x \in U}}B\left( U\right) = {\mathcal{B}}_{x} \) and therefore a function \( h : \mathcal{A} \rightarrow \mathcal{B} \) . If \( s \in A\left( U\right) \) then \( h \) maps the section \( \theta \left( s\right) \in \mathcal{A}\left( U\right) \) onto the section \( \theta \left( {h\left( s\right) }\right) \in \mathcal{B}\left( U\right) \) . Thus \( h \) is continuous (since the projections to \( U \) are local homeomorphisms and take this function to the identity map).
The group of all homomorphisms \( \mathcal{A} \rightarrow \mathcal{B} \) is denoted by \( \operatorname{Hom}\left( {\mathcal{A},\mathcal{B}}\right) \) .
2.1. Definition. A "subsheaf" A of a sheaf B is an open subspace of B such that \( {\mathcal{A}}_{x} = \mathcal{A} \cap {\mathcal{B}}_{x} \) is a subgroup of \( {\mathcal{B}}_{x} \) for all \( x \in X \) . (That is, \( \mathcal{A} \) is a subspace of \( \mathcal{B} \) that is a sheaf on \( X \) with the induced algebraic structure.)
If \( h : \mathcal{A} \rightarrow \mathcal{B} \) is a homomorphism of sheaves, then
\[
\operatorname{Ker}h = \{ \alpha \in \mathcal{A} \mid h\left( \alpha \right) = 0\}
\]
is a subsheaf of \( \mathcal{A} \) and \( \operatorname{Im}h \) is a subsheaf of \( \mathcal{B} \) . We define exact sequences of sheaves as usual; that is, the sequence \( \mathcal{A}\overset{f}{ \rightarrow }\mathcal{B}\overset{g}{ \rightarrow }\mathcal{C} \) of sheaves is exact if \( \operatorname{Im}f = \operatorname{Ker}g \) . Note that such a sequence of sheaves is exact \( \Leftrightarrow \) each \( {\mathcal{A}}_{x} \rightarrow {\mathcal{B}}_{x} \rightarrow {\mathcal{C}}_{x} \) is exact. \( {}^{4} \) Since \( \mathop{\lim }\limits_{ \rightarrow } \) is an exact functor, it follows that the functor \( A \mapsto {\mathcal{{Rea}}}^{\prime }\left( A\right) \), from presheaves to sheaves, is exact.
Let \( h : A\overset{f}{ \rightarrow }B\overset{g}{ \rightarrow }C \) be homomorphisms of presheaves. The induced sequence \( \mathcal{A}\overset{{f}^{\prime }}{ \rightarrow }\mathcal{B}\overset{{g}^{\prime }}{ \rightarrow }\mathcal{C} \) of generated sheaves will be exact if and only if \( \theta \circ g \circ f = 0 \) and the following condition holds: For each open \( U \subset X \) , \( x \in U \), and \( s \in B\left( U\right) \) such that \( g\left( s\right) = 0 \), there exists a neighborhood \( V \subset U \) of \( x \) such that \( s \mid V = f\left( t\right) \) for some \( t \in A\left( V\right) \) . This is an elementary fact resulting from properties of direct limits and from the fact that \( \mathcal{A} \rightarrow \) \( \mathcal{B} \rightarrow \mathcal{C} \) is exact \( \Leftrightarrow {\mathcal{A}}_{x} \rightarrow {\mathcal{B}}_{x} \rightarrow {\mathcal{C}}_{x} \) is exact for all \( x \in X \) . It will be used repeatedly. Note that the condition \( \theta \circ g \circ f = 0 \) is equivalent to the statement that for each \( s \in A\left( U\right) \) and \( x \in U \), there is a neighborhood \( V \subset U \) of \( x \) such that \( g\left( {f\left( {s \mid V}\right) }\right) = 0 \), i.e., that \( \left( {g \circ f}\right) \left( s\right) \) is "locally zero."
2.2. Proposition. If \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) is an exact sequence of sheaves, then the induced sequence
\[
0 \rightarrow {\mathcal{A}}^{\prime }\left( Y\right) \rightarrow \mathcal{A}\left( Y\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( Y\right)
\]
is exact for all \( Y \subset X \) .
Proof. Since the restriction of this sequence to \( Y \) is still exact, it suffices to prove the statement in the case \( Y = X \) . The fact that the sequence of sections o |
1004_(GTM170)Sheaf Theory | 6 | thcal{C}}_{x} \) is exact for all \( x \in X \) . It will be used repeatedly. Note that the condition \( \theta \circ g \circ f = 0 \) is equivalent to the statement that for each \( s \in A\left( U\right) \) and \( x \in U \), there is a neighborhood \( V \subset U \) of \( x \) such that \( g\left( {f\left( {s \mid V}\right) }\right) = 0 \), i.e., that \( \left( {g \circ f}\right) \left( s\right) \) is "locally zero."
2.2. Proposition. If \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) is an exact sequence of sheaves, then the induced sequence
\[
0 \rightarrow {\mathcal{A}}^{\prime }\left( Y\right) \rightarrow \mathcal{A}\left( Y\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( Y\right)
\]
is exact for all \( Y \subset X \) .
Proof. Since the restriction of this sequence to \( Y \) is still exact, it suffices to prove the statement in the case \( Y = X \) . The fact that the sequence of sections over \( X \) has order two and the exactness at \( {\mathcal{A}}^{\prime }\left( X\right) \) are obvious
\( {}^{4} \) Caution: an exact sequence of sheaves is not necessarily an exact sequence of pre-sheaves. See Proposition 2.2 and Example 2.3.
Categorical readers might check that these definitions give notions equivalent to those based on the fact that sheaves and presheaves form abelian categories. (look at stalks). We can assume that \( {\mathcal{A}}^{\prime } \) is a subspace of \( \mathcal{A} \) . Then a section \( s \in \mathcal{A}\left( X\right) \) going to \( 0 \in {\mathcal{A}}^{\prime \prime }\left( X\right) \) must take values in the subspace \( {\mathcal{A}}^{\prime } \), as is seen by looking at stalks. But this just means that it comes from a section in \( {\mathcal{A}}^{\prime }\left( X\right) \) .
2.3. Example. This example shows that \( \mathcal{A}\left( Y\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( Y\right) \) need not be onto even if \( \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \) is onto. On the unit interval \( \mathbb{I} \) let \( \mathcal{A} \) be the sheaf \( \mathbb{I} \times {\mathbb{Z}}_{2} \) and \( {\mathcal{A}}^{\prime \prime } \) the sheaf with stalks \( {\mathbb{Z}}_{2} \) at \( \{ 0\} \) and \( \{ 1\} \) and zero otherwise. (There is only one possible topology in \( {\mathcal{A}}^{\prime \prime } \) .) The canonical map \( \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \) is onto (with kernel being the subsheaf \( {\mathcal{A}}^{\prime } = \left( {0,1}\right) \times {\mathbb{Z}}_{2} \cup \left\lbrack {0,1}\right\rbrack \times \{ 0\} \subset \mathbb{I} \times {\mathbb{Z}}_{2} \) ), but \( \mathcal{A}\left( \mathbb{I}\right) \approx {\mathbb{Z}}_{2} \) while \( {\mathcal{A}}^{\prime \prime }\left( \mathbb{I}\right) \approx {\mathbb{Z}}_{2} \oplus {\mathbb{Z}}_{2} \), so that \( \mathcal{A}\left( \mathbb{I}\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( \mathbb{I}\right) \) is not surjective. Also see Example 2.5 and Exercises 13, 14, and 15.
2.4. Definition. Let \( \mathcal{A} \) be a subsheaf of a sheaf \( \mathcal{B} \) . The "quotient sheaf" \( \mathcal{B}/\mathcal{A} \) is defined to be
\[
\mathcal{B}/\mathcal{A} = \mathcal{P}\text{ heaf }\left( {U \mapsto \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) }\right) .
\]
The exact sequence of presheaves
\[
0 \rightarrow \mathcal{A}\left( U\right) \rightarrow \mathcal{B}\left( U\right) \rightarrow \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) \rightarrow 0
\]
(1)
induces a sequence \( 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{B}/\mathcal{A} \rightarrow 0 \) . On the stalks at \( x \) this is the direct limit of the sequences (1) for \( U \) ranging over the open neighborhoods of \( x \) . This sequence of stalks is exact since direct limits preserve exactness. Therefore, \( 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{B}/\mathcal{A} \rightarrow 0 \) is exact. \( {}^{5} \)
Suppose that \( 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{C} \rightarrow 0 \) is an exact sequence of sheaves. We may regard \( \mathcal{A} \) as a subsheaf of \( \mathcal{B} \) . The exact sequence
\[
0 \rightarrow \mathcal{A}\left( U\right) \rightarrow \mathcal{B}\left( U\right) \rightarrow \mathcal{C}\left( U\right)
\]
provides a monomorphism \( \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) \rightarrow \mathcal{C}\left( U\right) \) of presheaves and hence a homomorphism of sheaves \( \mathcal{B}/\mathcal{A} \rightarrow \mathcal{C} \), and the diagram
\[
\begin{array}{l} 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{B}/\mathcal{A} \rightarrow 0 \\ 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{C} \rightarrow 0 \end{array}
\]
commutes. It follows, by looking at stalks, that \( \mathcal{B}/\mathcal{A} \rightarrow \mathcal{C} \) is an isomorphism.
2.5. Example. Consider the sheaf \( \mathcal{C} \) of germs of continuous real-valued functions on \( X = {\mathbb{R}}^{2} - \{ 0\} \) . Let \( Z \) be the subsheaf of germs of locally constant functions with values the integer multiples of \( {2\pi } \) . Then \( Z \) can be regarded as a subsheaf of \( \mathcal{C} \) . (Note that \( Z \) is a constant sheaf.) The polar angle \( \theta \) is locally defined (ambiguously) as a section of \( \mathcal{C} \), but it
\( {}^{5} \) Note, however, that \( \left( {\mathcal{B}/\mathcal{A}}\right) \left( U\right) \neq \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) \) in general. is not a global section. It does define (unambiguously) a section of the quotient sheaf \( \mathcal{C}/Z \) . This gives another example of an exact sequence \( 0 \rightarrow Z \rightarrow \mathcal{C} \rightarrow \mathcal{C}/Z \rightarrow 0 \) of sheaves for which the sequence of sections is not right exact. Note that \( \mathcal{C}/Z \) can be interpreted as the sheaf of germs of continuous functions on \( X \) with values in the circle group \( {\mathbb{S}}^{1} \) . Note also that \( Z\left( X\right) \) is the group of constant functions on \( X \) with values in \( {2\pi }\mathbb{Z} \) and hence is isomorphic to \( \mathbb{Z};\mathcal{C}\left( X\right) \) is the group of continuous real valued functions \( X \rightarrow \mathbb{R} \) ; and \( \left( {\mathcal{C}/Z}\right) \left( X\right) \) is the group of continuous functions \( X \rightarrow {\mathbb{S}}^{1} \) . The sequence
\[
0 \rightarrow Z\left( X\right) \rightarrow \mathcal{C}\left( X\right) \overset{j}{ \rightarrow }\left( {\mathcal{C}/Z}\right) \left( X\right) \xrightarrow[]{\text{ deg }}\mathbb{Z} \rightarrow 0
\]
is exact by covering space theory, and so \( \operatorname{Coker}j \approx \mathbb{Z} \) .
2.6. Let \( A \) be a locally closed subspace of \( X \) and let \( \mathcal{B} \) be a sheaf on \( A \) . It is easily seen (since \( A \) is locally closed) that there is a unique topology on the point set \( \mathcal{B} \cup \left( {X\times \{ 0\} }\right) \) such that \( \mathcal{B} \) is a subspace and the projection onto \( X \) is a local homeomorphism (we identify \( A \times \{ 0\} \) with the zero section of \( \mathcal{B} \) ). With this topology and the canonical algebraic structure, \( \mathcal{B} \cup \left( {X\times \{ 0\} }\right) \) is a sheaf on \( X \) denoted by
\[
{\mathcal{B}}^{X} = \mathcal{B} \cup \left( {X\times \{ 0\} }\right)
\]
Thus \( {\mathcal{B}}^{X} \) is the unique sheaf on \( X \) inducing \( \mathcal{B} \) on \( A \) and 0 on \( X - A \) . Clearly, \( \mathcal{B} \mapsto {\mathcal{B}}^{X} \) is an exact functor. The sheaf \( {\mathcal{B}}^{X} \) is called the extension of \( \mathcal{B} \) by zero.
Now let \( \mathcal{A} \) be a sheaf on \( X \) and let \( A \subset X \) be locally closed. We define
\[
{\mathcal{A}}_{A} = {\left( \mathcal{A} \mid A\right) }^{X}
\]
For \( U \subset X \) open, \( {\mathcal{A}}_{U} \) is the subsheaf \( {\pi }^{-1}\left( U\right) \cup \left( {X\times \{ 0\} }\right) \) of \( \mathcal{A} \), while for \( F \subset X \) closed \( {\mathcal{A}}_{F} \) is the quotient sheaf \( {\mathcal{A}}_{F} = \mathcal{A}/{\mathcal{A}}_{X - F} \) . If \( A = U \cap F \) , then \( {\mathcal{A}}_{A} = {\left( {\mathcal{A}}_{U}\right) }_{F} = {\left( {\mathcal{A}}_{F}\right) }_{U}{}^{6} \)
In this notation, the sheaf \( {\mathcal{A}}^{\prime } \) of 2.3 is \( {\mathcal{A}}_{\left( 0,1\right) } \), and \( {\mathcal{A}}^{\prime \prime } \approx {\mathcal{A}}_{\{ 0,1\} } \) .
2.7. Example. Let \( {U}_{i} \) be the open disk of radius \( 1 - {2}^{-i} \) in \( X = {\mathbb{D}}^{n} \), the unit disk in \( {\mathbb{R}}^{n} \) . Put \( {A}_{1} = {U}_{1} \) and \( {A}_{i} = {U}_{i} - {U}_{i - 1} \) for \( i > 1 \) . Note that for \( i > 1,{A}_{i} \approx {\mathbb{S}}^{n - 1} \times (0,1\rbrack \) is locally closed in \( X \) . Using the notation of 2.6, put
\[
{\mathcal{L}}_{1} = {\mathbb{Z}}_{{U}_{1}}
\]
\[
{\mathcal{L}}_{2} = {\mathcal{L}}_{1} \cup 2{\mathbb{Z}}_{{U}_{2}}
\]
\[
{\mathcal{L}}_{3} = {\mathcal{L}}_{2} \cup 4{\mathbb{Z}}_{{U}_{3}}
\]
\[
\text{...}
\]
\( {}^{6} \) Note that any locally closed subspace is the intersection of an open subspace with a closed subspace; see [19]. Let \( \mathcal{L} = \bigcup {\mathcal{L}}_{i} \subset \mathbb{Z} \) . Then the stalks of \( \mathcal{L} \) are 0 on \( \partial {\mathbb{D}}^{n} \) and are \( {2}^{i}\mathbb{Z} \) on \( {A}_{i} \) . This is an example of a fairly complicated subsheaf of the constant sheaf \( \mathbb{Z} \) on \( {\mathbb{D}}^{n} \) even in the case \( n = 1 \) . It is a counterexample to \( \lbrack {40} \), Remark II-2.9.3].
## 3 Direct and inverse images
Let \( f : X \rightarrow Y \) be a map and let \( \mathcal{A} \) be a sheaf on \( X \) . The presheaf \( U \mapsto \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) \) on \( Y \) clearly satisfies (S1) and (S2) and hence is a sheaf. This sheaf on \( Y \) is denoted by \( |
1004_(GTM170)Sheaf Theory | 7 | cal{L}}_{2} = {\mathcal{L}}_{1} \cup 2{\mathbb{Z}}_{{U}_{2}}
\]
\[
{\mathcal{L}}_{3} = {\mathcal{L}}_{2} \cup 4{\mathbb{Z}}_{{U}_{3}}
\]
\[
\text{...}
\]
\( {}^{6} \) Note that any locally closed subspace is the intersection of an open subspace with a closed subspace; see [19]. Let \( \mathcal{L} = \bigcup {\mathcal{L}}_{i} \subset \mathbb{Z} \) . Then the stalks of \( \mathcal{L} \) are 0 on \( \partial {\mathbb{D}}^{n} \) and are \( {2}^{i}\mathbb{Z} \) on \( {A}_{i} \) . This is an example of a fairly complicated subsheaf of the constant sheaf \( \mathbb{Z} \) on \( {\mathbb{D}}^{n} \) even in the case \( n = 1 \) . It is a counterexample to \( \lbrack {40} \), Remark II-2.9.3].
## 3 Direct and inverse images
Let \( f : X \rightarrow Y \) be a map and let \( \mathcal{A} \) be a sheaf on \( X \) . The presheaf \( U \mapsto \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) \) on \( Y \) clearly satisfies (S1) and (S2) and hence is a sheaf. This sheaf on \( Y \) is denoted by \( {f}_{\mathcal{A}} \) and is called the direct image of \( \mathcal{A}.{}^{7} \)
Thus we have
\[
f\mathcal{A}\left( U\right) = \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right)
\]
(2)
By 2.2 it is clear that \( \mathcal{A} \mapsto f\mathcal{A} \) is a left exact covariant functor. The direct image is not generally right exact, and in fact, the theory of sheaves is largely concerned with the right derived functors of the direct image functor.
For the map \( \varepsilon : X \rightarrow \star \), where \( \star \) is the one point space, the direct image \( \varepsilon \mathcal{A} \) is just the group \( \Gamma \left( \mathcal{A}\right) = \mathcal{A}\left( X\right) \) (regarded as a sheaf on \( \star \) ). Consequently, the direct image functor \( \mathcal{A} \mapsto f\mathcal{A} \) is a generalization of the global section functor \( \Gamma \) .
Now let \( \mathcal{B} \) be a sheaf on \( Y \) . The inverse image \( {f}^{ * }\mathcal{B} \) of \( \mathcal{B} \) is the sheaf on \( X \) defined by
\[
{f}^{ * }\mathcal{B} = \{ \langle x, b\rangle \in X \times \mathcal{B} \mid f\left( x\right) = \pi \left( b\right) \}
\]
where \( \pi : \mathcal{B} \rightarrow Y \) is the canonical projection. The projection \( {f}^{ * }\mathcal{B} \rightarrow Y \) is given by \( \langle x, b\rangle \mapsto x \) . To check that \( {f}^{ * }\mathcal{B} \) is indeed a sheaf, we note that if \( U \subset Y \) is an open neighborhood of \( f\left( x\right) \) and \( s : U \rightarrow \mathcal{B} \) is a section of \( \mathcal{B} \) with \( s\left( {f\left( x\right) }\right) = b \), then the neighborhood \( \left( {{f}^{-1}\left( U\right) \times s\left( U\right) }\right) \cap {f}^{ * }\mathcal{B} \) of \( \langle x, b\rangle \in {f}^{ * }\mathcal{B} \) is precisely \( \left\{ {\left\langle {{x}^{\prime },{sf}\left( {x}^{\prime }\right) }\right\rangle \mid {x}^{\prime } \in {f}^{-1}\left( U\right) }\right\} \) and hence maps homeomorphically onto \( {f}^{-1}\left( U\right) \) . The group structure on \( {\left( {f}^{ * }\mathcal{B}\right) }_{x} \) is defined so that the one-to-one correspondence
\[
{f}_{x}^{ * } : {\mathcal{B}}_{f\left( x\right) }\overset{ \approx }{ \rightarrow }{\left( {f}^{ * }\mathcal{B}\right) }_{x}
\]
(3)
defined by \( {f}_{x}^{ * }\left( b\right) = \langle x, b\rangle \), is an isomorphism. It is easy to check that the group operations are continuous.
We have already remarked that if \( s : U \rightarrow \mathcal{B} \) is a section, then \( x \mapsto \) \( \langle x, s\left( {f\left( x\right) }\right) \rangle = {f}_{x}^{ * }\left( {s\left( {f\left( x\right) }\right) }\right) \) is a section of \( {f}^{ * }\mathcal{B} \) over \( {f}^{-1}\left( U\right) \) . Thus we have the canonical homomorphism
\[
{f}_{U}^{ * } : \mathcal{B}\left( U\right) \rightarrow \left( {{f}^{ * }\mathcal{B}}\right) \left( {{f}^{-1}\left( U\right) }\right)
\]
(4)
defined by \( {f}_{U}^{ * }\left( s\right) \left( x\right) = {f}_{x}^{ * }\left( {s\left( {f\left( x\right) }\right) }\right) \) .
---
\( {}^{7} \) For a generalization of the direct image see IV-3.
---
From (3) it follows that \( {f}^{ * } \) is an exact functor.
Note that for an inclusion \( i : X \hookrightarrow Y \) and a sheaf \( \mathcal{B} \) on \( Y \), we have \( \mathcal{B} \mid X \approx {i}^{ * }\mathcal{B} \), as the reader is asked to detail in Exercise 1.
3.1. Example. Consider the constant sheaf \( \mathbb{Z} \) on \( X = \left\lbrack {0,1}\right\rbrack \) and its restriction \( \mathcal{L} = \mathbb{Z} \mid \left( {0,1}\right) \) . Let \( i : \left( {0,1}\right) \hookrightarrow X \) be the inclusion. Then \( i\mathcal{L} \approx \mathbb{Z} \), because for \( U \) a small open interval about 0 or 1, we have that \( i\mathcal{L}\left( U\right) = \mathcal{L}\left( {U \cap \left( {0,1}\right) }\right) \approx \mathbb{Z} \) . Also, \( {\mathcal{L}}^{X} = {\mathbb{Z}}_{\left( 0,1\right) } \) . Therefore, \( i\mathcal{L} ≉ {\mathcal{L}}^{X} \) in general.
However, for an inclusion \( i : F \hookrightarrow X \) of a closed subspace and for any sheaf \( \mathcal{L} \) on \( F \), it is true that \( i\mathcal{L} \approx {\mathcal{L}}^{X} \), as the reader can verify. (This is essentially Exercise 2.)
3.2. Example. Consider the constant sheaf \( \mathcal{A} \) with stalks \( \mathbb{Z} \) on \( \mathbb{R} - \{ 0\} \) and let \( i : \mathbb{R} - \{ 0\} \hookrightarrow \mathbb{R} \) . Then \( i\mathcal{A} \) has stalk \( \mathbb{Z} \oplus \mathbb{Z} \) at 0 and stalk \( \mathbb{Z} \) elsewhere, because, for example, \( i\mathcal{A}\left( {-\varepsilon ,\varepsilon }\right) = \mathcal{A}\left( {\left( {-\varepsilon ,0}\right) \cup \left( {0,\varepsilon }\right) }\right) \approx \mathbb{Z} \oplus \mathbb{Z} \) . A local section over a connected neighborhood of 0 taking value \( \left( {n, m}\right) \in {\left( i\mathcal{A}\right) }_{0} \) at 0 is \( n \in {\left( i\mathcal{A}\right) }_{x} \) for \( x < 0 \) and is \( m \in {\left( i\mathcal{A}\right) }_{x} \) for \( x > 0 \) .
3.3. Example. Consider the constant sheaf \( \mathcal{A} \) with stalks \( \mathbb{Z} \) on \( X = {\mathbb{S}}^{1} \) ; let \( Y = \left\lbrack {-1,1}\right\rbrack \) and let \( \pi : X \rightarrow Y \) be the projection. Then \( \pi \mathcal{A} \) has stalks \( \mathbb{Z} \) at -1 and at 1 but has stalks \( {\left( \pi \mathcal{A}\right) }_{x} \approx \mathbb{Z} \oplus \mathbb{Z} \) for \( - 1 < x < 1 \) . The reader should try to understand the topology connecting these stalks. For example, is it true that \( \pi \mathcal{A} \approx \mathbb{Z} \oplus {\mathbb{Z}}_{\left( -1,1\right) } \), as defined in Section 5? Is there a sheaf on \( Y \) that is "locally isomorphic" to \( \pi \mathcal{A} \) but not isomorphic to it?
3.4. Example. Let \( Z \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = {\mathbb{S}}^{1} \) and let \( {Z}^{t} \) denote the "twisted" sheaf with stalks \( \mathbb{Z} \) on \( X \) (i.e., \( {Z}^{t} = \left\lbrack {0,1}\right\rbrack \times \mathbb{Z} \) modulo the identifications \( \left( {0 \times n}\right) \sim \left( {1 \times - n}\right) ) \) . Let \( f : X \rightarrow X \) be the covering map of degree 2 . Then \( {fZ} \) is the sheaf on \( X \) with stalks \( \mathbb{Z} \oplus \mathbb{Z} \) twisted by the exchange of basis elements in the stalks. Also, \( {f}^{ * }{Z}^{t} \approx Z \) since it is the locally constant sheaf with stalks \( \mathbb{Z} \) on \( X \) twisted twice, which is no twist at all. Note that \( {fZ} \) has both \( Z \) and \( {Z}^{t} \) as subsheaves. The corresponding quotient sheaves are \( \left( {fZ}\right) /Z \approx {Z}^{t} \) and \( \left( {fZ}\right) /{Z}^{t} \approx Z \) . However, \( {fZ} ≉ Z \oplus {Z}^{t} \) (defined in Section 5).
3.5. Example. Let \( X \) and \( Z \) be as in Example 3.4 but let \( f : X \rightarrow X \) be the covering map of degree 3 . Then \( {fZ} \) is the locally constant sheaf with stalks \( \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z} \) twisted by the cyclic permutation of factors. Thus \( {fZ} \) has the constant sheaf \( Z \) as a subsheaf (the "diagonal") with quotient sheaf the locally constant sheaf with stalks \( \mathbb{Z} \oplus \mathbb{Z} \) twisted by the, essentially unique, nontrivial automorphism of period 3 .
3.6. Example. Let \( \pi : {\mathbb{S}}^{n} \rightarrow {\mathbb{{RP}}}^{n} \) be the canonical double covering. Then \( \pi \mathbb{Z} \) is a twisted sheaf with stalks \( \mathbb{Z} \oplus \mathbb{Z} \) on projective space, analogous to the sheaf \( {fZ} \) of Example 3.4. It contains the constant sheaf \( \mathbb{Z} \) as a subsheaf with quotient sheaf \( {Z}^{t} \), a twisted integer sheaf. If \( n \) is even, so that \( {\mathbb{{RP}}}^{n} \) is nonorientable, \( {Z}^{t} \) is just the orientation sheaf \( {\mathcal{O}}_{n} \) of Example 1.11. \( \diamond \)
## 4 Cohomomorphisms
Throughout this section we let \( f : X \rightarrow Y \) be a given map.
4.1. Definition. If \( A \) and \( B \) are presheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : B \rightsquigarrow A \) is a collection of homomorphisms \( {k}_{U} : B\left( U\right) \rightarrow A\left( {{f}^{-1}\left( U\right) }\right) \), for \( U \) open in \( Y \), compatible with restrictions.
4.2. Definition. If \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) is a collection of homomorphisms \( {k}_{x} : {\mathcal{B}}_{f\left( x\right) } \rightarrow {\mathcal{A}}_{x} \) for each \( x \in X \) such that for any section \( s \in \mathcal{B}\left( U\right) \) the function \( x \mapsto {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) \) is a section of \( \mathcal{A} \) over \( {f}^{-1}\left( U\right) \) (i.e., this function is continuous). \( {}^{8} \)
An \( f \) -cohomomorphism of sheaves induces an \( f \) -cohomomorphism of presheaves by putting \( {k}_{U}\left( s |
1004_(GTM170)Sheaf Theory | 8 | presheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : B \rightsquigarrow A \) is a collection of homomorphisms \( {k}_{U} : B\left( U\right) \rightarrow A\left( {{f}^{-1}\left( U\right) }\right) \), for \( U \) open in \( Y \), compatible with restrictions.
4.2. Definition. If \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) is a collection of homomorphisms \( {k}_{x} : {\mathcal{B}}_{f\left( x\right) } \rightarrow {\mathcal{A}}_{x} \) for each \( x \in X \) such that for any section \( s \in \mathcal{B}\left( U\right) \) the function \( x \mapsto {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) \) is a section of \( \mathcal{A} \) over \( {f}^{-1}\left( U\right) \) (i.e., this function is continuous). \( {}^{8} \)
An \( f \) -cohomomorphism of sheaves induces an \( f \) -cohomomorphism of presheaves by putting \( {k}_{U}\left( s\right) \left( x\right) = {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) \) where \( U \subset Y \) is open and \( s \in \mathcal{B}\left( U\right) \) . Conversely, an \( f \) -cohomomorphism of presheaves \( k : B \rightsquigarrow A \) induces, for \( x \in X \), a homomorphism
\[
{k}_{x} : {\mathcal{B}}_{f\left( x\right) } = \mathop{\lim }\limits_{ \rightarrow }B\left( U\right) \rightarrow \mathop{\lim }\limits_{ \rightarrow }A\left( {{f}^{-1}U}\right) \rightarrow {\mathcal{A}}_{x}
\]
[where \( U \) ranges over neighborhoods of \( f\left( x\right) \) ]. Then for \( \theta : B\left( U\right) \rightarrow \mathcal{B}\left( U\right) \) the canonical homomorphism and for \( s \in B\left( U\right) \), we have
\[
\theta \left( {{k}_{U}\left( s\right) }\right) \left( x\right) = {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right)
\]
so that \( \left\{ {k}_{x}\right\} \) is an \( f \) -cohomomorphism of sheaves \( \mathcal{B} \rightsquigarrow \mathcal{A} \) (generated by \( B \) and \( A \) ).
For any sheaf \( \mathcal{B} \) on \( Y \), the collection \( {f}^{ * } = \left\{ {f}_{x}^{ * }\right\} \) of (3) defines an \( f \) - cohomomorphism
\[
{f}^{ * } : \mathcal{B} ⤳ {f}^{ * }\mathcal{B}
\]
If \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) is any \( f \) -cohomomorphism, let \( {h}_{x} : {\left( {f}^{ * }\mathcal{B}\right) }_{x} \rightarrow {\mathcal{A}}_{x} \) be defined by \( {h}_{x} = {k}_{x} \circ {\left( {f}_{x}^{ * }\right) }^{-1} \) . Together, the homomorphisms \( {h}_{x} \) define a function \( h : {f}^{ * }\mathcal{B} \rightarrow \mathcal{A} \) . For \( s \in \mathcal{B}\left( U\right) \), the equation
\[
h\left( {{f}_{U}^{ * }\left( s\right) \left( x\right) }\right) = h\left( {{f}_{x}^{ * }\left( {s\left( {f\left( x\right) }\right) }\right) }\right) = {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) ,
\]
together with the fact that the \( {f}_{U}^{ * }\left( s\right) \) form a basis for the topology of \( {f}^{ * }\mathcal{B} \) , implies that \( h \) is continuous. Thus any \( f \) -cohomomorphism \( k \) admits a unique factorization
\[
k : \mathcal{B}\overset{{f}^{ * }}{ \hookrightarrow }{f}^{ * }\mathcal{B}\overset{h}{ \rightarrow }\mathcal{A}
\]
\( h \) being a homomorphism.
---
\( {}^{8} \) Note that an \( f \) -cohomomorphism \( \mathcal{B} ⤳ \mathcal{A} \) is not generally a function, since it is multiply valued unless \( f \) is one-to-one, and it is not defined everywhere unless \( f \) is onto. Of course, cohomomorphisms are the morphisms in the category of all sheaves on all spaces.
---
Similarly, for any sheaf \( \mathcal{A} \) on \( X \), the definition (2) provides an \( f \) - cohomomorphism \( f : f\mathcal{A} \rightarrow \mathcal{A} \) . Since \( {f}_{U} : f\mathcal{A}\left( U\right) \rightarrow \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) \) is an isomorphism, it is clear that any \( f \) -cohomomorphism \( k \) admits a unique factorization
\[
k : \mathcal{B}\overset{j}{ \rightarrow }f\mathcal{A}\overset{f}{ \rightarrow }\mathcal{A}
\]
(i.e., \( {k}_{U} = {f}_{U}{j}_{U} \) ), where \( j \) is a homomorphism.
Thus to each \( f \) -cohomomorphism \( k \) there correspond unique homomorphisms \( h : {f}^{ * }\mathcal{B} \rightarrow \mathcal{A} \) and \( j : \mathcal{B} \rightarrow f\mathcal{A} \) . This correspondence is additive and natural in \( \mathcal{A} \) and \( \mathcal{B} \) . Therefore, denoting the group of all \( f \) - cohomomorphisms from \( \mathcal{B} \) to \( \mathcal{A} \) by \( f \) -cohom \( \left( {\mathcal{B},\mathcal{A}}\right) \), we have produced the following natural isomorphisms of functors:
\[
\operatorname{Hom}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \approx f\text{-cohom}\left( {\mathcal{B},\mathcal{A}}\right) \approx \operatorname{Hom}\left( {\mathcal{B}, f\mathcal{A}}\right) .
\]
Leaving out the middle term, we shall let \( \varphi \) denote this natural isomorphism
\[
\varphi : \operatorname{Hom}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \overset{ \approx }{ \rightarrow }\operatorname{Hom}\left( {\mathcal{B}, f\mathcal{A}}\right)
\]
(5)
of functors. \( {}^{9} \)
Taking \( \mathcal{A} = {f}^{ * }\mathcal{B} \), we obtain the homomorphism
\[
\beta = \varphi \left( 1\right) : \mathcal{B} \rightarrow f{f}^{ * }\mathcal{B}
\]
(6)
and taking \( \mathcal{B} = f\mathcal{A} \), we obtain the homomorphism
\[
\alpha = {\varphi }^{-1}\left( 1\right) : {f}^{ * }f\mathcal{A} \rightarrow \mathcal{A}.
\]
(7)
If \( h : {f}^{ * }\mathcal{B} \rightarrow \mathcal{A} \) is any homomorphism, then the naturality of \( \varphi \) implies that the diagram
\[
\mathrm{{Hom}}\left( {{f}^{ * }\mathcal{B},{f}^{ * }\mathcal{B}}\right) \;\overset{\varphi }{ \rightarrow }\;\mathrm{{Hom}}\left( {\mathcal{B}, f{f}^{ * }\mathcal{B}}\right)
\]
\[
\operatorname{Hom}\left( {{f}^{ * }\mathcal{B}, h}\right) \downarrow \operatorname{Hom}\left( {\mathcal{B}, f\left( h\right) }\right)
\]
\[
\operatorname{Hom}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \;\overset{\varphi }{ \rightarrow }\;\operatorname{Hom}\left( {\mathcal{B}, f\mathcal{A}}\right)
\]
commutes. That is,
\[
\varphi \left( h\right) = f\left( h\right) \circ \varphi \left( 1\right) = f\left( h\right) \circ \beta
\]
(8)
which means that \( \varphi \left( h\right) \) is the composition
\[
\mathcal{B}\overset{\beta }{ \rightarrow }f{f}^{ * }\mathcal{B}\xrightarrow[]{f\left( h\right) }f\mathcal{A}
\]
---
\( {}^{9} \) The existence of such a natural isomorphism means that \( {f}^{ * } \) and \( f \) are "adjoint functors."
---
Similarly, if \( j : \mathcal{B} \rightarrow f\mathcal{A} \) is any homomorphism, then the diagram
\[
\operatorname{Hom}\left( {{f}^{ * }f\mathcal{A},\mathcal{A}}\right) \overset{\varphi }{ \rightarrow }\operatorname{Hom}\left( {f\mathcal{A}, f\mathcal{A}}\right)
\]
\[
\operatorname{Hom}\left( {{f}^{ * }\left( j\right) ,\mathcal{A}}\right) \downarrow \operatorname{Hom}\left( {j, f\mathcal{A}}\right)
\]
\[
\mathrm{{Hom}}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \;\overset{\varphi }{ \rightarrow }\;\mathrm{{Hom}}\left( {\mathcal{B}, f\mathcal{A}}\right)
\]
commutes, whence
\[
{\varphi }^{-1}\left( j\right) = {\varphi }^{-1}\left( 1\right) \circ {f}^{ * }\left( j\right) = \alpha \circ {f}^{ * }\left( j\right)
\]
(9)
is the composition
\[
{f}^{ * }\mathcal{B}\xrightarrow[]{{f}^{ * }\left( j\right) }{f}^{ * }f\mathcal{A}\overset{\alpha }{ \rightarrow }\mathcal{A}
\]
In particular, applying (9) to \( j = \beta : \mathcal{B} \rightarrow f{f}^{ * }\mathcal{B} \), we obtain that \( \alpha \circ {f}^{ * }\left( \beta \right) = {\varphi }^{-1}\left( \beta \right) = 1 \) . That is, the composition
\[
{f}^{ * }\mathcal{B}\xrightarrow[]{{f}^{ * }\left( \beta \right) }{f}^{ * }f{f}^{ * }\mathcal{B}\overset{\alpha }{ \rightarrow }{f}^{ * }\mathcal{B}
\]
is the identity. Thus \( {f}^{ * }\left( \beta \right) \) is a monomorphism, and since \( {\left( {f}^{ * }\mathcal{B}\right) }_{x} = {\mathcal{B}}_{f\left( x\right) } \) , it follows that
\[
\beta : \mathcal{B} \rightarrow f{f}^{ * }\mathcal{B}
\]
is a monomorphism provided that \( f : X \rightarrow Y \) is surjective.
In the next chapter we shall apply this to the special case in which \( f : {X}_{d} \rightarrow X \) is the identity, where \( {X}_{d} \) denotes \( X \) with the discrete topology. In this case
\[
\left( {f{f}^{ * }\mathcal{B}}\right) \left( U\right) = \mathop{\prod }\limits_{{x \in U}}{\mathcal{B}}_{x}
\]
is the group of "serrations" of \( \mathcal{B} \) over \( U \), where a serration is a possibly discontinuous cross section of \( \mathcal{B} \mid U \) . Then \( \beta : \mathcal{B}\left( U\right) \rightarrow \left( {f{f}^{ * }\mathcal{B}}\right) \left( U\right) \) is just the inclusion of the group of (continuous) sections in that of serrations.
4.3. We conclude this section with a remark on cohomomorphisms in quotient sheaves. Let \( {\mathcal{A}}^{\prime } \) be a subsheaf of a sheaf \( \mathcal{A} \) on \( X \) and \( {\mathcal{B}}^{\prime } \) a subsheaf of \( \mathcal{B} \) on \( Y \) . Let \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) be an \( f \) -cohomomorphism that takes \( {\mathcal{B}}^{\prime } \) into \( {\mathcal{A}}^{\prime } \) . Then \( k \) clearly induces an \( f \) -cohomomorphism
\[
\mathcal{B}\left( U\right) /{\mathcal{B}}^{\prime }\left( U\right) \sim \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) /{\mathcal{A}}^{\prime }\left( {{f}^{-1}\left( U\right) }\right)
\]
of presheaves, which, in turn, induces an \( f \) -cohomomorphism \( \mathcal{B}/{\mathcal{B}}^{\prime } \sim \) \( \mathcal{A}/{\mathcal{A}}^{\prime } \) of the generated sheaves.
## 5 Algebraic constructions
In this section we shall consider covariant functors \( F\left( {{G}_{1},{G}_{2},\ldots }\right) \) of several variables from the category of abelian groups to itself. (More generally, one may consider covariant functors from the category of "diagrams of ab |
1004_(GTM170)Sheaf Theory | 9 | sheaf \( \mathcal{A} \) on \( X \) and \( {\mathcal{B}}^{\prime } \) a subsheaf of \( \mathcal{B} \) on \( Y \) . Let \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) be an \( f \) -cohomomorphism that takes \( {\mathcal{B}}^{\prime } \) into \( {\mathcal{A}}^{\prime } \) . Then \( k \) clearly induces an \( f \) -cohomomorphism
\[
\mathcal{B}\left( U\right) /{\mathcal{B}}^{\prime }\left( U\right) \sim \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) /{\mathcal{A}}^{\prime }\left( {{f}^{-1}\left( U\right) }\right)
\]
of presheaves, which, in turn, induces an \( f \) -cohomomorphism \( \mathcal{B}/{\mathcal{B}}^{\prime } \sim \) \( \mathcal{A}/{\mathcal{A}}^{\prime } \) of the generated sheaves.
## 5 Algebraic constructions
In this section we shall consider covariant functors \( F\left( {{G}_{1},{G}_{2},\ldots }\right) \) of several variables from the category of abelian groups to itself. (More generally, one may consider covariant functors from the category of "diagrams of abelian groups of a given shape" to the category of abelian groups.) For general illustrative purposes we shall take the case of a functor of two variables.
We may also consider \( F \) as a functor from the category of presheaves on \( X \) to itself in the canonical way (since \( F \) is covariant). That is, we let
\[
F\left( {A, B}\right) \left( U\right) = F\left( {A\left( U\right), B\left( U\right) }\right) ,
\]
for presheaves \( A \) and \( B \) on \( X \) . The sheaf generated by the presheaf \( F\left( {A, B}\right) \) will be denoted by \( \mathcal{F}\left( {A, B}\right) = \mathcal{R} \) has \( \left( {F\left( {A, B}\right) }\right) \) . In particular, if \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \) then \( \mathcal{F}\left( {\mathcal{A},\mathcal{B}}\right) = \mathcal{{Mean}}(U \mapsto F\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) = \) \( F\left( {\mathcal{A}\left( U\right) ,\mathcal{B}\left( U\right) }\right) ){.}^{10} \)
Now suppose that the functor \( F \) commutes with direct limits. That is, suppose that the canonical map \( \underline{\lim }F\left( {{G}_{\alpha },{H}_{\alpha }}\right) \rightarrow F\left( {\underline{\lim }{G}_{\alpha },\underline{\lim }{H}_{\alpha }}\right) \) is an isomorphism for direct systems \( \left\{ {G}_{\alpha }\right\} \) and \( \left\{ {H}_{\alpha }\right\} \) of abelian groups. Let \( \mathcal{A} \) and \( \mathcal{B} \) denote the sheaves generated by the presheaves \( A \) and \( B \) respectively. Then for \( U \) ranging over the neighborhoods of \( x \in X \), we have \( \lim F\left( {A, B}\right) \left( U\right) = \lim F\left( {A\left( U\right), B\left( U\right) }\right) \approx F\left( {\lim A\left( U\right) ,\lim B\left( U\right) }\right) = \) \( F\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right) \) so that we have the natural isomorphism
\[
\mathcal{F}{\left( A, B\right) }_{x} \approx F\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right)
\]
(10)
when \( F \) commutes with direct limits.
More generally, consider the natural maps \( A\left( U\right) \rightarrow \mathcal{A}\left( U\right) \) and \( B\left( U\right) \rightarrow \) \( \mathcal{B}\left( U\right) \) . These give rise to a homomorphism \( F\left( {A, B}\right) \rightarrow F\left( {\mathcal{A},\mathcal{B}}\right) \) of pre-sheaves and hence to a homomorphism
\[
\mathcal{F}\left( {A, B}\right) \rightarrow \mathcal{F}\left( {\mathcal{A},\mathcal{B}}\right)
\]
(11)
of the generated sheaves. If \( U \) ranges over the neighborhoods of \( x \in X \) , then the diagram
\[
\underline{\lim }F\left( {A\left( U\right), B\left( U\right) }\right) \; \rightarrow \;\underline{\lim }F\left( {\mathcal{A}\left( U\right) ,\mathcal{B}\left( U\right) }\right)
\]
\[
\downarrow \; \downarrow
\]
\[
F\left( {\underline{\lim }A\left( U\right) ,\underline{\lim }B\left( U\right) }\right) \rightarrow F\left( {\underline{\lim }\mathcal{A}\left( U\right) ,\underline{\lim }\mathcal{B}\left( U\right) }\right)
\]
commutes. The bottom homomorphism is an isomorphism by definition of \( \mathcal{A} \) and \( \mathcal{B} \) . The top homomorphism is the restriction of (11) to the stalks at \( x \) . The vertical maps are isomorphisms when \( F \) commutes with direct limits. Thus we see that (11) is an isomorphism of sheaves provided that \( F \) commutes with direct limits. That is, in this case,  naturally.
\( {}^{10} \) Our notation in some of the specific examples to follow will differ from the notation we are using in the general discussion.
We shall now discuss several explicit cases, starting with the tensor product. If \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \), we let
\[
\mathcal{A} \otimes \mathcal{B} = \mathcal{H}\text{ half }\left( {U \mapsto \mathcal{A}\left( U\right) \otimes \mathcal{B}\left( U\right) }\right) .
\]
Since \( \otimes \) commutes with direct limits, we have the natural isomorphism
\[
{\left( \mathcal{A} \otimes \mathcal{B}\right) }_{x} \approx {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{x}
\]
by (10). Since \( \otimes \) is right exact for abelian groups, it will also be right exact for sheaves since exactness is a stalkwise property.
The following terminology will be useful:
5.1. Definition. An exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) of sheaves on \( X \) is said to be "pointwise split" if \( 0 \rightarrow {\mathcal{A}}_{x}^{\prime } \rightarrow {\mathcal{A}}_{x} \rightarrow {\mathcal{A}}_{x}^{\prime \prime } \rightarrow 0 \) splits for each \( x \in X \) .
This condition clearly implies that \( 0 \rightarrow {\mathcal{A}}^{\prime } \otimes \mathcal{B} \rightarrow \mathcal{A} \otimes \mathcal{B} \rightarrow {\mathcal{A}}^{\prime \prime } \otimes \mathcal{B} \rightarrow 0 \) is exact for every sheaf \( \mathcal{B} \) on \( X \) .
Our second example concerns the torsion product. We shall use \( G * H \) to denote \( \operatorname{Tor}\left( {G, H}\right) \) . For sheaves \( \mathcal{A} \) and \( \mathcal{B} \) on \( X \) we let
\[
\mathcal{A} * \mathcal{B} = \mathcal{P}\text{ heaf }\left( {U \mapsto \mathcal{A}\left( U\right) * \mathcal{B}\left( U\right) }\right) .
\]
We have that
\[
\left( {\mathcal{A} * \mathcal{B}}\right) x \approx {\mathcal{A}}_{x} * {\mathcal{B}}_{x}
\]
since the torsion product \( * \) commutes with direct limits.
Let \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) be an exact sequence of sheaves. Then for each open set \( U \subset X \), we have the exact sequence
\[
0 \rightarrow {\mathcal{A}}^{\prime }\left( U\right) * \mathcal{B}\left( U\right) \rightarrow \mathcal{A}\left( U\right) * \mathcal{B}\left( U\right) \rightarrow \left( {\mathcal{A}\left( U\right) /{\mathcal{A}}^{\prime }\left( U\right) }\right) * \mathcal{B}\left( U\right)
\]
\[
\rightarrow {\mathcal{A}}^{\prime }\left( U\right) \otimes \mathcal{B}\left( U\right) \rightarrow \mathcal{A}\left( U\right) \otimes \mathcal{B}\left( U\right) \rightarrow \left( {\mathcal{A}\left( U\right) /{\mathcal{A}}^{\prime }\left( U\right) }\right) \otimes \mathcal{B}\left( U\right) \rightarrow 0
\]
of presheaves on \( X \), where \( \mathcal{B} \) is any sheaf. Now \( {\mathcal{A}}^{\prime \prime } \) is canonically isomorphic to \( \mathcal{P} \) heaf \( \left( {U \mapsto \mathcal{A}\left( U\right) /{\mathcal{A}}^{\prime }\left( U\right) }\right) \) . Thus this sequence of presheaves generates the exact sequence
\( 0 \rightarrow {\mathcal{A}}^{\prime } * \mathcal{B} \rightarrow \mathcal{A} * \mathcal{B} \rightarrow {\mathcal{A}}^{\prime \prime } * \mathcal{B} \rightarrow {\mathcal{A}}^{\prime } \otimes \mathcal{B} \rightarrow \mathcal{A} \otimes \mathcal{B} \rightarrow {\mathcal{A}}^{\prime \prime } \otimes \mathcal{B} \rightarrow 0 \)
(12)
of sheaves on \( X{.}^{11} \)
Before passing on to other examples of our general considerations, we shall introduce some further notation concerned with tensor and torsion
\( {}^{11} \) It should be noted that this is a special case of a general fact. Namely, if \( \left\{ {F}_{n}\right\} \) is an exact connected sequence of functors of abelian groups (as above), then the induced sequence of functors \( \left\{ {\mathcal{F}}_{n}\right\} \) on the category of sheaves to itself is also exact and connected.
products. If \( X \) and \( Y \) are spaces and \( {\pi }_{X} : X \times Y \rightarrow X,{\pi }_{Y} : X \times Y \rightarrow Y \) are the projections, then for sheaves \( \mathcal{A} \) on \( X \) and \( \mathcal{B} \) on \( Y \) we define the total tensor product \( \mathcal{A}\widehat{ \otimes }\mathcal{B} \) to be the sheaf
\[
\mathcal{A}\widehat{ \otimes }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) \otimes \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right)
\]
on \( X \times Y \) . Similarly, the total torsion product is defined to be
\[
\mathcal{A}\widehat{ * }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) * \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right)
\]
Clearly, we have natural isomorphisms
\[
{\left( \mathcal{A}\widehat{ \otimes }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{y}
\]
\[
{\left( \mathcal{A}\widehat{ * }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} * {\mathcal{B}}_{y}
\]
Another special case of our general discussion is provided by the direct sum functor. Thus, if \( \left\{ {\mathcal{A}}_{\alpha }\right\} \) is a family of sheaves on \( X \), we let
\[
\oplus {\mathcal{A}}_{\alpha } = \text{ Heaf }\left( {U \mapsto \bigoplus \left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) }\right) .
\]
Since direct sums commute with direct limits, we have that
\[
\left( {{\left( {\bigoplus }_{\alpha }{\mathcal{A}}_{\alpha }\right) }_{x} \approx {\bigoplus }_{\ |
1004_(GTM170)Sheaf Theory | 10 | Y \) . Similarly, the total torsion product is defined to be
\[
\mathcal{A}\widehat{ * }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) * \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right)
\]
Clearly, we have natural isomorphisms
\[
{\left( \mathcal{A}\widehat{ \otimes }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{y}
\]
\[
{\left( \mathcal{A}\widehat{ * }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} * {\mathcal{B}}_{y}
\]
Another special case of our general discussion is provided by the direct sum functor. Thus, if \( \left\{ {\mathcal{A}}_{\alpha }\right\} \) is a family of sheaves on \( X \), we let
\[
\oplus {\mathcal{A}}_{\alpha } = \text{ Heaf }\left( {U \mapsto \bigoplus \left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) }\right) .
\]
Since direct sums commute with direct limits, we have that
\[
\left( {{\left( {\bigoplus }_{\alpha }{\mathcal{A}}_{\alpha }\right) }_{x} \approx {\bigoplus }_{\alpha }{\left( {\mathcal{A}}_{\alpha }\right) }_{x}}\right)
\]
In the case of the direct product, we note that the presheaf \( U \mapsto \prod \left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) \) satisfies (S1) and (S2) on page 6 and therefore is a sheaf. It is denoted by \( \prod {\mathcal{A}}_{\alpha } \) . However, direct products do not generally commute with direct limits, and in fact, \( {\left( \prod {\mathcal{A}}_{\alpha }\right) }_{x} ≉ \prod {\left( {\mathcal{A}}_{\alpha }\right) }_{x} \) in general. [For example let \( {\mathcal{A}}_{i} = {\mathbb{Z}}_{\lbrack 0,1/i)} \subset \mathbb{Z} \) for \( i \geq 1 \) on \( X = \left\lbrack {0,1}\right\rbrack \) . Then for \( U = \lbrack 0,1/n) \) , we have that \( {\mathcal{A}}_{i}\left( U\right) = 0 \) for \( i > n \), and so \( \mathop{\prod }\limits_{{i = 1}}^{\infty }\left( {{\mathcal{A}}_{i}\left( U\right) }\right) = {\mathbb{Z}}^{n} \), whence \( {\left( \prod {\mathcal{A}}_{i}\right) }_{\{ 0\} } \approx \underline{\lim }{\mathbb{Z}}^{n} = {\mathbb{Z}}^{\infty } \), the countable direct sum of copies of \( \mathbb{Z} \) . However, \( \prod {\left( {\mathcal{A}}_{i}\right) }_{\{ 0\} } \approx \mathop{\prod }\limits_{{i = 1}}^{\infty }\mathbb{Z} \), which is uncountable. For another example, let \( {\mathcal{B}}_{n} = {\mathbb{Z}}_{\{ 1/n\} } \) . Then \( \prod {\left( {\mathcal{B}}_{n}\right) }_{\{ 0\} } = 0 \) ; but \( {\left( \prod {\mathcal{B}}_{n}\right) }_{\{ 0\} } \neq 0 \) since the sections \( {s}_{n} \in {\mathcal{B}}_{n}\left( X\right) \) that are 1 at \( 1/n \) give a section \( s = \prod {s}_{n} \) of the product that is not zero in any neighborhood of 0 and hence has nonzero germ at 0 .]
For a finite number of variables (or, generally, for locally finite families), direct sums and direct products of sheaves coincide. For two variables (for example) \( \mathcal{A} \) and \( \mathcal{B} \), the direct sum is denoted by \( \mathcal{A} \oplus \mathcal{B} \) . (Note that \( \mathcal{A}\bigtriangleup \mathcal{B} \) is the underlying topological space of \( \mathcal{A} \oplus \mathcal{B} \) .) The notation \( \mathcal{A} \times \mathcal{B} \) is reserved for the cartesian product of \( \mathcal{A} \) and \( \mathcal{B} \), which with coordinatewise addition is a sheaf on \( X \times Y \) when \( \mathcal{A} \) is a sheaf on \( X \) and \( \mathcal{B} \) is one on \( Y \) . Note that \( \mathcal{A} \times \mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) \oplus \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right) \) .
Note that for \( X = Y \), we have that \( \mathcal{A} \oplus \mathcal{B} = \left( {\mathcal{A} \times \mathcal{B}}\right) \mid \Delta \), where \( \Delta \) is the diagonal of \( X \times X \), identified with \( X \), and similarly that \( \mathcal{A} \otimes \mathcal{B} = \left( {\mathcal{A}\widehat{ \otimes }\mathcal{B}}\right) \mid \Delta \) and \( \mathcal{A} * \mathcal{B} = \left( {\mathcal{A}\widehat{ * }\mathcal{B}}\right) |\Delta . \)
Our next example is given by a functor on a category of "diagrams." Let \( A \) be a directed set. Consider direct systems \( \left\{ {{G}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \) (where \( {\pi }_{\alpha ,\beta } \) : \( \left. {{G}_{\beta } \rightarrow {G}_{\alpha }\text{is defined for}\alpha > \beta \text{in}A\text{and satisfies}{\pi }_{\alpha ,\beta }{\pi }_{\beta ,\gamma } = {\pi }_{\alpha ,\gamma }}\right) \) of abelian groups based on the directed set \( A \) . Let \( F \) be the (covariant) functor that assigns to each such direct system \( \left\{ {{G}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \) its direct limit
\[
F\left( \left\{ {{G}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \right) = \underline{\lim }{G}_{\alpha }
\]
Now let \( \left\{ {{\mathcal{A}}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \) be a direct system of sheaves based on \( A \) . Then we define
\[
\mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } = \text{ Sheaf }\left( {U \mapsto \mathop{\lim }\limits_{ \rightarrow }{}_{\alpha }{\mathcal{A}}_{\alpha }\left( U\right) }\right) .
\]
There are the compatible maps \( {\mathcal{A}}_{\alpha }\left( U\right) \rightarrow {\underline{\lim }}_{\alpha }{\mathcal{A}}_{\alpha }\left( U\right) \) that induce canonical homomorphisms \( {\pi }_{\beta } : {\mathcal{A}}_{\beta } \rightarrow {\underline{\lim }}_{\alpha }{\mathcal{A}}_{\alpha } \) such that \( {\pi }_{\beta } = {\pi }_{\alpha } \circ {\pi }_{\alpha ,\beta } \) whenever \( \alpha > \beta \) . Since direct limits commute with one another, we have that
\[
{\left( \underline{\lim }{\mathcal{A}}_{\alpha }\right) }_{x} \approx \underline{\lim }{\left( {\mathcal{A}}_{\alpha }\right) }_{x}
\]
Now suppose that \( \mathcal{A} \) is another sheaf on \( X \) and that we have a family of homomorphisms \( {h}_{\alpha } : {\mathcal{A}}_{\alpha } \rightarrow \mathcal{A} \) that are compatible in the sense that \( {h}_{\beta } = {h}_{\alpha } \circ {\pi }_{\alpha ,\beta } \) whenever \( \alpha > \beta \) . These induce compatible maps \( {\mathcal{A}}_{\alpha }\left( U\right) \rightarrow \) \( \mathcal{A}\left( U\right) \) for all open \( U \) and hence a homomorphism \( \underline{\lim }\left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) \rightarrow \mathcal{A}\left( U\right) \) of presheaves. In turn this induces a homomorphism
\[
h : \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \rightarrow \mathcal{A}
\]
of the generated sheaves such that \( h \circ {\pi }_{\alpha } = {h}_{\alpha } \) for all \( \alpha \) . That is, the direct limit of sheaves satisfies the "universal property" of direct limits.
In particular, if \( \left\{ {\mathcal{A}}_{\alpha }\right\} \) and \( \left\{ {\mathcal{B}}_{\alpha }\right\} \) are direct systems based on the same directed set, then the homomorphisms \( {\mathcal{A}}_{\alpha } \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \) and \( {\mathcal{B}}_{\alpha } \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha } \) induce compatible homomorphisms \( {\mathcal{A}}_{\alpha } \otimes {\mathcal{B}}_{\alpha } \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \otimes \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha } \) and hence a homomorphism \( \mathop{\lim }\limits_{ \rightarrow }\left( {{\mathcal{A}}_{\alpha } \otimes {\mathcal{B}}_{\alpha }}\right) \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \otimes \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha } \) . On stalks this is an isomorphism since tensor products and direct limits commute. Thus it follows that this is an isomorphism
\[
\lambda : \mathop{\lim }\limits_{ \rightarrow }\left( {{\mathcal{A}}_{\alpha } \otimes {\mathcal{B}}_{\alpha }}\right) \overset{ \approx }{ \rightarrow }\mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \otimes \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha }.
\]
(13)
The functor \( \operatorname{Hom}\left( {G, H}\right) \) on abelian groups is covariant in only one of its variables, so that the general discussion does not apply. However, note that every homomorphism \( \mathcal{A} \rightarrow \mathcal{B} \) of sheaves induces a homomorphism
\[
\mathcal{A}\left| {U \rightarrow \mathcal{B}}\right| U
\]
Thus we see that the functor
\[
U \mapsto \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right)
\]
(14)
defines a presheaf on \( X \) . We define
\[
\mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) = \mathcal{H}\text{ heaf }\left( {U \mapsto \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) }\right) .
\]
It is clear that the presheaf (14) satisfies (S1) and (S2), so that
\[
\mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) \approx \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) .
\]
It is important to note that the last equation does not apply in general to sections over nonopen subspaces, and in particular that
\[
\operatorname{Hom}{\left( \mathcal{A},\mathcal{B}\right) }_{x} ≉ \operatorname{Hom}\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right)
\]
in general. For example, let \( \mathcal{B} \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = \) \( \left\lbrack {0,1}\right\rbrack \) and let \( \mathcal{A} = {\mathcal{B}}_{\{ 0\} } \), which has stalk \( \mathbb{Z} \) over \( \{ 0\} \) and stalks \( 0 \) elsewhere. Then \( \mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) \approx \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) = 0 \) for the open sets of the form \( U = \lbrack 0,\varepsilon ) \), and hence \( {\mathcal{{Hom}\left( {A, B}\right) }}_{\{ 0\} } = 0 \), whereas \( {\mathcal{A}}_{\{ 0\} } = \mathbb{Z} = {\mathcal{B}}_{\{ 0\} } \) , whence \( \mathrm{{Hom}}\left( {{\mathcal{A}}_{\{ 0\} },{\mathcal{B}}_{\{ 0\} }}\right) \approx \mathrm{{Hom}}\left( {\dot{\mathbb{Z}},\dot |
1004_(GTM170)Sheaf Theory | 11 | ns over nonopen subspaces, and in particular that
\[
\operatorname{Hom}{\left( \mathcal{A},\mathcal{B}\right) }_{x} ≉ \operatorname{Hom}\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right)
\]
in general. For example, let \( \mathcal{B} \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = \) \( \left\lbrack {0,1}\right\rbrack \) and let \( \mathcal{A} = {\mathcal{B}}_{\{ 0\} } \), which has stalk \( \mathbb{Z} \) over \( \{ 0\} \) and stalks \( 0 \) elsewhere. Then \( \mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) \approx \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) = 0 \) for the open sets of the form \( U = \lbrack 0,\varepsilon ) \), and hence \( {\mathcal{{Hom}\left( {A, B}\right) }}_{\{ 0\} } = 0 \), whereas \( {\mathcal{A}}_{\{ 0\} } = \mathbb{Z} = {\mathcal{B}}_{\{ 0\} } \) , whence \( \mathrm{{Hom}}\left( {{\mathcal{A}}_{\{ 0\} },{\mathcal{B}}_{\{ 0\} }}\right) \approx \mathrm{{Hom}}\left( {\dot{\mathbb{Z}},\dot{\mathbb{Z}}}\right) \approx \mathbb{Z}. \)
If \( \mathcal{R} \) is a sheaf of rings on \( X \) and if \( \mathcal{A} \) and \( \mathcal{B} \) are \( \mathcal{R} \) -modules, then one can define, in a similar manner, the sheaves
\[
\mathcal{A}{ \otimes }_{\mathcal{R}}\mathcal{B},\;{\mathcal{{Int}}}_{n}^{\mathcal{R}}(\mathcal{A},\mathcal{B}),\;\mathcal{{Hom}}{}_{\mathcal{R}}(\mathcal{A},\mathcal{B}).
\]
## 6 Supports
A paracompact space is a Hausdorff space with the property that every open covering has an open, locally finite, refinement. The following facts are well known (see [34], [53] and [19]):
(1) Every paracompact space is normal.
(2) A metric space is paracompact.
(3) A closed subspace of a paracompact space is paracompact.
(4) If \( \left\{ {U}_{\alpha }\right\} \) is a locally finite open cover of a normal space \( X \), then there is an open cover \( \left\{ {V}_{\alpha }\right\} \) of \( X \) such that \( {\bar{V}}_{\alpha } \subset {U}_{\alpha } \) .
(5) A locally compact space is paracompact \( \Leftrightarrow \) it is a disjoint union of open, \( \sigma \) -compact subspaces.
A space is called hereditarily paracompact if every open subspace is paracompact. It is easily seen that this implies that every subspace is paracompact. Of course, metric spaces are hereditarily paracompact.
6.1. Definition. A "family of supports" on \( X \) is a family \( \Phi \) of closed subsets of \( X \) such that:
(1) a closed subset of a member of \( \Phi \) is a member of \( \Phi \) ;
(2) \( \Phi \) is closed under finite unions.
\( \Phi \) is said to be a "paracompactifying" family of supports if in addition:
(3) each element of \( \Phi \) is paracompact;
(4) each element of \( \Phi \) has a (closed) neighborhood which is in \( \Phi \) .
We define the extent \( E\left( \Phi \right) \) of a family of supports to be the union of the members of \( \Phi \) . Note that \( E\left( \Phi \right) \) is open when \( \Phi \) is paracompactifying.
The family of all compact subsets of \( X \) is denoted by \( c \) . It is para-compactifying if \( X \) is locally compact. We use 0 to denote the family of supports whose only member is the empty set \( \varnothing \) . It is customary to "denote" the family of all closed subsets of \( X \) by the absence of a symbol, and we shall also use \( {cld} \) to denote this family.
Recall that for \( s \in \mathcal{A}\left( X\right) ,\left| s\right| = \{ x \in X \mid s\left( x\right) \neq 0\} \) denotes the support of the section \( s \) . Now if \( A \) is a presheaf on \( X \) and \( s \in A\left( X\right) \), we put \( \left| s\right| = \left| {\theta \left( s\right) }\right| \), where \( \theta : A\left( X\right) \rightarrow \mathcal{A}\left( X\right) \) is the canonical map, \( \mathcal{A} \) being the sheaf generated by \( A \) .
Note that for \( s \in A\left( X\right), x \notin \left| s\right| \Leftrightarrow (s \mid U = 0 \) for some neighborhood \( U \) of \( x \) ).
If \( \mathcal{A} \) is a sheaf on \( X \), we put
\[
{\Gamma }_{\Phi }\left( \mathcal{A}\right) = \{ s \in \mathcal{A}\left( X\right) \mid \left| s\right| \in \Phi \}
\]
Then \( {\Gamma }_{\Phi }\left( \mathcal{A}\right) \) is a subgroup of \( \mathcal{A}\left( X\right) \), and for an exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \) \( \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \), the sequence
\[
0 \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime }\right) \rightarrow {\Gamma }_{\Phi }\left( \mathcal{A}\right) \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime \prime }\right)
\]
is exact.
For a presheaf \( A \) on \( X \) we put \( {A}_{\Phi }\left( X\right) = \{ s \in A\left( X\right) \left| \;\right| s| \in \Phi \} {.}^{12} \)
6.2. Theorem. Let \( A \) be a presheaf on \( X \) that is conjunctive for coverings of \( X \) and let \( \mathcal{A} \) be the sheaf generated by \( A \) . Then for any paracompactifying family \( \Phi \) of supports on \( X \), the sequence
\[
0 \rightarrow {A}_{0}\left( X\right) \rightarrow {A}_{\Phi }\left( X\right) \overset{\theta }{ \rightarrow }{\Gamma }_{\Phi }\left( \mathcal{A}\right) \rightarrow 0
\]
is exact.
Proof. The only nontrivial part is that \( \theta \) is surjective. Let \( s \in {\Gamma }_{\Phi }\left( \mathcal{A}\right) \) and let \( U \) be an open neighborhood of \( \left| s\right| \) with \( \bar{U} \) paracompact. By covering \( \bar{U} \) and then restricting to \( U \), we can find a covering \( \left\{ {U}_{\alpha }\right\} \) of \( U \) that is locally finite in \( X \) and such that there exist \( {s}_{\alpha } \in A\left( {U}_{\alpha }\right) \) with \( \theta \left( {s}_{\alpha }\right) = s \mid {U}_{\alpha } \) . Similarly, we can find a covering \( \left\{ {V}_{\alpha }\right\} \) of \( U \) with \( U \cap {\bar{V}}_{\alpha } \subset {U}_{\alpha } \) . Add \( X - \left| s\right| \) to the collection \( \left\{ {V}_{\alpha }\right\} \), giving a locally finite covering of \( X \), and use the zero section for the corresponding \( {s}_{\alpha } \) .
\( {}^{12} \) The \( \Gamma \) notation will be used only for sheaves and not for presheaves.
For \( x \in X \) let \( I\left( x\right) = \left\{ {\alpha \mid x \in {\bar{V}}_{\alpha }}\right\} \), a finite set. For each \( x \in X \) there is a neighborhood \( W\left( x\right) \) such that \( y \in W\left( x\right) \Rightarrow I\left( y\right) \subset I\left( x\right) \) and such that \( W\left( x\right) \subset {V}_{\alpha } \) for each \( \alpha \in I\left( x\right) \) .
If \( \alpha \in I\left( x\right) \), then \( \theta \left( {s}_{\alpha }\right) \left( x\right) = s\left( x\right) \) . Since \( I\left( x\right) \) is finite, we may further assume that \( W\left( x\right) \) is so small that \( {s}_{\alpha } \mid W\left( x\right) \) is independent of \( \alpha \in I\left( x\right) \) [since \( {\mathcal{A}}_{x} = \mathop{\lim }\limits_{ \rightarrow }A\left( N\right), N \) ranging over the neighborhoods of \( x \) ].
Let \( {s}_{x} \in A\left( {W\left( x\right) }\right) \) be the common value of \( {s}_{\alpha } \mid W\left( x\right) \) for \( \alpha \in I\left( x\right) \) .
Suppose that \( x, y \in X \) and \( z \in W\left( x\right) \cap W\left( y\right) \) . Let \( \alpha \in I\left( z\right) \subset I\left( x\right) \cap I\left( y\right) \) . Then \( {s}_{x} = {s}_{\alpha }\left| {W\left( y\right) \text{, so that}{s}_{x}}\right| W\left( x\right) \cap W\left( y\right) = {s}_{y} \mid W\left( x\right) \cap W\left( y\right) \) . Since \( A \) is conjunctive for coverings of \( X \), there is a \( t \in A\left( X\right) \) such that \( t \mid W\left( x\right) = {s}_{x} \) for all \( x \in X \) . Clearly \( \theta \left( t\right) = s \), and by definition, \( \left| t\right| = \left| s\right| \in \Phi \) .
Note that \( {A}_{0}\left( U\right) = 0 \) for all open \( U \subset X \Leftrightarrow A \) is a monopresheaf.
6.3. Definition. If \( A \subset X \) and \( \Phi \) is a family of supports on \( X \), then \( \Phi \cap A \) denotes the family \( \{ K \cap A \mid K \in \Phi \} \) of supports on \( A \), and \( \Phi \mid A \) denotes the family \( \{ K \mid K \subset A \) and \( K \in \Phi \} \) of supports on \( A \) or on \( X{.}^{13} \)
If \( X, Y \) are spaces with support families \( \Phi \) and \( \Psi \) respectively, then \( \Phi \times \Psi \) denotes the family on \( X \times Y \) of all closed subsets of sets of the form \( K \times L \) with \( K \in \Phi \) and \( L \in \Psi \) .
If \( f : X \rightarrow Y \) and \( \Psi \) is a family on \( Y \), then \( {f}^{-1}\Psi \) denotes the family on \( X \) of all closed subsets of sets of the form \( {f}^{-1}K \) for \( K \in \Psi \) .
6.4. Example. For the purposes of this example, let us use the subscript \( Y \) on the family of supports \( {cld} \) or \( c \) to indicate the space to which these symbols apply. (In other places we let the context determine this.) Let \( X = \mathbb{R} \) and \( A = \left( {0,1}\right) \) . Then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {cl}{d}_{X} \mid A = {c}_{A} \) . Also, \( {c}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A}. \)
If instead, \( X = (0,1\rbrack \), then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A} \), while \( {cl}{d}_{X} \mid A \) is the family of closed subsets of \( A \) bounded away from 1. Also, \( {c}_{X} \cap A \) is the family of closed subsets of \( A \) bounded away from 0 .
If \( X = \mathbb{R} = Y \), then the family \( {c}_{X} \times {c}_{Y} = {c}_{X \times Y} \), while \( {c}_{X} \times {cl}{d}_{Y} \) is the family of all closed subsets of \( X \times Y \) whose projection to \( X \) is bounded (but the projection need not be closed). This is the same as the family \( {\pi }_{X}^{-1}{c}_{X} \) . Also, \( {cl}{d}_{X} \times {cl}{d}_{Y} = {cl}{d}_{X \times Y} = {\pi }_{X}^{-1}{cl}{d}_{X} \) .
For any map \( f : X \rightarrow Y \), the family \( {f}^{-1}{c}_{Y} |
1004_(GTM170)Sheaf Theory | 12 | and \( A = \left( {0,1}\right) \) . Then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {cl}{d}_{X} \mid A = {c}_{A} \) . Also, \( {c}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A}. \)
If instead, \( X = (0,1\rbrack \), then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A} \), while \( {cl}{d}_{X} \mid A \) is the family of closed subsets of \( A \) bounded away from 1. Also, \( {c}_{X} \cap A \) is the family of closed subsets of \( A \) bounded away from 0 .
If \( X = \mathbb{R} = Y \), then the family \( {c}_{X} \times {c}_{Y} = {c}_{X \times Y} \), while \( {c}_{X} \times {cl}{d}_{Y} \) is the family of all closed subsets of \( X \times Y \) whose projection to \( X \) is bounded (but the projection need not be closed). This is the same as the family \( {\pi }_{X}^{-1}{c}_{X} \) . Also, \( {cl}{d}_{X} \times {cl}{d}_{Y} = {cl}{d}_{X \times Y} = {\pi }_{X}^{-1}{cl}{d}_{X} \) .
For any map \( f : X \rightarrow Y \), the family \( {f}^{-1}{c}_{Y} \) can be thought of as the family of (closed) "basewise compact" sets. In IV-5 we shall define what can be thought of as the family of "fiberwise compact" sets.
6.5. Proposition. If \( \Phi \) is a paracompactifying family of supports on \( X \) and if \( Y \subset X \) is locally closed, then \( \Phi \mid Y \) is a paracompactifying family of supports on \( Y \) .
Proof. For \( Y = U \cap F \) with \( U \) open and \( F \) closed, we have that \( \Phi \mid Y = \) \( \left( {\Phi \mid U}\right) \mid \left( {U \cap F}\right) \), so that it suffices to consider the two cases \( Y \) open and \( Y \) closed. These cases are obvious.
---
\( {}^{13} \) Note that \( \Phi \mid F = \Phi \cap F \) for \( F \) closed.
---
6.6. Proposition. Let \( A \subset X \) be locally closed, let \( \Phi \) be a family of supports on \( X \), and let \( \mathcal{B} \) be a sheaf on \( A \) . Then the restriction of sections \( \Gamma \left( {\mathcal{B}}^{X}\right) \rightarrow \Gamma \left( {{\mathcal{B}}^{X} \mid A}\right) = \Gamma \left( \mathcal{B}\right) \) induces an isomorphism
\[
{\Gamma }_{\Phi }\left( {\mathcal{B}}^{X}\right) \overset{ \approx }{ \rightarrow }{\Gamma }_{\Phi \mid A}\left( \mathcal{B}\right)
\]
Similarly, for a sheaf \( \mathcal{A} \) on \( X \), the restriction of sections induces an isomorphism
\[
{\Gamma }_{\Phi }\left( {\mathcal{A}}_{A}\right) \overset{ \approx }{ \rightarrow }{\Gamma }_{\Phi \mid A}\left( {\mathcal{A} \mid A}\right)
\]
Proof. A section \( s \in {\Gamma }_{\Phi }\left( {\mathcal{B}}^{X}\right) \) must have support in \( A \) since \( {\mathcal{B}}^{X} \) vanishes outside of \( A \) . Thus \( \left| s\right| \in \Phi \mid A \) . Moreover, \( s \mid A \) can be zero only if \( s \) is zero. Now suppose that \( t \in {\Gamma }_{\Phi \mid A}\left( \mathcal{B}\right) \), and let \( s : X \rightarrow {\mathcal{B}}^{X} \) be the extension of \( t \) by zero. It suffices to show that \( s \) is continuous. Since \( s \) coincides with the zero section on the open set \( X - \left| t\right| \), it suffices to restrict our attention to the neighborhood of any point \( x \in \left| t\right| \) . Let \( v \in {\mathcal{B}}^{X}\left( U\right) \) be a section of \( {\mathcal{B}}^{X} \) with \( v\left( x\right) = t\left( x\right) = s\left( x\right) \) . We may assume, by changing the open neighborhood \( U \) of \( x \), that \( v\left| {U \cap A = t}\right| U \cap A \) . But \( v \) must vanish on \( U - A \) , so that \( v = s \mid U \) . Hence \( s \) is continuous on \( U \), and this completes the proof of the first statement. The second statement is immediate from the identity \( {\left( \mathcal{A} \mid A\right) }^{X} = {\mathcal{A}}_{A} \)
6.7. In this book the \( \Gamma \) notation will be used only for the group of global sections. Thus the group of sections over \( A \subset X \) of a sheaf \( \mathcal{A} \) on \( X \) is denoted by \( \Gamma \left( {\mathcal{A} \mid A}\right) \) . In the literature, but not here, it is often denoted by \( \Gamma \left( {A,\mathcal{A}}\right) \) . Of course, for the case of a support family \( \Phi \) on \( X \), there are at least two variations: \( {\Gamma }_{\Phi \cap A}\left( {\mathcal{A} \mid A}\right) \) and \( {\Gamma }_{\Phi \mid A}\left( {\mathcal{A} \mid A}\right) \) .
## 7 Classical cohomology theories
As examples of the use of Theorem 6.2 and also for future reference we will briefly describe the "classical" singular, Alexander-Spanier, de Rham, and Čech cohomology theories.
## Alexander-Spanier cohomology
Let \( G \) be a fixed abelian group. For \( U \subset X \) open let \( {A}^{p}\left( {U;G}\right) \) be the group of all functions \( f : {U}^{p + 1} \rightarrow G \) under pointwise addition. Then the functor \( U \mapsto {A}^{p}\left( {U;G}\right) \) is a conjunctive presheaf on \( X \) . [For if \( {f}_{\alpha } : {U}_{\alpha }^{p + 1} \rightarrow \) \( G \) are functions such that \( {f}_{\alpha } \) and \( {f}_{\beta } \) agree on \( {U}_{\alpha }^{p + 1} \cap {U}_{\beta }^{p + 1} \), then define \( f : {U}^{p + 1} \rightarrow G \), where \( U = \bigcup {U}_{\alpha } \), by \( f\left( x\right) = {f}_{\alpha }\left( x\right) \) if \( x \in {U}_{\alpha }^{p + 1} \) and \( f\left( x\right) \) arbitrary if \( \left. {x \notin {U}_{\alpha }^{p + 1}\text{for any}\alpha \text{.}}\right\rbrack \) Let \( {\mathcal{A}}^{p}\left( {X;G}\right) = \mathcal{R} \) eq \( \left( {U \mapsto {A}^{p}\left( {U;G}\right) }\right) \) .
The differential (or "coboundary") \( d : {A}^{p}\left( {U;G}\right) \rightarrow {A}^{p + 1}\left( {U;G}\right) \) is defined by
\[
{df}\left( {{x}_{0},\ldots ,{x}_{p + 1}}\right) = \mathop{\sum }\limits_{{i = 0}}^{{p + 1}}{\left( -1\right) }^{i}f\left( {{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p + 1}}\right) ,
\]
where \( f : {U}^{p + 1} \rightarrow G \) .
Now \( d \) is a homomorphism of presheaves and \( {d}^{2} = 0 \) . Thus \( d \) induces a differential
\[
d : {\mathcal{A}}^{p}\left( {X;G}\right) \rightarrow {\mathcal{A}}^{p + 1}\left( {X;G}\right)
\]
with \( {d}^{2} = 0 \) .
The classical definition of Alexander-Spanier cohomology with supports in the family \( \Phi \) is
\[
{AS}{H}_{\Phi }^{p}\left( {X;G}\right) = {H}^{p}\left( {{A}_{\Phi }^{ * }\left( {X;G}\right) /{A}_{0}^{ * }\left( {X;G}\right) }\right) .
\]
Note that \( {A}_{0}^{p}\left( {X;G}\right) \) is the set of all functions \( {X}^{p + 1} \rightarrow G \) that vanish in some neighborhood of the diagonal. Thus two functions \( f, g : {X}^{p + 1} \rightarrow G \) represent the same element of \( {A}^{p}\left( {X;G}\right) /{A}_{0}^{p}\left( {X;G}\right) \Leftrightarrow \) they coincide in some neighborhood of the diagonal. \( {}^{14} \)
Thus Theorem 6.2 implies that if \( \Phi \) is a paracompactifying family of supports, then there is a natural isomorphism 
(15)
There is a "cup product" \( \cup : {A}^{p}\left( {U;{G}_{1}}\right) \otimes {A}^{q}\left( {U;{G}_{2}}\right) \rightarrow {A}^{p + q}\left( {U;{G}_{1} \otimes {G}_{2}}\right) \) given by the Alexander-Whitney formula
\[
\left( {f \cup g}\right) \left( {{x}_{0},\ldots ,{x}_{p + q}}\right) = f\left( {{x}_{0},\ldots ,{x}_{p}}\right) \otimes g\left( {{x}_{p},\ldots ,{x}_{p + q}}\right)
\]
with \( d\left( {f \cup g}\right) = {df} \cup g + {\left( -1\right) }^{p}f \cup {dg} \) and \( \left| {f \cup g}\right| \subset \left| f\right| \cap \left| g\right| \) . This induces products
\[
\cup : {\mathcal{A}}^{p}\left( {X;{G}_{1}}\right) \otimes {\mathcal{A}}^{q}\left( {X;{G}_{2}}\right) \rightarrow {\mathcal{A}}^{p + q}\left( {X;{G}_{1} \otimes {G}_{2}}\right) ,
\]
\[
{\Gamma }_{\Phi }\left( {{\mathcal{A}}^{p}\left( {X;{G}_{1}}\right) }\right) \otimes {\Gamma }_{\Psi }\left( {{\mathcal{A}}^{q}\left( {X;{G}_{2}}\right) }\right) \rightarrow {\Gamma }_{\Phi \cap \Psi }\left( {{\mathcal{A}}^{p + q}\left( {X;{G}_{1} \otimes {G}_{2}}\right) }\right)
\]
and
\[
{H}^{p}\left( {{\Gamma }_{\Phi }\left( {{\mathcal{A}}^{q}\left( {X;{G}_{1}}\right) }\right) }\right) \otimes {H}^{q}\left( {{\Gamma }_{\Psi }\left( {{\mathcal{A}}^{q}\left( {X;{G}_{2}}\right) }\right) }\right) \rightarrow {H}^{p + q}\left( {{\Gamma }_{\Phi \cap \Psi }\left( {{\mathcal{A}}^{ * }\left( {X;{G}_{1} \otimes {G}_{2}}\right) }\right) }\right) ;
\]
i.e.,
\[
{}_{AS}{H}_{\Phi }^{p}\left( {X;{G}_{1}}\right) \otimes {}_{AS}{H}_{\Psi }^{q}\left( {X;{G}_{2}}\right) \rightarrow {}_{AS}{H}_{\Phi \cap \Psi }^{p + q}\left( {X;{G}_{1} \otimes {G}_{2}}\right) .
\]
In particular, for a base ring \( L \) with unit and an \( L \) -module \( G,{\mathcal{A}}^{0}\left( {X;L}\right) \) is a sheaf of rings with unit, and each \( {\mathcal{A}}^{n}\left( {X;G}\right) \) is an \( {\mathcal{A}}^{0}\left( {X;L}\right) \) -module.
\( {}^{14} \) Note that it is the taking of the quotient by the elements of empty support that brings the topology of \( X \) into the cohomology groups, since \( {A}^{ * }\left( {X;G}\right) \) itself is totally independent of the topology.
## Singular cohomology
Let \( \mathcal{A} \) be a locally constant sheaf on \( X \) . (Classically \( \mathcal{A} \) is called a "bundle of coefficients.") For \( U \subset X \), let \( {S}^{p}\left( {U;\mathcal{A}}\right) \) be the group of singular \( p \) - cochains of \( U \) with values in \( \mathcal{A} \) . That is, an element \( f \in {S}^{p}\left( {U;\mathcal{A}}\right) \) is a function that assigns to each singular \( p \) -simplex \( \sigma : {\Delta }_{p} \rightarrow U \) of \( U \), a cross section \( f\left( \sigma \right) \in \Gamma \left( {{\sigma }^{ * }\left( \mathcal{A}\right) }\right) \), where \( {\Delta }_{p} \) denotes the standard \( p \) -simplex.
Since \( \mathcal{A} \) is locally constant and \( {\Delta }_{p} \) is simply connected, \( {\sigma }^{ * }\left( \mathcal{A}\right) \) is a constant sheaf on \( {\Delta }_{p} \) (as \( {\sigma }^{ * }\left( \mathcal{A}\rig |
1004_(GTM170)Sheaf Theory | 13 | ups, since \( {A}^{ * }\left( {X;G}\right) \) itself is totally independent of the topology.
## Singular cohomology
Let \( \mathcal{A} \) be a locally constant sheaf on \( X \) . (Classically \( \mathcal{A} \) is called a "bundle of coefficients.") For \( U \subset X \), let \( {S}^{p}\left( {U;\mathcal{A}}\right) \) be the group of singular \( p \) - cochains of \( U \) with values in \( \mathcal{A} \) . That is, an element \( f \in {S}^{p}\left( {U;\mathcal{A}}\right) \) is a function that assigns to each singular \( p \) -simplex \( \sigma : {\Delta }_{p} \rightarrow U \) of \( U \), a cross section \( f\left( \sigma \right) \in \Gamma \left( {{\sigma }^{ * }\left( \mathcal{A}\right) }\right) \), where \( {\Delta }_{p} \) denotes the standard \( p \) -simplex.
Since \( \mathcal{A} \) is locally constant and \( {\Delta }_{p} \) is simply connected, \( {\sigma }^{ * }\left( \mathcal{A}\right) \) is a constant sheaf on \( {\Delta }_{p} \) (as \( {\sigma }^{ * }\left( \mathcal{A}\right) \) is just the induced bundle on \( {\Delta }_{p} \) ). It follows that we can define the coboundary operator
\[
d : {S}^{p}\left( {U;\mathcal{A}}\right) \rightarrow {S}^{p + 1}\left( {U;\mathcal{A}}\right)
\]
by \( {df}\left( \tau \right) = f\left( {\partial \tau }\right) \in \Gamma \left( {{\tau }^{ * }\left( \mathcal{A}\right) }\right) \) .
Let \( {\mathcal{P}}^{p}\left( {X;\mathcal{A}}\right) = \mathcal{R} \) has \( /\left( {U \mapsto {S}^{p}\left( {U;\mathcal{A}}\right) }\right) \) with the induced differential. The presheaf \( {S}^{p}\left( {\bullet ;\mathcal{A}}\right) \) is conjunctive since if \( \left\{ {U}_{\alpha }\right\} \) is a collection of open sets with union \( U \) and if \( f\left( \sigma \right) \) is defined whenever \( \sigma \) is a singular simplex in some \( {U}_{\alpha } \) with value that is independent of the particular index \( \alpha \), then just define \( f\left( \sigma \right) = 0 \) (or anything) if \( \sigma \notin {U}_{\alpha } \) for any \( \alpha \), and this extends \( f \) to be an element of \( {S}^{p}\left( {U;\mathcal{A}}\right) \) .
The classical definition of singular cohomology (with the local coefficients \( \mathcal{A} \) and supports in \( \Phi \) ) is
\[
\Delta {H}_{\Phi }^{p}\left( {X;\mathcal{A}}\right) = {H}^{p}\left( {{S}_{\Phi }^{ * }\left( {X;\mathcal{A}}\right) }\right) .
\]
However, it is a well-known consequence of the operation of subdivision that
\[
{H}^{p}\left( {{S}_{0}^{ * }\left( {X;\mathcal{A}}\right) }\right) = 0\;\text{ for all }p.
\]
[We indicate the proof: Let \( \mathfrak{U} = \left\{ {U}_{\alpha }\right\} \) be a covering of \( X \) by open sets and let \( {S}^{p}\left( {\mathfrak{U};\mathcal{A}}\right) \) be the group of singular cochains based on \( \mathfrak{U} \) -small singular simplices. Then a subdivision argument shows that the surjection
\[
{j}_{\mathfrak{U}} : {S}^{ * }\left( {X;\mathcal{A}}\right) \rightarrow {S}^{ * }\left( {\mathfrak{U};\mathcal{A}}\right)
\]
induces a cohomology isomorphism. Therefore, if we let \( {K}_{\mathfrak{U}}^{ * } = \operatorname{Ker}{j}_{\mathfrak{U}} \), then \( {H}^{ * }\left( {K}_{u}^{ * }\right) = 0 \) by the long exact cohomology sequence induced by the short exact cochain sequence \( 0 \rightarrow {K}_{\mathfrak{U}}^{ * } \rightarrow {S}^{ * }\left( {X;\mathcal{A}}\right) \rightarrow {S}^{ * }\left( {\mathfrak{U};\mathcal{A}}\right) \rightarrow 0 \) . However, \( {S}_{0}^{ * }\left( {X;\mathcal{A}}\right) = \bigcup {K}_{\mathfrak{U}}^{ * } = \mathop{\lim }\limits_{ \rightarrow }{K}_{\mathfrak{U}}^{ * } \), so that
\[
\left. {{H}^{ * }\left( {{S}_{0}^{ * }\left( {X;\mathcal{A}}\right) }\right) = {H}^{ * }\left( {\underline{\lim }{K}_{\mathfrak{U}}^{ * }}\right) \approx \underline{\lim }{H}^{ * }\left( {K}_{\mathfrak{U}}^{ * }\right) = 0.}\right\rbrack
\]
Therefore, if \( \Phi \) is paracompactifying, then the exact sequence
\[
0 \rightarrow {S}_{0}^{ * } \rightarrow {S}_{\Phi }^{ * } \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{S}}^{ * }\right) \rightarrow 0
\]
of 6.2 yields the isomorphism
\[
\Delta {H}_{\Phi }^{p}\left( {X;\mathcal{A}}\right) \approx {H}^{p}\left( {{\Gamma }_{\Phi }\left( {{\mathcal{S}}^{ * }\left( {X;\mathcal{A}}\right) }\right) }\right) .
\]
(16)
As with the case of Alexander-Spanier cohomology, the singular cup product makes \( {\mathcal{P}}^{0}\left( {X;L}\right) \) into a sheaf of rings and each \( {\mathcal{P}}^{n}\left( {X;\mathcal{A}}\right) \) into an \( {\mathcal{P}}^{0}\left( {X;L}\right) \) -module, where \( \mathcal{A} \) is a locally constant sheaf of \( L \) -modules.
Remark: If \( X \) is a differentiable manifold and we let \( {S}^{ * }\left( {U;\mathcal{A}}\right) \) be the complex of singular cochains based on \( {C}^{\infty } \) singular simplices, a similar discussion applies.
## de Rham cohomology
Let \( X \) be a differentiable manifold and let \( {\Omega }^{p}\left( U\right) \) be the group of differential \( p \) -forms on \( U \) with \( d : {\Omega }^{p}\left( U\right) \rightarrow {\Omega }^{p + 1}\left( U\right) \) being the exterior derivative. \( {}^{15} \)
The de Rham cohomology group of \( X \) is defined to be
\[
{\Omega }_{\Omega }{H}_{\Phi }^{p}\left( X\right) = {H}^{p}\left( {{\Omega }_{\Phi }^{ * }\left( X\right) }\right) .
\]
However, the presheaf \( U \mapsto {\Omega }^{p}\left( U\right) \) is a conjunctive monopresheaf and hence is a sheaf. Thus, trivially, we have 
(17)
for any family \( \Phi \) of supports.
## Čech cohomology
Let \( \mathfrak{U} = \left\{ {{U}_{\alpha };\alpha \in I}\right\} \) be an open covering of a space \( X \) indexed by a set \( I \) and let \( G \) be a presheaf on \( X \) . Then an \( n \) -cochain \( c \) of \( \mathfrak{U} \) is a function defined on ordered \( \left( {n + 1}\right) \) -tuples \( \left( {{\alpha }_{0},\ldots ,{\alpha }_{n}}\right) \) of members of \( I \) such that \( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n}} = {U}_{{\alpha }_{0}} \cap \cdots \cap {U}_{{\alpha }_{n}} \neq \varnothing \) with value
\[
c\left( {{\alpha }_{0},\ldots ,{\alpha }_{n}}\right) \in G\left( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n}}\right)
\]
These form a group denoted by \( {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \) . An open set \( V \) of \( X \) is covered by \( \mathfrak{U} \cap V = \left\{ {{U}_{\alpha } \cap V;\alpha \in I}\right\} \) . Thus we have the cochain group \( {\check{C}}^{n}\left( {\mathfrak{U} \cap V;G}\right) \) , and the assignment \( V \mapsto {\check{C}}^{n}\left( {\mathfrak{U} \cap V;G}\right) \) gives a presheaf on \( X \), and hence a sheaf
\[
{\check{\mathcal{C}}}^{n}\left( {\mathfrak{U};G}\right) = \mathcal{P}\text{ heaf }\left( {V \mapsto {\check{C}}^{n}\left( {\mathfrak{U} \cap V;G}\right) }\right) .
\]
Thus it makes sense to speak of the support \( \left| c\right| \) of a cochain, i.e., \( \left| c\right| = \) \( \left| {\theta \left( c\right) }\right| \), where \( \theta : {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \rightarrow \Gamma \left( {{\check{\mathcal{C}}}^{n}\left( {\mathfrak{U};G}\right) }\right) \) . This defines the cochain group \( {\check{C}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \) for a family \( \Phi \) of supports on \( X \) . The coboundary operator \( d : {\check{C}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{n + 1}\left( {\mathfrak{U};G}\right) \) is defined by
\[
{dc}\left( {{\alpha }_{0},\ldots ,{\alpha }_{n + 1}}\right) = \mathop{\sum }\limits_{{i = 0}}^{{n + 1}}{\left( -1\right) }^{i}c\left( {{\alpha }_{0},\ldots ,\widehat{{\alpha }_{i}},\ldots ,{\alpha }_{n + 1}}\right) \mid {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n + 1}}.
\]
---
\( {}^{15} \) See, for example, \( \left\lbrack {{19}\text{, Chapters II and V}}\right\rbrack \) .
---
It is easy to see that \( {d}^{2} = 0 \) and so there are the cohomology groups
\[
{\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) = {H}^{n}\left( {{\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) }\right)
\]
A refinement of \( \mathfrak{U} \) is another open covering \( \mathfrak{V} = \left\{ {{V}_{\beta };\beta \in J}\right\} \) together with a function (called a refinement projection) \( \varphi : J \rightarrow I \) such that \( {V}_{\beta } \subset {U}_{\varphi \left( \beta \right) } \) for all \( \beta \in J \) . This yields a chain map \( {\varphi }^{ * } : {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{V};G}\right) \) by
\[
{\varphi }^{ * }\left( c\right) \left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{n}\right) }\right) |{V}_{{\beta }_{0},\ldots ,{\beta }_{n}}.
\]
If \( \psi : J \rightarrow I \) is another refinement projection, then the functions \( D \) : \( {\check{C}}_{\Phi }^{n + 1}\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \) given by
\[
{Dc}\left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{i}\right) ,\psi \left( {\beta }_{i}\right) ,\ldots ,\psi \left( {\beta }_{n}\right) }\right) \mid {V}_{{\beta }_{0},\ldots ,{\beta }_{n}}
\]
provide a chain homotopy between \( {\varphi }^{ * } \) and \( {\psi }^{ * } \) . Therefore, there is a homomorphism
\[
{j}_{\mathfrak{V},\mathfrak{U}}^{n} : {\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \rightarrow {\check{H}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right)
\]
induced by \( {\varphi }^{ * } \) but independent of the particular refinement projection \( \varphi \) used to define it. Thus we can define the Čech cohomology group as
\[
{\check{H}}_{\Phi }^{n}\left( {X;G}\right) = {\underline{\lim }}_{\mathfrak{U}}{\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\ri |
1004_(GTM170)Sheaf Theory | 14 | ightarrow {\check{C}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \) given by
\[
{Dc}\left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{i}\right) ,\psi \left( {\beta }_{i}\right) ,\ldots ,\psi \left( {\beta }_{n}\right) }\right) \mid {V}_{{\beta }_{0},\ldots ,{\beta }_{n}}
\]
provide a chain homotopy between \( {\varphi }^{ * } \) and \( {\psi }^{ * } \) . Therefore, there is a homomorphism
\[
{j}_{\mathfrak{V},\mathfrak{U}}^{n} : {\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \rightarrow {\check{H}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right)
\]
induced by \( {\varphi }^{ * } \) but independent of the particular refinement projection \( \varphi \) used to define it. Thus we can define the Čech cohomology group as
\[
{\check{H}}_{\Phi }^{n}\left( {X;G}\right) = {\underline{\lim }}_{\mathfrak{U}}{\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right)
\]
Since it does not affect the direct limit to restrict the coverings to a cofinal set of coverings, it is legitimate to restrict attention to coverings \( \mathfrak{U} = \left\{ {{U}_{x};x \in X}\right\} \) such that \( x \in {U}_{x} \) for all \( x \) . In this case there is a canonical refinement projection, the identity map \( X \rightarrow X \), for a refinement \( \mathfrak{V} = \left\{ {{V}_{x};x \in X}\right\} ,{V}_{x} \subset {U}_{x} \) . Thus there is a canonical chain map
\[
{\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{V};G}\right)
\]
and so it is legitimate to pass to the limit and define the Čech cochain group
\[
{\check{C}}_{\Phi }^{ * }\left( {X;G}\right) = {\underline{\lim }}_{\mathfrak{U}}{\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right)
\]
Since the direct limit functor is exact, it commutes with cohomology, i.e., there is a canonical isomorphism
\[
{\check{H}}_{\Phi }^{ * }\left( {X;G}\right) \approx {H}^{ * }\left( {{\check{C}}_{\Phi }^{ * }\left( {X;G}\right) }\right)
\]
which we shall regard as equality.
We shall study this further in Chapter III. For the present, let us restrict attention to the case in which \( G \) is an abelian group regarded as a constant presheaf. We wish to define a natural homomorphism from the Alexander-Spanier groups to the Čech groups. If \( f : {X}^{n + 1} \rightarrow G \) is an Alexander-Spanier cochain and \( \mathfrak{U} = \left\{ {{U}_{x};x \in X}\right\} \) is a covering of \( X \), then \( f \) induces
an element \( {f}_{\mathfrak{U}} \in {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \) by putting \( {f}_{\mathfrak{U}}\left( {{x}_{0},\ldots ,{x}_{n}}\right) = f\left( {{x}_{0},\ldots ,{x}_{n}}\right) \) when \( {U}_{{x}_{0},\ldots ,{x}_{n}} \neq \varnothing \) . Consequently, \( f \) induces \( {f}_{\infty } = \underline{\lim }{f}_{\mathfrak{U}} \in {\check{C}}^{ * }\left( {X;G}\right) \) . We claim that \( \left| {f}_{\infty }\right| = \left| f\right| \) . Indeed,
\( x \notin \left| {f}_{\infty }\right| \Leftrightarrow \exists \mathfrak{U}, V, x \in V \) and \( {f}_{\mathfrak{U}}|V = 0 \) in \( \check{C}\left( {\mathfrak{U} \cap V;G}\right) \)
\[
\Leftrightarrow \;\exists \mathfrak{U}, V,\;x \in V \ni {x}_{0},\ldots ,{x}_{n} \in V\; \Rightarrow \;{f}_{\mathfrak{U}}\left( {{x}_{0},\ldots ,{x}_{n}}\right) = 0
\]
\[
\Leftrightarrow \;\exists W,\;x \in W \ni {x}_{0},\ldots ,{x}_{n} \in W\; \Rightarrow \;f\left( {{x}_{0},\ldots ,{x}_{n}}\right) = 0
\]
\[
\Leftrightarrow \;\exists W,\;x \in W \ni f|{W}^{n + 1} = 0
\]
\[
\Leftrightarrow x \notin \left| f\right| \text{. }
\]
Also, \( \left| {f}_{\infty }\right| = \varnothing \Leftrightarrow {f}_{\infty } = 0 \) . Now, given \( g \in {\check{C}}^{ * }\left( {X;G}\right), g \) comes from some \( {g}_{\mathfrak{U}} \in {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \), and it is clear that \( {g}_{\mathfrak{U}} \) extends arbitrarily to an Alexander-Spanier cochain \( g \) . It follows that \( f \mapsto {f}_{\infty } \) induces an isomorphism
\[
{A}_{\Phi }^{n}\left( {X;G}\right) /{A}_{0}^{n}\left( {X;G}\right) \overset{ \approx }{ \rightarrow }{\check{C}}_{\Phi }^{n}\left( {X;G}\right)
\]
whence
\[
{A}_{S}{H}_{\Phi }^{n}\left( {X;G}\right) \approx {\check{H}}_{\Phi }^{n}\left( {X;G}\right)
\]
(18)
for all spaces \( X \) and families \( \Phi \) of supports on \( X \) .
Now, the Čech cohomology groups are not altered by restriction to any cofinal system of coverings. Therefore, if \( X \) is compact, we can restrict the coverings used to finite coverings. Similarly, if \( X \) is paracompact, we can restrict attention to locally finite coverings.
Finally, if the covering dimension \( {}^{16}\operatorname{covdim}X = n < \infty \) then we can restrict attention to locally finite coverings \( \mathfrak{U} = \left\{ {{U}_{\alpha };\alpha \in I}\right\} \) such that \( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n + 1}} = \varnothing \) for distinct \( {\alpha }_{i} \) . Now, \( {\check{C}}^{ * }\left( {\mathfrak{U};G}\right) \) is the ordered simplicial cochain complex \( {C}^{ * }\left( {N\left( \mathfrak{U}\right) ;G}\right) \) of an \( n \) -dimensional abstract simplicial complex, namely the nerve \( N\left( \mathfrak{U}\right) \) of \( {\mathfrak{U}}^{.17} \) For \( c \in {\check{C}}^{p}\left( {\mathfrak{U};G}\right) \) we have that \( \left| c\right| = \bigcup \left\{ {{\bar{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} \mid c\left( {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right) \neq 0}\right\} \), since \( \mathfrak{U} \) is locally finite. Thus
\[
{\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) = \{ c \in {\check{C}}^{ * }\left( {\mathfrak{U};G}\right) \;|\;\exists K \in \Phi \ni c\left( {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right) = 0\text{ if }{\overline{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} ⊄ K\}
\]
\[
= \mathop{\lim }\limits_{{K \in \Phi }}\left\{ {c \in {\check{C}}^{ * }\left( {\mathfrak{U};G}\right) \mid c\left( {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right) = 0\text{ if }{\overline{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} ⊄ K}\right\}
\]
\[
= {\underline{\lim }}_{K \in \Phi }{C}^{ * }\left( {N\left( \mathfrak{U}\right) ,{N}_{K}\left( \mathfrak{U}\right) ;G}\right) ,
\]
where \( {N}_{K}\left( \mathfrak{U}\right) = \left\{ {\left\{ {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right\} \in N\left( \mathfrak{U}\right) \mid {\bar{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} ⊄ K}\right\} \), which is a sub-complex of \( N\left( \mathfrak{U}\right) \) . But \( {C}^{ * }\left( {N\left( \mathfrak{U}\right) ,{N}_{K}\left( \mathfrak{U}\right) ;G}\right) \) is chain equivalent to the corresponding oriented simplicial cochain complex that vanishes above degree \( n \) . Therefore \( {\check{H}}_{\Phi }^{p}\left( {\mathfrak{U};G}\right) = 0 \) for \( p > n \), whence \( {\check{H}}_{\Phi }^{p}\left( {X;G}\right) = 0 \) for \( p > n \) . Consequently,
\[
{}_{AS}{H}_{\Phi }^{p}\left( {X;G}\right) = 0\text{ for }p > \operatorname{covdim}X.
\]
(19)
\( {}^{16} \) The covering dimension of \( X \) is the least integer \( n \) (or \( \infty \) ) such that every covering of \( X \) has a refinement for which no point is contained in more than \( n + 1 \) distinct members of the covering.
\( {}^{17} \) This has the members of \( I \) as vertices and the subsets \( \left\{ {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right\} \subset I \), where \( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} \neq \varnothing \), as the \( p \) -simplices.
## Singular homology
Even though the definition of singular cohomology requires a locally constant sheaf as coefficients, \( {}^{18} \) one can define singular homology with coefficients in an arbitrary sheaf \( \mathcal{A} \) . To do this, define the group of singular \( n \) -chains by
\[
{S}_{n}\left( {X;\mathcal{A}}\right) = {\bigoplus }_{\sigma }\Gamma \left( {{\sigma }^{ * }\mathcal{A}}\right)
\]
where the sum ranges over all singular simplices \( \sigma : {\Delta }_{n} \rightarrow X \) of \( X \) . If \( {F}_{i} \) : \( {\Delta }_{n - 1} \rightarrow {\Delta }_{n} \) is the \( i \) th face map, then we have the induced homomorphism
\[
{\eta }_{i} : \Gamma \left( {{\sigma }^{ * }\mathcal{A}}\right) \rightarrow \Gamma \left( {{F}_{i}^{ * }{\sigma }^{ * }\mathcal{A}}\right) = \Gamma \left( {{\left( \sigma \circ {F}_{i}\right) }^{ * }\mathcal{A}}\right)
\]
of Section 4, and so the boundary operator
\[
\partial : {S}_{n}\left( {X;\mathcal{A}}\right) \rightarrow {S}_{n - 1}\left( {X;\mathcal{A}}\right)
\]
can be defined by
\[
\partial s = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}{\eta }_{i}\left( s\right)
\]
for \( s \in \Gamma \left( {{\sigma }^{ * }\mathcal{A}}\right) \) .
When \( \mathcal{A} \) is locally constant, then this, and the case of cohomology, is equivalent to Steenrod's definition of (co)homology with "local coefficients"; see [75] for the definition of the latter.
The functor \( U \mapsto {S}_{n}\left( {U;\mathcal{A}}\right) \) is covariant, and so it is not a presheaf. Thus it has a different nature than do the cohomology theories. See, however, Exercise 12 for a different description of singular homology that has a closer relationship to the cohomology theories.
## Exercises
1. (c) If \( \mathcal{A} \) is a sheaf on \( X \) and \( i : B \hookrightarrow X \), show that \( {i}^{ * }\mathcal{A} \approx \mathcal{A} \mid B \) .
2. (c) If \( \mathcal{B} \) is a sheaf on \( B \) and \( i : B \hookrightarrow X \), show that \( \left( {i\mathcal{B}}\right) \mid B \approx \mathcal{B} \) .
3. Let \( \left\{ {{B}_{\alpha },{\pi }_{\alpha ,\beta }}\right\} \) be a direct system of presheaves [that is, for \( U \subset X \) , \( \left\{ {{B}_{\alpha }\left( U\right) ,{\pi }_{\alpha ,\beta }\left( U\right) }\right\} \) is a direct system of groups such that the \( {\pi }_{\alpha ,\beta } \) ’s commute with restrictions]. Let \( B = \mathop{\lim }\limits_{ \rightarrow }{B}_{\alpha } \) denote the |
1004_(GTM170)Sheaf Theory | 15 | \mathcal{A}}\right) \) is covariant, and so it is not a presheaf. Thus it has a different nature than do the cohomology theories. See, however, Exercise 12 for a different description of singular homology that has a closer relationship to the cohomology theories.
## Exercises
1. (c) If \( \mathcal{A} \) is a sheaf on \( X \) and \( i : B \hookrightarrow X \), show that \( {i}^{ * }\mathcal{A} \approx \mathcal{A} \mid B \) .
2. (c) If \( \mathcal{B} \) is a sheaf on \( B \) and \( i : B \hookrightarrow X \), show that \( \left( {i\mathcal{B}}\right) \mid B \approx \mathcal{B} \) .
3. Let \( \left\{ {{B}_{\alpha },{\pi }_{\alpha ,\beta }}\right\} \) be a direct system of presheaves [that is, for \( U \subset X \) , \( \left\{ {{B}_{\alpha }\left( U\right) ,{\pi }_{\alpha ,\beta }\left( U\right) }\right\} \) is a direct system of groups such that the \( {\pi }_{\alpha ,\beta } \) ’s commute with restrictions]. Let \( B = \mathop{\lim }\limits_{ \rightarrow }{B}_{\alpha } \) denote the presheaf \( U \mapsto \mathop{\lim }\limits_{ \rightarrow }{B}_{\alpha }\left( U\right) \) . Let \( {\mathcal{B}}_{\alpha } = \mathcal{R} \) heaf \( \left( {B}_{\alpha }\right) \) and \( \mathcal{B} = \mathcal{R} \) heaf \( \left( B\right) \) . Show that \( \mathcal{B} \) and \( \underline{\lim }{\mathcal{B}}_{\alpha } \) are canonically isomorphic.
4. (c) A sheaf \( \mathcal{P} \) on \( X \) is called projective if the following commutative diagram, with exact row, can always be completed as indicated:
---
\( {}^{18} \) This will be generalized by another method in Chapter III.
---
Show that the constant sheaf \( \mathbb{Z} \) on the unit interval is not the quotient of a projective sheaf. (Thus there are not "sufficiently many projectives" in the category of sheaves.)
More generally, show that on a locally connected Hausdorff space without isolated points the only projective sheaf is 0 .
5. Show that the tensor product of two sheaves satisfies the universal property of tensor products. That is, if \( \mathcal{A},\mathcal{B} \), and \( \mathcal{C} \) are sheaves on \( X \) and if \( f : \mathcal{A}\Delta \mathcal{B} \rightarrow \mathcal{C} \) is a map that commutes with the projections onto \( X \) and is bilinear on each stalk, then there exists a unique homomorphism \( h : \mathcal{A} \otimes \mathcal{B} \rightarrow \mathcal{C} \) such that \( f = {hk} \), where \( k : \mathcal{A}\bigtriangleup \mathcal{B} \rightarrow \mathcal{A} \otimes \mathcal{B} \) takes \( \left( {a, b}\right) \in {\mathcal{A}}_{x} \times {\mathcal{B}}_{x} = {\left( \mathcal{A}\bigtriangleup \mathcal{B}\right) }_{x} \) into \( a \otimes b \in {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{x} = {\left( \mathcal{A} \otimes \mathcal{B}\right) }_{x}. \)
Treat the direct sum in a similar manner.
6. Show that the functor \( \operatorname{Hom}\left( {\bullet , \bullet }\right) \) of sheaves is left exact.
7. Let \( f : X \rightarrow Y \) and let \( \mathcal{R} \) be a sheaf of rings on \( Y \) . Show that the natural equivalence (5) of Section 4 restricts to a natural equivalence
\[
\varphi : {\operatorname{Hom}}_{{f}^{ * }\mathcal{R}}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \overset{ \approx }{ \rightarrow }{\operatorname{Hom}}_{\mathcal{R}}\left( {\mathcal{B}, f\mathcal{A}}\right)
\]
where \( \mathcal{B} \) is an \( \mathcal{R} \) -module and \( \mathcal{A} \) is an \( {f}^{ * }\mathcal{R} \) -module. [The \( \mathcal{R} \) -module structure of \( f\mathcal{A} \) is given by the composition
\[
\left. {\mathcal{R}\left( U\right) \otimes \left( {f\mathcal{A}}\right) \left( U\right) \rightarrow \left( {{f}^{ * }\mathcal{R}}\right) \left( {{f}^{-1}U}\right) \otimes \mathcal{A}\left( {{f}^{-1}U}\right) \rightarrow \mathcal{A}\left( {{f}^{-1}U}\right) .}\right\rbrack
\]
8. (c) Let \( f : X \rightarrow Y \) and let \( \mathcal{A} \) be a sheaf on \( X \) . Show that \( {\Gamma }_{\Phi }\left( {f\mathcal{A}}\right) = \) \( {\Gamma }_{{f}^{-1}\Phi }\left( \mathcal{A}\right) \), under the defining equality \( \left( {f\mathcal{A}}\right) \left( Y\right) = \mathcal{A}\left( X\right) \), for any family \( \Phi \) of supports on \( Y \) . [Also, see IV-3.]
9. (c) Let \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) be an exact sequence of sheaves on a locally connected Hausdorff space \( X \) . Suppose that \( {\mathcal{A}}^{\prime } \) and \( {\mathcal{A}}^{\prime \prime } \) are locally constant and that the stalks of \( {\mathcal{A}}^{\prime \prime } \) are finitely generated (over some constant base ring). Show that \( \mathcal{A} \) is also locally constant. [Hint: For \( x \in X \) find a neighborhood \( U \) such that \( \mathcal{A}\left( U\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( U\right) \) is surjective and such that \( {\mathcal{A}}^{\prime }\left( U\right) \rightarrow {\mathcal{A}}_{y}^{\prime } \) and \( {\mathcal{A}}^{\prime \prime }\left( U\right) \rightarrow {\mathcal{A}}_{y}^{\prime \prime } \) are isomorphisms for every \( y \in U \) .]
10. (c) Show by example that Exercise 9 does not hold without the condition that the stalks of \( {\mathcal{A}}^{\prime \prime } \) are finitely generated.
11. (c) If \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) is an exact sequence of constant sheaves on \( X \), show that the sequence
\[
0 \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime }\right) \rightarrow {\Gamma }_{\Phi }\left( \mathcal{A}\right) \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime \prime }\right) \rightarrow 0
\]
is exact for any family \( \Phi \) of supports on \( X \) .
12. (c) Let \( {\Delta }_{ * }\left( {X, A}\right) \) [respectively, \( {\Delta }_{ * }^{c}\left( {X, A}\right) \) ] be the chain complex of locally finite (respectively, finite) singular chains of \( X \) modulo those chains in \( A \) . Show that the homomorphism of generated sheaves induced by the obvious homomorphism
\[
{\Delta }_{ * }^{c}\left( {X, X - U}\right) \hookrightarrow {\Delta }_{ * }\left( {X, X - U}\right)
\]
of presheaves is an isomorphism. Denote this generated sheaf by \( {\Delta }_{ * } \) . Show that the presheaf \( U \mapsto {\Delta }_{ * }\left( {X, X - U}\right) \) (which generates \( {\Delta }_{ * } \) ) is a mono-presheaf and that it is conjunctive for coverings of \( X \) . Deduce that
\[
\theta : {\Delta }_{ * }\left( X\right) \rightarrow \Gamma \left( {\Delta }_{ * }\right)
\]
is an isomorphism when \( X \) is paracompact. [Note, however, that \( U \mapsto \) \( {\Delta }_{ * }\left( {X, X - U}\right) \) is not fully conjunctive and hence is not itself a sheaf.] Also show that \( \theta \) induces an isomorphism
\[
{\Delta }_{ * }^{c}\left( X\right) \overset{ \approx }{ \rightarrow }{\Gamma }_{c}\left( {\Delta }_{ * }\right)
\]
(This provides another approach to the definition of singular homology with coefficients in a sheaf, by putting \( {\Delta }_{ * }^{c}\left( {X;\mathcal{A}}\right) = {\Gamma }_{c}\left( {{\Delta }_{ * } \otimes \mathcal{A}}\right) \) .)
13. Let \( X \) be the complex line (real 2-dimensional) and let \( C \) denote the constant sheaf of complex numbers. Let \( \mathcal{A} \) be the sheaf of germs of complex analytic functions on \( X \) . Show that
\[
0 \rightarrow C\overset{i}{ \rightarrow }\mathcal{A}\overset{d}{ \rightarrow }\mathcal{A} \rightarrow 0
\]
is exact, where \( i \) is the canonical inclusion and \( d \) is differentiation. For \( U \subset X \) open show that \( d : \mathcal{A}\left( U\right) \rightarrow \mathcal{A}\left( U\right) \) need not be surjective. For which open sets \( U \) is it surjective?
14. (c) Let \( X \) be the unit circle in the plane. Let \( \mathbb{R} \) denote the constant sheaf of real numbers; \( \mathcal{D} \) the sheaf of germs of continuously differentiable real-valued functions on \( X \) ; and \( \mathcal{C} \) the sheaf of germs of continuous real-valued functions on \( X \) . Show that \( 0 \rightarrow \mathbb{R}\overset{i}{ \rightarrow }\mathcal{D}\overset{d}{ \rightarrow }\mathcal{C} \rightarrow 0 \) is exact, where \( d \) is differentiation. Show that \( \operatorname{Coker}\{ d : \mathcal{D}\left( X\right) \rightarrow \mathcal{C}\left( X\right) \} \) is isomorphic to the group of real numbers.
15. (c) Let \( X \) be the real line. Let \( \mathcal{F} \) be the sheaf of germs of all integer-valued functions on \( X \) and let \( i : \mathbb{Z} \hookrightarrow \mathcal{F} \) be the canonical inclusion. Let \( \mathcal{G} \) be the quotient sheaf of \( \mathcal{F} \) by \( \mathbb{Z} \) . Show that \( \mathcal{F}\left( X\right) \rightarrow \mathcal{G}\left( X\right) \) is surjective, while \( \operatorname{Coker}\left\{ {{\Gamma }_{c}\left( \mathcal{F}\right) \rightarrow {\Gamma }_{c}\left( \mathcal{G}\right) }\right\} \approx \mathbb{Z}. \)
16. Show that there are natural isomorphisms
\[
\operatorname{Hom}\left( {{\bigoplus }_{\lambda }{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {{\mathcal{A}}_{\lambda },\mathcal{B}}\right)
\]
and
\[
\operatorname{Hom}\left( {\mathcal{A},\mathop{\prod }\limits_{\lambda }{\mathcal{B}}_{\lambda }}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {\mathcal{A},{\mathcal{B}}_{\lambda }}\right)
\]
17. Prove or disprove that there is the following natural isomorphism of functors of sheaves \( \mathcal{A},\mathcal{B} \), and \( \mathcal{C} \) on \( X \) :
\[
\mathcal{H}\text{on}\left( {\mathcal{A},\mathcal{H}\text{on}\left( {\mathcal{B},\mathcal{C}}\right) }\right) \approx \mathcal{H}\text{on}\left( {\mathcal{A} \otimes \mathcal{B},\mathcal{C}}\right) .
\]
18. (c) If \( f : A \rightarrow X \) is a map with \( f\left( A\right) \) dense in \( X |
1004_(GTM170)Sheaf Theory | 16 | \left( \mathcal{G}\right) }\right\} \approx \mathbb{Z}. \)
16. Show that there are natural isomorphisms
\[
\operatorname{Hom}\left( {{\bigoplus }_{\lambda }{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {{\mathcal{A}}_{\lambda },\mathcal{B}}\right)
\]
and
\[
\operatorname{Hom}\left( {\mathcal{A},\mathop{\prod }\limits_{\lambda }{\mathcal{B}}_{\lambda }}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {\mathcal{A},{\mathcal{B}}_{\lambda }}\right)
\]
17. Prove or disprove that there is the following natural isomorphism of functors of sheaves \( \mathcal{A},\mathcal{B} \), and \( \mathcal{C} \) on \( X \) :
\[
\mathcal{H}\text{on}\left( {\mathcal{A},\mathcal{H}\text{on}\left( {\mathcal{B},\mathcal{C}}\right) }\right) \approx \mathcal{H}\text{on}\left( {\mathcal{A} \otimes \mathcal{B},\mathcal{C}}\right) .
\]
18. (c) If \( f : A \rightarrow X \) is a map with \( f\left( A\right) \) dense in \( X \) and \( \mathcal{M} \) is a subsheaf of a constant sheaf on \( X \), then show that the canonical map \( \beta : \mathcal{M} \rightarrow f{f}^{ * }\mathcal{M} \) of Section 4 is a monomorphism. Also, give an example showing that this is false for arbitrary sheaves \( \mathcal{M} \) on \( X \) .
19. (c) For a given point \( x \) in the Hausdorff space \( X \), let \( x \) also denote the family \( \{ \{ x\} ,\varnothing \} \) of supports on \( X \) . Show that \( {}_{AS}{H}_{x}^{n}\left( {X;G}\right) = 0 \) for all \( n > 0 \) . (Compare II-18.)
## Chapter II Sheaf Cohomology
In this chapter we shall define the sheaf-theoretic cohomology theory and shall develop many of its basic properties.
The cohomology groups of a space with coefficients in a sheaf are defined in Section 2 using the canonical resolution of a sheaf due to Godement. In Section 3 it is shown that the category of sheaves contains "enough injectives," and it follows from the results of Sections 4 and 5 that the sheaf cohomology groups are just the right derived functors of the left exact functor \( \Gamma \) that assigns to a sheaf its group of sections.
A sheaf \( \mathcal{A} \) is said to be acyclic if the higher cohomology groups with coefficients in \( \mathcal{A} \) are zero. Such sheaves provide a means of "computing" cohomology in particular situations. In Sections 5 and 9 some important classes of acyclic sheaves are defined and investigated.
In Section 6 we prove a theorem concerning the existence and uniqueness of extensions of a natural transformation of functors (of several variables) to natural transformations of "connected systems" of functors. This result is applied in Section 7 to define, and to give axioms for, the cup product in sheaf cohomology theory. These sections are central to our treatment of many of the fundamental consequences of sheaf theory.
The cohomology homomorphism induced by a map is defined in Section 8. The relationship between the cohomology of a subspace and that of its neighborhoods is investigated in Section 10, and the important notion of "tautness" of a subspace is introduced there.
In Section 11 we prove the Vietoris mapping theorem and use it to prove that sheaf-theoretic cohomology, with constant coefficients, satisfies the invariance under homotopy property for general topological spaces.
Relative cohomology theory is introduced into sheaf theory in Section 12, and its properties, such as invariance under excision, are developed. In Section 13 we derive some exact sequences of the Mayer-Vietoris type.
Sections 14, 15, and 17 are concerned, almost exclusively, with locally compact spaces. In Section 14 we prove the "continuity" property, both for spaces and for coefficient sheaves. This property is an important feature of sheaf-theoretic cohomology that is not satisfied for such theories as singular cohomology. A general Künneth formula is derived in Section 15. Section 17 treats local connectivity in higher degrees. This section really has nothing to do with sheaf theory, but the results of this section are used repeatedly in later parts of the book.
In Section 16 we study the concept of cohomological dimension, which has important applications to other parts of the book. Section 18 contains definitions of "local cohomology groups" and of cohomology groups of the "ideal boundary."
If \( G \) is a finite group acting on a space \( X \) and if \( \pi : X \rightarrow X/G \) is the "orbit map," then \( \pi \) induces, as does any map, a homomorphism from the cohomology of \( X/G \) to that of \( X \) . In Section 19 the "transfer map," which takes the cohomology of \( X \) into that of \( X/G \), is defined. When \( G \) is cyclic of prime order we also obtain the exact sequences of P. A. Smith relating the cohomology of the fixed point set of \( G \) to that of \( X \) (a general Hausdorff space on which \( G \) acts).
In Sections 20 and 21 we define the Steenrod cohomology operations (the squares and \( p \) th powers) in sheaf cohomology and derive several of their properties. This material is not used elsewhere in the book.
All of the sections of this chapter, except for Sections 18 through 21, are used repeatedly in other parts of the book.
Most of Chapter III can be read after Section 9 of the present chapter.
## 1 Differential sheaves and resolutions
1.1. Definition. A "graded sheaf" \( {\mathcal{X}}^{ * } \) is a sequence \( \left\{ {\mathcal{L}}^{p}\right\} \) of sheaves, \( p \) ranging over the integers. A "differential sheaf" is a graded sheaf together with homomorphisms \( d : {\mathcal{L}}^{p} \rightarrow {\mathcal{L}}^{p + 1} \) such that \( {d}^{2} : {\mathcal{L}}^{p} \rightarrow {\mathcal{L}}^{p + 2} \) is zero for all \( p \) . A "resolution" of a sheaf \( \mathcal{A} \) is a differential sheaf \( {\mathcal{L}}^{ * } \) with \( {\mathcal{L}}^{p} = 0 \) for \( p < 0 \) together with an "augmentation" homomorphism \( \varepsilon : \mathcal{A} \rightarrow {\mathcal{L}}^{0} \) such that the sequence \( 0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{L}}^{0}\overset{d}{ \rightarrow }{\mathcal{L}}^{1}\overset{d}{ \rightarrow }{\mathcal{L}}^{2} \rightarrow \cdots \) is exact.
Similarly, one can define graded and differential presheaves.
Since exact sequences and direct limits commute, it follows that the functor Meal, assigning to a presheaf its associated sheaf, is an exact functor. Thus if \( A\overset{f}{ \rightarrow }B\overset{g}{ \rightarrow }C \) is a sequence of presheaves of order two [i.e., \( g\left( U\right) \circ f\left( U\right) = 0 \) for all \( U \) ] and if \( \mathcal{A}\overset{{f}^{\prime }}{ \rightarrow }\mathcal{B}\overset{{g}^{\prime }}{ \rightarrow }\mathcal{C} \) is the induced sequence of generated sheaves, then \( \operatorname{Im}{f}^{\prime } \) and \( \operatorname{Ker}{g}^{\prime } \) are generated respectively by the presheaves \( \operatorname{Im}f \) and \( \operatorname{Ker}g \) . Similarly, the sheaf \( \operatorname{Ker}{g}^{\prime }/\operatorname{Im}{f}^{\prime } \) is (naturally isomorphic to) the sheaf generated by the pre-sheaf \( \operatorname{Ker}g/\operatorname{Im}f : U \mapsto \operatorname{Ker}g\left( U\right) /\operatorname{Im}f\left( U\right) \) .
If \( {\mathcal{L}}^{ * } \) is a differential sheaf then we define its homology sheaf (or "derived sheaf") to be the graded sheaf \( {\mathcal{H}}^{ * }\left( {\mathcal{L}}^{ * }\right) \), where as usual,
\[
{\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) = \operatorname{Ker}\left( {d : {\mathcal{L}}^{p} \rightarrow {\mathcal{L}}^{p + 1}}\right) /\operatorname{Im}\left( {d : {\mathcal{L}}^{p - 1} \rightarrow {\mathcal{L}}^{p}}\right) .
\]
The preceding remarks show that \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) = \mathcal{R} \) eq. \( \left( {U \mapsto {H}^{p}\left( {{\mathcal{L}}^{ * }\left( U\right) }\right) }\right) \) , and in general, if \( {\mathcal{L}}^{ * } \) is generated by the differential presheaf \( {L}^{ * } \), then \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) = \mathcal{P} \) heaf \( \left( {U \mapsto {H}^{p}\left( {{L}^{ * }\left( U\right) }\right) }\right) \) .
Note that in general, \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) \left( U\right) ≉ {H}^{p}\left( {{\mathcal{L}}^{ * }\left( U\right) }\right) \) . [For example, if we let \( {\mathcal{L}}^{0} = {\mathcal{L}}^{1} \) be the "twisted" sheaf with stalks \( \mathbb{Z} \) on \( X = {\mathbb{S}}^{1} \) and let
\( {\mathcal{L}}^{2} = {\mathbb{Z}}_{2} \), the constant sheaf, then \( 0 \rightarrow {\mathcal{L}}^{0}\overset{2}{ \rightarrow }{\mathcal{L}}^{1} \rightarrow {\mathcal{L}}^{2} \rightarrow 0 \) is exact, so that \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) \left( X\right) = 0 \) for all \( p \) . However, \( {\mathcal{L}}^{0}\left( X\right) = 0 = {\mathcal{L}}^{1}\left( X\right) \) and \( \left. {{\mathcal{L}}^{2}\left( X\right) \approx {\mathbb{Z}}_{2}\text{, so that }{H}^{2}\left( {{\mathcal{L}}^{ * }\left( X\right) }\right) \approx {\mathbb{Z}}_{2}\text{. }}\right\rbrack \)
1.2. Example. In singular cohomology let \( G \) be the coefficient group (that is, the constant sheaf with stalk \( G \) ; this is no loss of generality since we are interested here in local matters).
We have the differential presheaf
\[
0 \rightarrow G \rightarrow {S}^{0}\left( {U;G}\right) \rightarrow {S}^{1}\left( {U;G}\right) \rightarrow \cdots ,
\]
(1)
where \( G \rightarrow {S}^{0}\left( {U;G}\right) \) is the usual augmentation. Here we regard \( G \) as the constant presheaf \( U \mapsto G\left( U\right) = G \) . This generates the differential sheaf
\[
0 \rightarrow G \rightarrow {\mathcal{S}}^{0}\left( {X;G}\right) \rightarrow {\mathcal{S}}^{1}\left( {X;G}\right) \rightarrow \cdots
\]
(2)
When is this exact? That is, when is \( {\mathcal{P}}^{ * }\left( {X;G}\right) \) a resolution of \( G \) ? Clearly this sequence is exact at \( G \) since (1) is. The homology |
1004_(GTM170)Sheaf Theory | 17 | {2}\left( {{\mathcal{L}}^{ * }\left( X\right) }\right) \approx {\mathbb{Z}}_{2}\text{. }}\right\rbrack \)
1.2. Example. In singular cohomology let \( G \) be the coefficient group (that is, the constant sheaf with stalk \( G \) ; this is no loss of generality since we are interested here in local matters).
We have the differential presheaf
\[
0 \rightarrow G \rightarrow {S}^{0}\left( {U;G}\right) \rightarrow {S}^{1}\left( {U;G}\right) \rightarrow \cdots ,
\]
(1)
where \( G \rightarrow {S}^{0}\left( {U;G}\right) \) is the usual augmentation. Here we regard \( G \) as the constant presheaf \( U \mapsto G\left( U\right) = G \) . This generates the differential sheaf
\[
0 \rightarrow G \rightarrow {\mathcal{S}}^{0}\left( {X;G}\right) \rightarrow {\mathcal{S}}^{1}\left( {X;G}\right) \rightarrow \cdots
\]
(2)
When is this exact? That is, when is \( {\mathcal{P}}^{ * }\left( {X;G}\right) \) a resolution of \( G \) ? Clearly this sequence is exact at \( G \) since (1) is. The homology of (1) is just the reduced singular cohomology group \( {}_{\Delta }{\widetilde{H}}^{ * }\left( {U;G}\right) \) . Thus (2) is exact \( \Leftrightarrow \) the sheaf \( {}_{\Delta }{\widetilde{\mathcal{H}}}^{ * }\left( {X;G}\right) = \mathcal{R} \) eaf \( \left( {U \mapsto {}_{\Delta }{\widetilde{H}}^{ * }\left( {U;G}\right) }\right) \) is trivial. This is the case \( \Leftrightarrow \)
\[
\mathop{\lim }\limits_{ \rightarrow }{}_{\Delta }{\widetilde{H}}^{ * }\left( {U;G}\right) = 0
\]
(3)
for all \( x \in X \), where \( U \) ranges over the neighborhoods of \( x \) . In the terminology of Spanier,(3) is the condition that the point \( x \) be "taut" with respect to singular cohomology with coefficients in \( G \) . Note that this condition is implied by the condition \( {HL}{C}_{\mathbb{Z}}^{\infty } \) . [A space \( X \) is said to be \( {HL}{C}_{L}^{n} \) (homologically locally connected) if for each \( x \in X \) and neighborhood \( \bar{U} \) of \( x \), there is a neighborhood \( V \subset U \) of \( x \), depending on \( p \), such that the homomorphism \( {}_{\Delta }{\widetilde{H}}_{p}\left( {V;L}\right) \rightarrow {}_{\Delta }{\widetilde{H}}_{p}\left( {U;L}\right) \) is trivial for \( p \leq n \) . Obviously any locally contractible space, and hence any manifold or CW-complex, is HLC.] An example of a space that does not satisfy this condition is the union \( X \) of circles of radius \( 1/n \) all tangent to the \( x \) -axis at the origin. It is clear that this is not \( {HL}{C}^{1} \), and at least in the case of rational coefficients, the sheaf \( {}_{\Delta }{\mathcal{H}}^{1}\left( {X;\mathbb{Q}}\right) \neq 0 \) for this space.
1.3. Example. The Alexander-Spanier presheaf \( {A}^{ * }\left( {\bullet ;G}\right) \) provides a differential presheaf
\[
0 \rightarrow G \rightarrow {A}^{0}\left( {U;G}\right) \rightarrow {A}^{1}\left( {U;G}\right) \rightarrow \cdots
\]
(4)
and hence a differential sheaf. However, in this case (4) is already exact. [For if \( f : {U}^{p + 1} \rightarrow G \) and \( {df} = 0 \), define \( g : {U}^{p} \rightarrow G \) by \( g\left( {{x}_{0},\ldots ,{x}_{p - 1}}\right) = \) \( f\left( {x,{x}_{0},\ldots ,{x}_{p - 1}}\right) \), where \( x \) is an arbitrary element of \( U \) . Then
\[
{dg}\left( {{x}_{0},\ldots ,{x}_{p}}\right) = \sum {\left( -1\right) }^{i}g\left( {{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p}}\right)
\]
\[
= \sum {\left( -1\right) }^{i}f\left( {x,{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p}}\right)
\]
\[
= \overline{f}\left( {{x}_{0},\ldots ,{x}_{p}}\right) - {df}\left( {x,{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p}}\right)
\]
\[
= f\left( {{x}_{0},\ldots ,{x}_{p}}\right)
\]
so that \( f = {dg} \) .]
Thus the Alexander-Spanier sheaf \( {\mathcal{A}}^{ * }\left( {X;G}\right) \) is always a resolution of \( G \) .
1.4. Example. The de Rham sheaf \( {\Omega }^{ * } \) on any differentiable manifold is a differential sheaf and has an augmentation \( \mathbb{R} \rightarrow {\Omega }^{0} \) defined by taking a real number \( r \) into the constant function on \( X \) with value \( r \) . Moreover, \( 0 \rightarrow \mathbb{R} \rightarrow {\Omega }^{0} \rightarrow {\Omega }^{1} \rightarrow \cdots \) is an exact sequence of sheaves, as follows from the Poincaré Lemma, which states that every closed differential form on euclidean space is exact; see [19, V-9.2]. Therefore, \( {\Omega }^{ * } \) is a resolution of \( \mathbb{R} \) .
## 2 The canonical resolution and sheaf cohomology
For any sheaf \( \mathcal{A} \) on \( X \) and open set \( U \subset X \) we let \( {C}^{0}\left( {U;\mathcal{A}}\right) \) be the collection of all functions (not necessarily continuous) \( f : U \rightarrow \mathcal{A} \) such that \( \pi \circ f \) is the identity on \( U,\pi : \mathcal{A} \rightarrow X \) being the canonical projection. Such possibly discontinuous sections are called serrations, a terminology introduced by Bourgin [10]. That is,
\[
{C}^{0}\left( {U;\mathcal{A}}\right) = \mathop{\prod }\limits_{{x \in U}}{\mathcal{A}}_{x}
\]
Under pointwise operations, this is a group, and the functor \( U \mapsto {C}^{0}\left( {U;\mathcal{A}}\right) \) is a conjunctive monopresheaf on \( X \) . Hence this presheaf is a sheaf, which will be denoted by \( {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \) . Note that if \( {X}_{d} \) denotes the point set of \( X \) with the discrete topology and if \( f : {X}_{d} \rightarrow X \) is the canonical map, then \( {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \approx f{f}^{ * }\mathcal{A} \), as already mentioned in I-4.
Inclusion of the collection of sections of \( \mathcal{A} \) in the collection of all ser-rations gives an inclusion \( \mathcal{A}\left( U\right) \hookrightarrow {C}^{0}\left( {U;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \left( U\right) \) and hence provides a natural monomorphism
\[
\varepsilon : \mathcal{A} \rightarrowtail {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) .
\]
(For \( f : {X}_{d} \rightarrow X \) as above, this inclusion coincides with the monomorphism \( \beta : \mathcal{A} \rightarrowtail f{f}^{ * }\mathcal{A} \) of (6) on page 15.)
If \( \Phi \) is a family of supports on \( X \), we put
\[
{C}_{\Phi }^{0}\left( {X;\mathcal{A}}\right) = {\Gamma }_{\Phi }\left( {{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) }\right)
\]
Then for any exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) of sheaves, the corresponding sequence of presheaves
\[
0 \rightarrow {C}^{0}\left( {\bullet ;{\mathcal{A}}^{\prime }}\right) \rightarrow {C}^{0}\left( {\bullet ;\mathcal{A}}\right) \rightarrow {C}^{0}\left( {\bullet ;{\mathcal{A}}^{\prime \prime }}\right) \rightarrow 0
\]
is obviously exact. Moreover, for any family \( \Phi \) of supports, the sequence
\[
0 \rightarrow {C}_{\Phi }^{0}\left( {\bullet ;{\mathcal{A}}^{\prime }}\right) \rightarrow {C}_{\Phi }^{0}\left( {\bullet ;\mathcal{A}}\right) \rightarrow {C}_{\Phi }^{0}\left( {\bullet ;{\mathcal{A}}^{\prime \prime }}\right) \rightarrow 0
\]
is exact. [To see that the last map is onto, we recall that \( f \in {C}_{\Phi }^{0}\left( {X;{\mathcal{A}}^{\prime \prime }}\right) \) can be regarded as a serration \( \widehat{f} : X \rightarrow {\mathcal{A}}^{\prime \prime } \) . Then \( \left| f\right| \) is the closure of \( \{ x \mid \widehat{f}\left( x\right) = 0\} \) . Clearly \( \widehat{f} \) is the image of a serration \( \widehat{g} \) of \( \mathcal{A} \) that vanishes wherever \( \widehat{f} \) vanishes. Thus \( \left. {\left| g\right| = \left| f\right| \in \Phi \text{.}}\right\rbrack \)
Let \( {\mathcal{Z}}^{1}\left( {X;\mathcal{A}}\right) = \operatorname{Coker}\left\{ {\varepsilon : \mathcal{A} \rightarrow {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) }\right\} \), so that the sequence
\[
0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \overset{\partial }{ \rightarrow }{\mathcal{J}}^{1}\left( {X;\mathcal{A}}\right) \rightarrow 0
\]
is exact. We also define, inductively,
\[
{\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;{\mathcal{J}}^{n}\left( {X;\mathcal{A}}\right) }\right)
\]
\[
{\mathcal{Z}}^{n + 1}\left( {X;\mathcal{A}}\right) = {\mathcal{Z}}^{1}\left( {X;{\mathcal{Z}}^{n}\left( {X;\mathcal{A}}\right) }\right)
\]
so that
\[
0 \rightarrow {\mathcal{J}}^{n}\left( {X;\mathcal{A}}\right) \overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) \overset{\partial }{ \rightarrow }{\mathcal{J}}^{n + 1}\left( {X;\mathcal{A}}\right) \rightarrow 0
\]
is exact. Let \( d = \varepsilon \circ \partial \) be the composition
\[
{\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) \overset{\partial }{ \rightarrow }{\mathcal{J}}^{n + 1}\left( {X;\mathcal{A}}\right) \overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{n + 1}\left( {X;\mathcal{A}}\right)
\]
Then the sequence
\[
0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{1}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{2}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }\ldots
\]
is exact. That is, \( {\mathcal{C}}^{ * }\left( {X;\mathcal{A}}\right) \) is a resolution of \( \mathcal{A} \) . It is called the canonical resolution of \( \mathcal{A} \) and is due to Godement [40].
This resolution satisfies the stronger property of being naturally "pointwise homotopically trivial." In fact, for \( x \in U \subset X \) consider the homomorphism \( {C}^{0}\left( {U;\mathcal{A}}\right) \rightarrow {\mathcal{A}}_{x} \) that assigns to a serration \( U \rightarrow \mathcal{A} \) its value at \( x \) . Passing to the limit over neighborhoods of \( x \), this induces a homomorphism \( {\eta }_{x} : {\mathcal{C}}^{0}{\left( X;\mathcal{A}\right) }_{x} \rightarr |
1004_(GTM170)Sheaf Theory | 18 |
\]
Then the sequence
\[
0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{1}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{2}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }\ldots
\]
is exact. That is, \( {\mathcal{C}}^{ * }\left( {X;\mathcal{A}}\right) \) is a resolution of \( \mathcal{A} \) . It is called the canonical resolution of \( \mathcal{A} \) and is due to Godement [40].
This resolution satisfies the stronger property of being naturally "pointwise homotopically trivial." In fact, for \( x \in U \subset X \) consider the homomorphism \( {C}^{0}\left( {U;\mathcal{A}}\right) \rightarrow {\mathcal{A}}_{x} \) that assigns to a serration \( U \rightarrow \mathcal{A} \) its value at \( x \) . Passing to the limit over neighborhoods of \( x \), this induces a homomorphism \( {\eta }_{x} : {\mathcal{C}}^{0}{\left( X;\mathcal{A}\right) }_{x} \rightarrow {\mathcal{A}}_{x} \) . Clearly \( {\eta }_{x} \circ \varepsilon : {\mathcal{A}}_{x} \rightarrow {\mathcal{A}}_{x} \) is the identity. Thus, defining \( {\nu }_{x} : {\mathcal{F}}^{1}{\left( X;\mathcal{A}\right) }_{x} \rightarrow {\mathcal{C}}^{0}{\left( X;\mathcal{A}\right) }_{x} \) by \( {\nu }_{x} \circ \partial = 1 - \varepsilon \circ {\eta }_{x} \) (which is unambiguous), we obtain the pointwise splitting
\[
{\mathcal{A}}_{x}\overset{\varepsilon }{\underset{{\eta }_{x}}{ \rightarrow }}{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \overset{\partial }{\underset{{\nu }_{x}}{ \rightarrow }}{\mathcal{J}}^{1}{\left( X;\mathcal{A}\right) }_{x}.
\]
Replacing \( \mathcal{A} \) by \( {\mathcal{F}}^{n}\left( {X;\mathcal{A}}\right) \) we obtain, generally, the splittings
\[
{\mathcal{F}}^{n}{\left( X;\mathcal{A}\right) }_{x}\overset{\varepsilon }{\underset{{\eta }_{x}}{ \rightarrow }}{\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) \overset{\partial }{\underset{{\nu }_{x}}{ \rightarrow }}{\mathcal{J}}^{n + 1}{\left( X;\mathcal{A}\right) }_{x}.
\]
[Note that as a consequence of these splittings, all these sheaves are torsion free (i.e., have torsion free stalks) when \( \mathcal{A} \) is torsion free.] Let \( {D}_{x} = {\nu }_{x} \circ {\eta }_{x} \) : \( {\mathcal{C}}^{n}{\left( X;\mathcal{A}\right) }_{x} \rightarrow {\mathcal{C}}^{n - 1}{\left( X;\mathcal{A}\right) }_{x} \), for \( n > 0 \) . Then on \( {\mathcal{C}}^{n}{\left( X;\mathcal{A}\right) }_{x} \) for \( n > 0 \) we have \( d{D}_{x} + {D}_{x}d = \varepsilon \partial {\nu }_{x}{\eta }_{x} + {\nu }_{x}{\eta }_{x}\varepsilon \partial = \varepsilon {\eta }_{x} + {\nu }_{x}\partial = 1 \), while on \( {\mathcal{C}}^{0}{\left( X;\mathcal{A}\right) }_{x} \) we have \( {D}_{x}d = {\nu }_{x}{\eta }_{x}\varepsilon \partial = {\nu }_{x}\partial = 1 - \varepsilon {\eta }_{x} \) . These three equations,
\[
\left\{ \begin{array}{ll} d{D}_{x} + {D}_{x}d = 1, & \text{ in positive degrees,} \\ {D}_{x}d = 1 - \varepsilon {\eta }_{x}, & \text{ in degree zero,} \\ {\eta }_{x}\varepsilon = 1, & \text{ on }\mathcal{A}, \end{array}\right.
\]
(5)
show that \( {\mathcal{C}}^{ * }{\left( X;\mathcal{A}\right) }_{x} \) is a homotopically trivial resolution of \( {\mathcal{A}}_{x} \), and this is what we mean when we say that \( {\mathcal{C}}^{ * }\left( {X;\mathcal{A}}\right) \) is pointwise homotopically trivial. Moreover, the \( {D}_{x} \) and \( {\eta }_{x} \) are natural in \( \mathcal{A} \) .
2.1. Lemma. If \( {\mathcal{A}}^{ * } \) is a pointwise homotopically trivial resolution of a sheaf \( \mathcal{A} \) on \( X\left( {\text{e.g.,}{\mathcal{A}}^{ * } = {\mathcal{C}}^{ * }\left( {X;\mathcal{A}}\right) }\right) \), then \( {\mathcal{A}}^{ * } \otimes \mathcal{B} \) is a resolution of \( \mathcal{A} \otimes \mathcal{B} \) for any sheaf \( \mathcal{B} \) on \( X \) .
Proof. The hypothesis means that there are the homomorphisms \( {D}_{x} \) : \( {\mathcal{A}}_{x}^{n} \rightarrow {\mathcal{A}}_{x}^{n - 1} \) for \( n > 0 \) and \( {\eta }_{x} : {\mathcal{A}}_{x}^{0} \rightarrow {\mathcal{A}}_{x} \) satisfying (5). Tensoring with \( \mathcal{B} \) preserves these equations and the result follows immediately.
Now suppose that \( \mathcal{R} \) is a sheaf of rings and that \( \mathcal{A} \) is an \( \mathcal{R} \) -module. Then \( {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \) is a \( {\mathcal{C}}^{0}\left( {X;\mathcal{R}}\right) \) -module and, a fortiori, it is an \( \mathcal{R} \) -module. Also note that \( \varepsilon : \mathcal{A} \rightarrow {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \) is an \( \mathcal{R} \) -module homomorphism. Thus \( {\mathfrak{F}}^{1}\left( {X;\mathcal{A}}\right) \) is an \( \mathcal{R} \) -module. By induction, each \( {\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) \) and \( {\mathfrak{F}}^{n}\left( {X;\mathcal{A}}\right) \) is an \( \mathcal{R} \) -module, and \( d \) is an \( \mathcal{R} \) -module homomorphism.
Since \( {\mathcal{F}}^{n}\left( {X;\mathcal{A}}\right) \) is an \( \mathcal{R} \) -module, when \( \mathcal{A} \) is an \( \mathcal{R} \) -module, it follows that \( {\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;{\mathcal{F}}^{n}\left( {X;\mathcal{A}}\right) }\right) \) is a \( {\mathcal{C}}^{0}\left( {X;\mathcal{R}}\right) \) -module. We remark, however, that \( d \) is not a \( {\mathcal{C}}^{0}\left( {X;\mathcal{R}}\right) \) -module homomorphism (for if it were, then it would turn out that the cohomology theory we are going to develop would all be trivial).
Since \( {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \) is an exact functor of \( \mathcal{A} \), so is \( {\mathcal{Z}}^{1}\left( {X;\mathcal{A}}\right) \) . By induction it follows that \( {\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) \) and \( {\mathcal{J}}^{n}\left( {X;\mathcal{A}}\right) \) are all exact functors of \( \mathcal{A} \) .
For a family \( \Phi \) of supports on \( X \) we put
\[
{C}_{\Phi }^{n}\left( {X;\mathcal{A}}\right) = {\Gamma }_{\Phi }\left( {{\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) }\right) = {C}_{\Phi }^{0}\left( {X;{\mathcal{Z}}^{n}\left( {X;\mathcal{A}}\right) }\right) .
\]
Since \( {C}_{\Phi }^{0}\left( {X; \bullet }\right) \) and \( {\mathcal{J}}^{n}\left( {X; \bullet }\right) \) are exact functors, it follows that
\[
{C}_{\Phi }^{n}\left( {X; \bullet }\right) \text{is an exact functor.}
\]
2.2. Definition. For a family \( \Phi \) of supports on \( X \) and for a sheaf \( \mathcal{A} \) on \( X \) we define
\[
{H}_{\Phi }^{n}\left( {X;\mathcal{A}}\right) = {H}^{n}\left( {{C}_{\Phi }^{ * }\left( {X;\mathcal{A}}\right) }\right) .
\]
Since \( 0 \rightarrow {\Gamma }_{\Phi }\left( \mathcal{A}\right) \rightarrow {\Gamma }_{\Phi }\left( {{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) }\right) \rightarrow {\Gamma }_{\Phi }\left( {{\mathcal{C}}^{1}\left( {X;\mathcal{A}}\right) }\right) \) is exact, we obtain the natural isomorphism
\[
{\Gamma }_{\Phi }\left( \mathcal{A}\right) \overset{ \approx }{ \rightarrow }{H}_{\Phi }^{0}\left( {X;\mathcal{A}}\right)
\]
of functors of \( \mathcal{A} \) .
From a short exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) of sheaves on \( X \) we obtain a short exact sequence
\[
0 \rightarrow {C}_{\Phi }^{ * }\left( {X;{\mathcal{A}}^{\prime }}\right) \rightarrow {C}_{\Phi }^{ * }\left( {X;\mathcal{A}}\right) \rightarrow {C}_{\Phi }^{ * }\left( {X;{\mathcal{A}}^{\prime \prime }}\right) \rightarrow 0
\]
of chain complexes and hence an induced long exact sequence
\[
\cdots \rightarrow {H}_{\Phi }^{p}\left( {X;{\mathcal{A}}^{\prime }}\right) \rightarrow {H}_{\Phi }^{p}\left( {X;\mathcal{A}}\right) \rightarrow {H}_{\Phi }^{p}\left( {X;{\mathcal{A}}^{\prime \prime }}\right) \overset{\delta }{ \rightarrow }{H}_{\Phi }^{p + 1}\left( {X;{\mathcal{A}}^{\prime }}\right) \rightarrow \cdots
\]
2.3. If \( \mathcal{A} \) is a sheaf on \( X \) and \( A \subset X \), and if \( \Psi \) is a family of supports on \( A \), then we shall often use the abbreviation
\[
{H}_{\Psi }^{p}\left( {A;\mathcal{A}}\right) = {H}_{\Psi }^{p}\left( {A;\mathcal{A} \mid A}\right)
\]
2.4. We shall now describe another type of canonical resolution, also due to Godement, which is of value in certain situations. Most of the details will be left to the reader since this resolution will play a minor role in this book. Let \( {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) \) be defined inductively by \( {\mathcal{F}}^{0}\left( {X;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \) and \( {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;{\mathcal{F}}^{p - 1}\left( {X;\mathcal{A}}\right) }\right) \) . Define \( {F}_{\Phi }^{p}\left( {X;\mathcal{A}}\right) = {\Gamma }_{\Phi }\left( {{\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) }\right) \) for any family \( \Phi \) of supports on \( X \) . Then both \( {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) \) and \( {F}^{p}\left( {X;\mathcal{A}}\right) \) are exact functors of \( \mathcal{A} \) . We shall define a differential \( \delta : {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) \rightarrow \) \( {\mathcal{F}}^{p + 1}\left( {X;\mathcal{A}}\right) \) that makes \( {\mathcal{F}}^{ * }\left( {X;\mathcal{A}}\right) \) into a resolution of \( \mathcal{A} \) . To do this we first give a description of \( {F}^{p}\left( {X;\mathcal{A}}\right) = \Gamma \left( {{\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) }\right) \) that is analogous to the definition of Alexander-Spanier cochains.
Denote by \( {M}^{p}\left( {X;\mathcal{A}}\right) \) the set of all functions defined on \( \left( {p + 1}\right) \) -tuples \( \left( {{x}_{0},{x}_{1},\ldots ,{x}_{p}}\right) \) of points in \( X \) such that \( f\left( {{x}_{0},\ldots ,{x}_{p}}\right) \in {\mathcal{A}}_{{x}_{p}} \) . We shall define an epimorphism
\[
{\psi }_{p} : {M}^{p}\left( {X;\mathcal{A}} |
1004_(GTM170)Sheaf Theory | 19 | {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) \) and \( {F}^{p}\left( {X;\mathcal{A}}\right) \) are exact functors of \( \mathcal{A} \) . We shall define a differential \( \delta : {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) \rightarrow \) \( {\mathcal{F}}^{p + 1}\left( {X;\mathcal{A}}\right) \) that makes \( {\mathcal{F}}^{ * }\left( {X;\mathcal{A}}\right) \) into a resolution of \( \mathcal{A} \) . To do this we first give a description of \( {F}^{p}\left( {X;\mathcal{A}}\right) = \Gamma \left( {{\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) }\right) \) that is analogous to the definition of Alexander-Spanier cochains.
Denote by \( {M}^{p}\left( {X;\mathcal{A}}\right) \) the set of all functions defined on \( \left( {p + 1}\right) \) -tuples \( \left( {{x}_{0},{x}_{1},\ldots ,{x}_{p}}\right) \) of points in \( X \) such that \( f\left( {{x}_{0},\ldots ,{x}_{p}}\right) \in {\mathcal{A}}_{{x}_{p}} \) . We shall define an epimorphism
\[
{\psi }_{p} : {M}^{p}\left( {X;\mathcal{A}}\right) \rightarrow {F}^{p}\left( {X;\mathcal{A}}\right)
\]
by induction on \( p \) . Let \( {\psi }_{0} \) be the identity. If \( {\psi }_{p - 1} \) has been defined, let \( f \in {M}^{p}\left( {X;\mathcal{A}}\right) \), and for each \( {x}_{0} \in X \), let \( {f}_{{x}_{0}} \in {M}^{p - 1}\left( {X;\mathcal{A}}\right) \) be defined by
\[
{f}_{{x}_{0}}\left( {{x}_{1},\ldots ,{x}_{p}}\right) = f\left( {{x}_{0},{x}_{1},\ldots ,{x}_{p}}\right) .
\]
The assignment
\[
{x}_{0} \mapsto {\psi }_{p - 1}\left( {f}_{{x}_{0}}\right) \left( {x}_{0}\right) \in {\mathcal{F}}^{p - 1}{\left( X;\mathcal{A}\right) }_{{x}_{0}}
\]
is a serration of \( {\mathcal{F}}^{p - 1}\left( {X;\mathcal{A}}\right) \) and hence defines an element of
\[
{C}^{0}\left( {X;{\mathcal{F}}^{p - 1}\left( {X;\mathcal{A}}\right) }\right) = {F}^{p}\left( {X;\mathcal{A}}\right) .
\]
We let \( {\psi }_{p}\left( f\right) \) be this element. Clearly, \( {\psi }_{p} \) is surjective.
By an induction on \( p \) it is easy to check that \( \operatorname{Ker}{\psi }_{p} \) consists of all elements \( f \in {M}^{p}\left( {X;\mathcal{A}}\right) \) such that for each \( \left( {q + 1}\right) \) -tuple \( \left( {{x}_{0},\ldots ,{x}_{q}}\right) \) there is a neighborhood \( U\left( {{x}_{0},\ldots ,{x}_{q}}\right) \) of \( {x}_{q} \) such that if
\[
{x}_{1} \in U\left( {x}_{0}\right)
\]
\[
{x}_{2} \in U\left( {{x}_{0},{x}_{1}}\right)
\]
\[
{x}_{p} \in U\left( {{x}_{0},\ldots ,{x}_{p - 1}}\right)
\]
then \( f\left( {{x}_{0},\ldots ,{x}_{p}}\right) = 0 \) .
We now define the differential \( \delta \) . If \( t \in {\mathcal{A}}_{x} \), let \( S\left( t\right) \) be any serration of \( \mathcal{A} \) that is continuous in some neighborhood of \( x \) and with \( S\left( t\right) \left( x\right) = t \) . [Note that \( S \) induces the canonical inclusion \( \varepsilon : \mathcal{A} \rightarrow {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) = {\mathcal{F}}^{0}\left( {X;\mathcal{A}}\right) \) . For \( f \in {M}^{p}\left( {X;\mathcal{A}}\right) \) let \( {\delta f} \in {M}^{p + 1}\left( {X;\mathcal{A}}\right) \) be defined by
\[
{\delta f}\left( {{x}_{0},\ldots ,{x}_{p + 1}}\right) = \mathop{\sum }\limits_{{i = 0}}^{p}{\left( -1\right) }^{i}f\left( {{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p + 1}}\right) + {\left( -1\right) }^{p + 1}S\left( {f\left( {{x}_{0},\ldots ,{x}_{p}}\right) }\right) \left( {x}_{p + 1}\right) .
\]
The reader may check that \( \delta \left( {\operatorname{Ker}{\psi }_{p}}\right) \subset \operatorname{Ker}{\psi }_{p + 1} \) and that \( {\delta \delta f} \in \operatorname{Ker}{\psi }_{p + 2} \) for all \( f \in {M}^{p}\left( {X;\mathcal{A}}\right) \) . Thus, we may define a differential \( \delta \) on \( {F}^{p}\left( {X;\mathcal{A}}\right) \) by \( {\psi }_{p + 1}\delta = \delta {\psi }_{p} \) . Note that on \( {F}^{p}\left( {X;\mathcal{A}}\right) ,\delta \) does not depend on the particular choice function \( S \) . On \( {F}^{p}\left( {X;\mathcal{A}}\right) \) we have that \( {\delta \delta } = 0 \) . These definitions are natural with respect to inclusions of open sets \( U \subset X \), and thus we obtain a differential \( \delta \) on \( {\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) = \mathcal{R} \) eaf \( \left( {U \mapsto {F}^{p}\left( {U;\mathcal{A}}\right) }\right) \) .
Consider the homomorphism \( {E}_{x} : {M}^{p}\left( {X;\mathcal{A}}\right) \rightarrow {M}^{p - 1}\left( {X;\mathcal{A}}\right) \), for \( p > \) 0, given by \( {E}_{x}\left( f\right) = {f}_{x} \) . We claim that this induces a homomorphism \( {D}_{x} : {\mathcal{F}}^{p}{\left( X;\mathcal{A}\right) }_{x} \rightarrow {\mathcal{F}}^{p - 1}{\left( X;\mathcal{A}\right) }_{x} \) . To see this, let \( {\theta }_{x}^{p} : {M}^{p}\left( {X;\mathcal{A}}\right) \rightarrow \) \( {\mathcal{F}}^{p}{\left( X;\mathcal{A}\right) }_{x} \) be the composition of \( {\psi }_{p} : {M}^{p}\left( {X;\mathcal{A}}\right) \rightarrow {F}^{p}\left( {X;\mathcal{A}}\right) \) with the restriction \( {F}^{p}\left( {X;\mathcal{A}}\right) = \Gamma \left( {{\mathcal{F}}^{p}\left( {X;\mathcal{A}}\right) }\right) \rightarrow {\mathcal{F}}^{p}{\left( X;\mathcal{A}\right) }_{x} \) . If \( f \in \operatorname{Ker}{\theta }_{x}^{p} \) then there exists a neighborhood \( U \) of \( x \) such that \( f \mid U \in {M}^{p}\left( {U;\mathcal{A}}\right) \) is in \( \operatorname{Ker}{\psi }_{p} \) . Thus there is a sequence of neighborhoods \( U\left( {{x}_{0},\ldots ,{x}_{q}}\right) \) in \( U \) as above such that if each \( {x}_{q} \in U\left( {{x}_{0},\ldots ,{x}_{q - 1}}\right) \), then \( f\left( {{x}_{0},\ldots ,{x}_{p}}\right) = 0 \) . Specializing to \( {x}_{0} = x \) we can cut \( U \) down so that \( U = U\left( x\right) \) . Put \( V\left( {{x}_{1},\ldots ,{x}_{q}}\right) = \) \( U\left( {x,{x}_{1}\ldots ,{x}_{q}}\right) \) . Then if \( {x}_{2} \in V\left( {x}_{1}\right) ,{x}_{3} \in V\left( {{x}_{1},{x}_{2}}\right) \), etc., we have that \( {f}_{x}\left( {{x}_{1},{x}_{2},\ldots ,{x}_{p}}\right) = f\left( {x,{x}_{1},\ldots ,{x}_{p}}\right) = 0 \), whence \( {E}_{x}\left( f\right) = {f}_{x} \in \operatorname{Ker}{\theta }_{x}^{p - 1} \) , as claimed.
Now, it is easy to compute that \( {E}_{x}\delta + \delta {E}_{x} = 1 \), whence \( {D}_{x}\delta + \delta {D}_{x} = 1 \) . We also have \( {\eta }_{x} : {\mathcal{F}}^{0}\left( {X;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \rightarrow \mathcal{A} \), as before, and the reader may check that \( {D}_{x}\delta = 1 - \varepsilon {\eta }_{x} \) . Thus, we have produced a pointwise splitting of \( {\mathcal{F}}^{ * }\left( {X;\mathcal{A}}\right) \) . In particular, this implies that \( {\mathcal{F}}^{ * }\left( {X;\mathcal{A}}\right) \) is a resolution of \( \mathcal{A} \) .
One advantage of this resolution is that it has a semisimplicial structure. The description above in terms of the Alexander-Spanier type of cochains has an analogue for \( {\mathcal{C}}^{ * }\left( {X;\mathcal{A}}\right) \) . For these facts we refer the reader to [40].
## 3 Injective sheaves
Let \( \mathcal{R} \) be a sheaf of rings with unit on \( X \) . All sheaves on \( X \) in this section will be \( \mathcal{R} \) -modules; homomorphisms will be \( \mathcal{R} \) -module homomorphisms and so on. An \( \mathcal{R} \) -module \( \mathcal{I} \) on \( X \) is said to be injective (with respect to \( \mathcal{R} \) ) if, for any subsheaf \( \mathcal{A} \) of a sheaf \( \mathcal{B} \) on \( X \) and for any homomorphism \( h : \mathcal{A} \rightarrow \mathcal{I} \) (of \( \mathcal{R} \) -modules) there exists an extension \( \mathcal{B} \rightarrow \mathcal{I} \) of \( h \) . That is, \( \mathcal{I} \) is injective if the contravariant functor
\[
{\operatorname{Hom}}_{\mathcal{R}}\left( {\bullet ,\mathcal{I}}\right)
\]
is right exact and hence exact (see I-Exercise 6).
In this section we shall show that there are "enough" injective sheaves in the category of \( \mathcal{R} \) -modules. \( {}^{1} \)
First, we need the following preliminary result:
3.1. Theorem. Let \( f : W \rightarrow X \) be a map and let \( \mathcal{L} \) be an \( {f}^{ * }\mathcal{R} \) -injective sheaf on \( W \) . Then \( f\mathcal{L} \) is an \( \mathcal{R} \) -injective sheaf on \( X \) .
Proof. We must show that the functor \( {\operatorname{Hom}}_{\mathcal{R}}\left( {\bullet, f\mathcal{L}}\right) \) is exact. But by I-Exercise 7, it is naturally equivalent to the functor \( {\operatorname{Hom}}_{{f}^{ * }\mathcal{R}}\left( {{f}^{ * }\left( \bullet \right) ,\mathcal{L}}\right) \) , which is exact since \( {f}^{ * } \) is exact and \( \mathcal{L} \) is \( {f}^{ * }\mathcal{R} \) -injective.
We shall now apply this result to the case in which \( W = {X}_{d} \), the discrete space with the same underlying point set as \( X \) . Here \( \mathcal{R} \) and \( {f}^{ * }\mathcal{R} \) have the same stalks, and it is clear that a sheaf \( \mathcal{L} \) on \( {X}_{d} \) is injective \( \Leftrightarrow \) each stalk \( {\mathcal{L}}_{x} \) is an injective \( {\mathcal{R}}_{x} \) -module. Thus 3.1 immediately yields the result that if \( I\left( x\right) \) is an injective \( {\mathcal{R}}_{x} \) -module for each \( x \in X \), then the sheaf \( \mathcal{I} \) on \( X \) defined by
\[
\mathcal{I}\left( U\right) = \mathop{\prod }\limits_{{x \in U}}I\left( x\right)
\]
(6)
with the obvious restriction maps, is an injective sheaf on \( X \), since it is just the direct image of the sheaf on \( {X}_{d} \) whose stalk at \( x \) is \( I\left( x\right) \) .
3.2. Theorem. Any sheaf \( \mathcal{A} \) on \( X \) is a subsheaf of some injective sheaf.
Proof. With the previous notation, let \( I\left( x\right) \) be some injective \( {\mathcal{R}}_{x} \) -module containing \( {\mathcal{A}}_{x} \) as a submodule. Then the inclusion
\[
\mathop{\prod }\limits_{{x \in U}}{\mathcal{A}}_{x} \hookrightarrow \mathop{\prod }\limits_{{x \in U}}I\left( x\right)
\]
provides a monomorphism \( {\mathcal{C}}^{ |
1004_(GTM170)Sheaf Theory | 20 | h stalk \( {\mathcal{L}}_{x} \) is an injective \( {\mathcal{R}}_{x} \) -module. Thus 3.1 immediately yields the result that if \( I\left( x\right) \) is an injective \( {\mathcal{R}}_{x} \) -module for each \( x \in X \), then the sheaf \( \mathcal{I} \) on \( X \) defined by
\[
\mathcal{I}\left( U\right) = \mathop{\prod }\limits_{{x \in U}}I\left( x\right)
\]
(6)
with the obvious restriction maps, is an injective sheaf on \( X \), since it is just the direct image of the sheaf on \( {X}_{d} \) whose stalk at \( x \) is \( I\left( x\right) \) .
3.2. Theorem. Any sheaf \( \mathcal{A} \) on \( X \) is a subsheaf of some injective sheaf.
Proof. With the previous notation, let \( I\left( x\right) \) be some injective \( {\mathcal{R}}_{x} \) -module containing \( {\mathcal{A}}_{x} \) as a submodule. Then the inclusion
\[
\mathop{\prod }\limits_{{x \in U}}{\mathcal{A}}_{x} \hookrightarrow \mathop{\prod }\limits_{{x \in U}}I\left( x\right)
\]
provides a monomorphism \( {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \rightarrowtail \mathcal{I} \) . Composing this with the canonical monomorphism \( \mathcal{A} \mapsto {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \) yields the desired monomorphism \( \mathcal{A} \rightarrowtail \mathcal{I} \) .
---
\( {}^{1} \) We shall make use of this fact only in the case in which \( \mathcal{R} \) is a constant sheaf.
---
|
1005_(GTM171)Riemannian Geometry | 0 |
Graduate Texts in Mathematics GTM
Peter Petersen
# Riemannian Geometry
Third Edition
Springer
## Graduate Texts in Mathematics
## Series Editors:
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
## Advisory Board:
Alejandro Adem, University of British Columbia
David Eisenbud, University of California, Berkeley & MSRI
Irene M. Gamba, The University of Texas at Austin
J.F. Jardine, University of Western Ontario
Jeffrey C. Lagarias, University of Michigan
Ken Ono, Emory University
Jeremy Quastel, University of Toronto
Fadil Santosa, University of Minnesota
Barry Simon, California Institute of Technology
Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study. More information about this series at http://www.springer.com/series/136 Riemannian Geometry
Third Edition
Peter Petersen
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA, USA
ISSN 0072-5285 ISSN 2197-5612 (electronic)
Graduate Texts in Mathematics
ISBN 978-3-319-26652-7 ISBN 978-3-319-26654-1 (eBook)
DOI 10.1007/978-3-319-26654-1
Library of Congress Control Number: 2015960754
Springer Cham Heidelberg New York Dordrecht London
(C) Springer Science+Business Media New York 1998
(C) Springer Science+Business Media, LLC 2006
(C) Springer International Publishing AG 2016
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper
Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
This book is intended as a comprehensive introduction to Riemannian geometry. The reader is assumed to have basic knowledge of standard manifold theory, including the theory of tensors, forms, and Lie groups. At times it is also necessary to have some familiarity with algebraic topology and de Rham cohomology. Specifically, we recommend that the reader be familiar with texts such as [15, 72] or [97, vol. 1]. On my web page, there are links to lecture notes on these topics as well as classical differential geometry (see [90] and [89]). It is also helpful if the reader has a nodding acquaintance with ordinary differential equations. For this, a text such as [74] is more than sufficient. More basic prerequisites are real analysis, linear algebra, and some abstract algebra. Differential geometry is and always has been an "applied discipline" within mathematics that uses many other parts of mathematics for its own purposes.
Most of the material generally taught in basic Riemannian geometry as well as several more advanced topics is presented in this text. The approach we have taken occasionally deviates from the standard path. Alongside the usual variational approach, we have also developed a more function-oriented methodology that likewise uses standard calculus together with techniques from differential equations. Our motivation for this treatment has been that examples become a natural and integral part of the text rather than a separate item that is sometimes minimized. Another desirable by-product has been that one actually gets the feeling that Hessians and Laplacians are intimately related to curvatures.
The book is divided into four parts:
Part I: Tensor geometry, consisting of chapters 1, 2, 3, and 4
Part II: Geodesic and distance geometry, consisting of chapters 5, 6, and 7
Part III: Geometry à la Bochner and Cartan, consisting of chapters 8, 9, and 10
Part IV: Comparison geometry, consisting of chapters 11 and 12
There are significant structural changes and enhancements in the third edition, so chapters no longer correspond to those of the first two editions. We offer a brief outline of each chapter below.
Chapter 1 introduces Riemannian manifolds, isometries, immersions, and submersions. Homogeneous spaces and covering maps are also briefly mentioned.
There is a discussion on various types of warped products. This allows us to give both analytic and geometric definitions of the basic constant curvature geometries. The Hopf fibration as a Riemannian submersion is also discussed in several places. Finally, there is a section on tensor notation.
Chapter 2 discusses both Lie and covariant derivatives and how they can be used to define several basic concepts such as the classical notions of Hessian, Laplacian, and divergence on Riemannian manifolds. Iterated derivatives and abstract derivations are discussed toward the end and used later in the text.
Chapter 3 develops all of the important curvature concepts and discusses a few simple properties. We also develop several important formulas that relate curvature and the underlying metric. These formulas can be used in many places as a replacement for the second variation formula.
Chapter 4 is devoted to calculating curvatures in several concrete situations such as spheres, product spheres, warped products, and doubly warped products. This is used to exhibit several interesting examples. In particular, we explain how the Riemannian analogue of the Schwarzschild metric can be constructed. There is a new section that explains warped products in general and how they are characterized. This is an important section for later developments as it leads to an interesting characterization of both local and global constant curvature geometries from both the warped product and conformal view point. We have a section on Lie groups. Here two important examples of left invariant metrics are discussed as well as the general formulas for the curvatures of biinvariant metrics. It is also explained how submersions can be used to create new examples with special focus on complex projective space. There are also some general comments on how submersions can be constructed using isometric group actions.
Chapter 5 further develops the foundational topics for Riemannian manifolds. These include the first variation formula, geodesics, Riemannian manifolds as metric spaces, exponential maps, geodesic completeness versus metric completeness, and maximal domains on which the exponential map is an embedding. The chapter includes a detailed discussion of the properties of isometries. This naturally leads to the classification of simply connected space forms. At a more basic level, we obtain metric characterizations of Riemannian isometries and submersions. These are used to show that the isometry group is a Lie group and to give a proof of the slice theorem for isometric group actions.
Chapter 6 contains three more foundational topics: parallel translation, Jacobi fields, and the second variation formula. Some of the classical results we prove here are the Hadamard-Cartan theorem, Cartan's center of mass construction in nonpositive curvature and why it shows that the fundamental group of such spaces is torsion-free, Preissman's theorem, Bonnet's diameter estimate, and Synge's lemma. At the end of the chapter, we cover the ingredients needed for the classical quarter pinched sphere theorem including Klingenberg's injectivity radius estimates and Berger's proof of this theorem. Sphere theorems are revisited in chapter 12.
Chapter 7 focuses on manifolds with lower Ricci curvature bounds. We discuss volume comparison and its uses. These include proofs of how Poincaré and Sobolev constants can be bounded and theorems about restrictions on fundamental groups for manifolds with lower Ricci curvature bounds. The strong maximum principle for continuous functions is developed. This result is first used in a warm-up exercise to prove Cheng's maximal diameter theorem. We then proceed to cover the Cheeger-Gromoll splitting theorem and its consequences for manifolds with nonnegative Ricci curvature.
Chapter 8 covers various aspects of symmetries on manifolds with emphasis on Killing fields. Here there is a further discussion on why the isometry group is a Lie group. The Bochner formulas for Killing fields are covered as well as a discussion on how the presence of Killing fields in positive sectional curvature can lead to topological restrictions. The latter is a fairly new area in Riemannian geometry.
Chapter 9 explains both the classical and more recent results that arise from the Bochner technique. We start with harmonic 1-forms as Bochner did and move on to general forms and other tensors such as the curvature tensor. We use an approach that considerably simplifies many of the tensor calculations in this subject (see, e.g., the first and second editions of this book). The idea is to consistently use how derivations act on tensors instead of using Clifford representations |
1005_(GTM171)Riemannian Geometry | 1 | the Cheeger-Gromoll splitting theorem and its consequences for manifolds with nonnegative Ricci curvature.
Chapter 8 covers various aspects of symmetries on manifolds with emphasis on Killing fields. Here there is a further discussion on why the isometry group is a Lie group. The Bochner formulas for Killing fields are covered as well as a discussion on how the presence of Killing fields in positive sectional curvature can lead to topological restrictions. The latter is a fairly new area in Riemannian geometry.
Chapter 9 explains both the classical and more recent results that arise from the Bochner technique. We start with harmonic 1-forms as Bochner did and move on to general forms and other tensors such as the curvature tensor. We use an approach that considerably simplifies many of the tensor calculations in this subject (see, e.g., the first and second editions of this book). The idea is to consistently use how derivations act on tensors instead of using Clifford representations. The Bochner technique gives many optimal bounds on the topology of closed manifolds with nonnegative curvature. In the spirit of comparison geometry, we show how Betti numbers of nonnegatively curved spaces are bounded by the prototypical compact flat manifold: the torus. More generally, we also show how the Bochner technique can be used to control the topology with more general curvature bounds. This requires a little more analysis, but is a fascinating approach that has not been presented in book form yet.
The importance of the Bochner technique in Riemannian geometry cannot be sufficiently emphasized. It seems that time and again, when people least expect it, new important developments come out of this philosophy.
Chapter 10 develops part of the theory of symmetric spaces and holonomy. The standard representations of symmetric spaces as homogeneous spaces or via Lie algebras are explained. There are several concrete calculations both specific and more general examples to get a feel for how curvatures behave. Having done this, we define holonomy for general manifolds and discuss the de Rham decomposition theorem and several corollaries of it. In particular, we show that holonomy irreducible symmetric spaces are Einstein and that their curvatures have the same sign as the Einstein constant. This theorem and the examples are used to indicate how one can classify symmetric spaces. Finally, we present a brief overview of how holonomy and symmetric spaces are related to the classification of holonomy groups. This is used, together with most of what has been learned up to this point, to give the Gallot and Meyer classification of compact manifolds with nonnegative curvature operator.
Chapter 11 focuses on the convergence theory of metric spaces and manifolds. First, we introduce the most general form of convergence: Gromov-Hausdorff convergence. This concept is often useful in many contexts as a way of getting a weak form of convergence. The real object here is to figure out what weak convergence implies in the presence of stronger side conditions. There is a section with a quick overview of Hölder spaces, Schauder's elliptic estimates, and harmonic coordinates. To facilitate the treatment of the stronger convergence ideas, we have introduced a norm concept for Riemannian manifolds. The main focus of the chapter is to prove the Cheeger-Gromov convergence theorem, which is called the Convergence Theorem of Riemannian Geometry, as well as Anderson's generalizations of this theorem to manifolds with bounded Ricci curvature.
Chapter 12 proves some of the more general finiteness theorems that do not fall into the philosophy developed in Chapter 11. To begin, we discuss generalized critical point theory and Toponogov's theorem. These two techniques are used throughout the chapter to establish all of the important theorems. First, we probe the mysteries of sphere theorems. These results, while often unappreciated by a larger audience, have been instrumental in developing most of the new ideas in the subject. Comparison theory, injectivity radius estimates, and Toponogov's theorem were first used in a highly nontrivial way to prove the classical quarter pinched sphere theorem of Rauch, Berger, and Klingenberg. Critical point theory was introduced by Grove and Shiohama to prove the diameter sphere theorem. Following the sphere theorems, we go through some of the major results of comparison geometry: Gromov's Betti number estimate, the Soul theorem of Cheeger and Gromoll, and the Grove-Petersen homotopy finiteness theorem.
At the end of most chapters, there is a short list of books and papers that cover and often expand on the material in the chapter. We have whenever possible attempted to refer just to books and survey articles. The reader is strongly urged to go from those sources back to the original papers as ideas are often lost in the modernization of most subjects. For more recent works, we also give journal references if the corresponding books or surveys do not cover all aspects of the original paper. One particularly exhaustive treatment of Riemannian Geometry for the reader who is interested in learning more is [12]. Other valuable texts that expand or complement much of the material covered here are \( \left\lbrack {{77},{97}}\right\rbrack \) and \( \left\lbrack {99}\right\rbrack \) . There is also a historical survey by Berger (see [11]) that complements this text very well.
Each chapter ends with a collection of exercises that are designed to reinforce the material covered, to establish some simple results that will be needed later, and also to offer alternative proofs of several results. The first six chapters have about 30 exercises each and there are \( {300} + \) in all. The reader should at least read and think about all of the exercises, if not actually solve all of them. There are several exercises that might be considered very challenging. These have been broken up into more reasonable steps and with occasional hints. Some instructors might want to cover some of the exercises in class.
A first course should definitely cover Chapters 3, 5, and 6 together with whatever one feels is necessary from Chapters 1, 2, and 4. I would definitely not recommend teaching every single topic covered in Chapters 1, 2, and 4. A more advanced course could consist of going through Chapter 7 and parts III or IV as defined earlier. These two parts do not depend in a serious way on each other. One can probably not cover the entire book in two semesters, but it should be possible to cover parts I, II, and III or alternatively I, II, and IV depending on one's inclination.
There are many people I would like to thank. First and foremost are those students who suffered through my continuing pedagogical experiments over the last 25 years. While using this text I always try different strategies every time I teach. Special thanks go to Victor Alvarez, Igor Belegradek, Marcel Berger, Timothy Carson, Gil Cavalcanti, Edward Fan, Hao Fang, John Garnett, or Her-shkovits, Ilkka Holopainen, Michael Jablonski, Lee Kennard, Mayer Amitai Landau, Peter Landweber, Pablo Lessa, Ciprian Manolescu, Geoffrey Mess, Jiayin Pan, Priyanka Rajan, Jacob Rooney, Yanir Rubinstein, Semion Shteingold, Jake Solomon, Chad Sprouse, Marc Troyanov, Gerard Walschap, Nik Weaver, Burkhard Wilking, Michael Williams, and Hung-Hsi Wu for their constructive criticism of parts of the book and mentioning various typos and other deficiencies in the first and second editions. I would especially like to thank Joseph Borzellino for his very careful reading of this text. Finally, I would like to thank Robert Greene, Karsten Grove, Gregory Kallo, and Fred Wilhelm for all the discussions on geometry we have had over the years.
## Contents
1 Riemannian Metrics 1
1.1 Riemannian Manifolds and Maps 2
1.2 The Volume Form 7
1.3 Groups and Riemannian Manifolds 8
1.3.1 Isometry Groups 8
1.3.2 Lie Groups 10
1.3.3 Covering Maps. 11
1.4 Local Representations of Metrics. 12
1.4.1 Einstein Summation Convention 12
1.4.2 Coordinate Representations 14
1.4.3 Frame Representations 15
1.4.4 Polar Versus Cartesian Coordinates. 18
1.4.5 Doubly Warped Products. 22
1.4.6 Hopf Fibrations 23
1.5 Some Tensor Concepts 26
1.5.1 Type Change 26
1.5.2 Contractions. 29
1.5.3 Inner Products of Tensors 30
1.5.4 Positional Notation 31
1.6 Exercises. 32
2 Derivatives 41
2.1 Lie Derivatives 42
2.1.1 Directional Derivatives 42
2.1.2 Lie Derivatives 42
2.1.3 Lie Derivatives and the Metric . 48
2.1.4 Lie Groups 50
2.2 Connections 51
2.2.1 Covariant Differentiation 51
2.2.2 Covariant Derivatives of Tensors 57
2.3 Natural Derivations. 62
2.3.1 Endomorphisms as Derivations 62
2.3.2 Derivatives 64
2.4 The Connection in Tensor Notation 65
2.5 Exercises. 70
3 Curvature 77
3.1 Curvature 77
3.1.1 The Curvature Tensor 78
3.1.2 The Curvature Operator 82
3.1.3 Sectional Curvature. 83
3.1.4 Ricci Curvature 85
3.1.5 Scalar Curvature 86
3.1.6 Curvature in Local Coordinates. 89
3.2 The Equations of Riemannian Geometry 90
3.2.1 Curvature Equations 90
3.2.2 Distance Functions 95
3.2.3 The Curvature Equations for Distance Functions 97
3.2.4 Jacobi Fields 98
3.2.5 Parallel Fields 101
3.2.6 Conjugate Points. 101
3.3 Further Study 103
3.4 Exercises. 103
4 Examples 115
4.1 Computational Simplifications 115
4.2 Warped Products 116
4.2.1 Spheres . 117
4.2.2 Product Spheres. 117
4.2.3 Rotationally Symmetric Metrics 118
4.2.4 Doubly Warped Products. 124
4.2.5 The Schwarzschild Metric 125
4.3 Warped Products in General 126
4.3.1 Basic Constructions 127
4.3.2 General Characterization 129
4.3.3 Conformal Representations of Warped Products. 132
4.3.4 Singular Points. 137
4.4 Metrics on Lie Groups 138
4.4.1 Generalities on Left-invariant Metrics. 138
4.4.2 Hyperbolic Space as a Lie Group 141
4.4.3 Berger Spheres. 143
4.5 Riemannian Submersions 144
4.5.1 Riemannian Submersions and Curva |
1005_(GTM171)Riemannian Geometry | 2 | Scalar Curvature 86
3.1.6 Curvature in Local Coordinates. 89
3.2 The Equations of Riemannian Geometry 90
3.2.1 Curvature Equations 90
3.2.2 Distance Functions 95
3.2.3 The Curvature Equations for Distance Functions 97
3.2.4 Jacobi Fields 98
3.2.5 Parallel Fields 101
3.2.6 Conjugate Points. 101
3.3 Further Study 103
3.4 Exercises. 103
4 Examples 115
4.1 Computational Simplifications 115
4.2 Warped Products 116
4.2.1 Spheres . 117
4.2.2 Product Spheres. 117
4.2.3 Rotationally Symmetric Metrics 118
4.2.4 Doubly Warped Products. 124
4.2.5 The Schwarzschild Metric 125
4.3 Warped Products in General 126
4.3.1 Basic Constructions 127
4.3.2 General Characterization 129
4.3.3 Conformal Representations of Warped Products. 132
4.3.4 Singular Points. 137
4.4 Metrics on Lie Groups 138
4.4.1 Generalities on Left-invariant Metrics. 138
4.4.2 Hyperbolic Space as a Lie Group 141
4.4.3 Berger Spheres. 143
4.5 Riemannian Submersions 144
4.5.1 Riemannian Submersions and Curvatures . 144
4.5.2 Riemannian Submersions and Lie Groups 147
4.5.3 Complex Projective Space 148
4.5.4 Berger-Cheeger Perturbations. 151
4.6 Further Study 153
4.7 Exercises. 153
5 Geodesics and Distance 165
5.1 Mixed Partials 166
5.2 Geodesics. 170
5.3 The Metric Structure of a Riemannian Manifold 176
5.4 First Variation of Energy 182
5.5 Riemannian Coordinates 186
5.5.1 The Exponential Map. 187
5.5.2 Short Geodesics Are Segments 190
5.5.3 Properties of Exponential Coordinates 192
5.6 Riemannian Isometries. 196
5.6.1 Local Isometries 196
5.6.2 Constant Curvature Revisited 199
5.6.3 Metric Characterization of Maps 201
5.6.4 The Slice Theorem 204
5.7 Completeness. 210
5.7.1 The Hopf-Rinow Theorem 210
5.7.2 Warped Product Characterization 212
5.7.3 The Segment Domain 215
5.7.4 The Injectivity Radius 219
5.8 Further Study 220
5.9 Exercises. 220
6 Sectional Curvature Comparison I 231
6.1 The Connection Along Curves 231
6.1.1 Vector Fields Along Curves 232
6.1.2 Third Partials. 233
6.1.3 Parallel Transport 236
6.1.4 Jacobi Fields 237
6.1.5 Second Variation of Energy 239
6.2 Nonpositive Sectional Curvature 241
6.2.1 Manifolds Without Conjugate Points 241
6.2.2 The Fundamental Group in Nonpositive Curvature 242
6.3 Positive Curvature. 250
6.3.1 The Diameter Estimate 250
6.3.2 The Fundamental Group in Even Dimensions 252
6.4 Basic Comparison Estimates 254
6.4.1 Riccati Comparison. 254
6.4.2 The Conjugate Radius 257
6.5 More on Positive Curvature. 259
6.5.1 The Injectivity Radius in Even Dimensions 259
6.5.2 Applications of Index Estimation 261
6.6 Further Study 266
6.7 Exercises. 266
7 Ricci Curvature Comparison. 275
7.1 Volume Comparison. 276
7.1.1 The Fundamental Equations 276
7.1.2 Volume Estimation 278
7.1.3 The Maximum Principle 280
7.1.4 Geometric Laplacian Comparison 284
7.1.5 The Segment, Poincaré, and Sobolev Inequalities 285
7.2 Applications of Ricci Curvature Comparison 293
7.2.1 Finiteness of Fundamental Groups 293
7.2.2 Maximal Diameter Rigidity 295
7.3 Manifolds of Nonnegative Ricci Curvature 298
7.3.1 Rays and Lines. 298
7.3.2 Busemann Functions 301
7.3.3 Structure Results in Nonnegative Ricci Curvature 304
7.4 Further Study 307
7.5 Exercises. 307
8 Killing Fields 313
8.1 Killing Fields in General. 313
8.2 Killing Fields in Negative Ricci Curvature 318
8.3 Killing Fields in Positive Curvature 320
8.4 Exercises. 329
9 The Bochner Technique . 333
9.1 Hodge Theory 334
9.2 1-Forms. 336
9.2.1 The Bochner Formula 336
9.2.2 The Vanishing Theorem 337
9.2.3 The Estimation Theorem 338
9.3 Lichnerowicz Laplacians. 342
9.3.1 The Connection Laplacian 343
9.3.2 The Weitzenböck Curvature. 343
9.3.3 Simplification of \( \operatorname{Ric}\left( T\right) \) 345
9.4 The Bochner Technique in General 347
9.4.1 Forms. 347
9.4.2 The Curvature Tensor 348
9.4.3 Symmetric \( \left( {0,2}\right) \) -Tensors 349
9.4.4 Topological and Geometric Consequences 351
9.4.5 Simplification of \( g\left( {\operatorname{Ric}\left( T\right), T}\right) \) 354
9.5 Further Study 358
9.6 Exercises. 359
10 Symmetric Spaces and Holonomy 365
10.1 Symmetric Spaces . 366
10.1.1 The Homogeneous Description 366
10.1.2 Isometries and Parallel Curvature. 368
10.1.3 The Lie Algebra Description 370
10.2 Examples of Symmetric Spaces 376
10.2.1 The Compact Grassmannian 377
10.2.2 The Hyperbolic Grassmannian. 379
10.2.3 Complex Projective Space Revisited 380
10.2.4 \( \mathrm{{SL}}\left( n\right) /\mathrm{{SO}}\left( n\right) \) 382
10.2.5 Lie Groups 383
10.3 Holonomy 383
10.3.1 The Holonomy Group 383
10.3.2 Rough Classification of Symmetric Spaces 386
10.3.3 Curvature and Holonomy 387
10.4 Further Study 392
10.5 Exercises. 392
11 Convergence. 395
11.1 Gromov-Hausdorff Convergence . 396
11.1.1 Hausdorff Versus Gromov Convergence 396
11.1.2 Pointed Convergence 401
11.1.3 Convergence of Maps 401
11.1.4 Compactness of Classes of Metric Spaces 402
11.2 Hölder Spaces and Schauder Estimates 405
11.2.1 Hölder Spaces. 405
11.2.2 Elliptic Estimates 407
11.2.3 Harmonic Coordinates. 409
11.3 Norms and Convergence of Manifolds 413
11.3.1 Norms of Riemannian Manifolds 413
11.3.2 Convergence of Riemannian Manifolds 414
11.3.3 Properties of the Norm 415
11.3.4 The Harmonic Norm 418
11.3.5 Compact Classes of Riemannian Manifolds. 421
11.3.6 Alternative Norms 424
11.4 Geometric Applications. 426
11.4.1 Ricci Curvature 426
11.4.2 Volume Pinching. 430
11.4.3 Sectional Curvature. 432
11.4.4 Lower Curvature Bounds. 434
11.4.5 Curvature Pinching 436
11.5 Further Study 439
11.6 Exercises. 440
12 Sectional Curvature Comparison II 443
12.1 Critical Point Theory 444
12.2 Distance Comparison 449
12.3 Sphere Theorems . 457
12.4 The Soul Theorem. 461
12.5 Finiteness of Betti Numbers 470
12.6 Homotopy Finiteness. 480
12.7 Further Study 488
12.8 Exercises. 488
Bibliography 491
Index . 495
|
1006_(GTM172)Classical Topics in Complex Function Theory | 0 |
Gradua in Math
# Reinho
Class ILON Graduate Texts in Mathematics 172
Editorial Board
S. Axler F.W. Gehring P.R. Halmos
Springer Science+Business Media, LLC
## Graduate Texts in Mathematics
1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 Oxtoby. Measure and Category. 2nd ed.
3 Schaefer. Topological Vector Spaces.
4 Hilton/Stammbach. A Course in Homological Algebra. 2nd ed.
5 MAC LANE. Categories for the Working Mathematician.
6 Hughes/Piper. Projective Planes.
7 Serre. A Course in Arithmetic.
8 TAKEUTI/ZARING. Axiomatic Set Theory.
9 Humphreys. Introduction to Lie Algebras and Representation Theory.
10 Cohen. A Course in Simple Homotopy Theory.
11 Conway. Functions of One Complex Variable I. 2nd ed.
12 BEALS. Advanced Mathematical Analysis.
13 Anderson/Fuller. Rings and Categories of Modules. 2nd ed.
14 Golubitsky/Guillemin. Stable Mappings and Their Singularities.
15 Berberian. Lectures in Functional Analysis and Operator Theory.
16 WINTER. The Structure of Fields.
17 Rosenblatt. Random Processes. 2nd ed.
18 Halmos. Measure Theory.
19 Halmos. A Hilbert Space Problem Book. 2nd ed.
20 Husemoller. Fibre Bundles. 3rd ed.
21 Humphreys. Linear Algebraic Groups.
22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic.
23 Greub. Linear Algebra. 4th ed.
24 Holmes. Geometric Functional Analysis and Its Applications.
25 Hewitt/Stromberg. Real and Abstract Analysis.
26 Manes. Algebraic Theories.
27 Kelley. General Topology.
28 Zariski/Samuel. Commutative Algebra. Vol.I.
29 Zariski/Samuel. Commutative Algebra. Vol.II.
30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts.
31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra.
32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory.
33 Hirsch. Differential Topology.
34 SPITZER. Principles of Random Walk. 2nd ed.
35 Wermer. Banach Algebras and Several Complex Variables. 2nd ed.
36 Kelley/Namioka et al. Linear Topological Spaces.
37 Monk. Mathematical Logic.
38 Grauert/Fritzsche. Several Complex Variables.
39 Arveson. An Invitation to \( {C}^{ * } \) -Algebras.
40 Kemeny/Snell/Knapp. Denumerable Markov Chains. 2nd ed.
41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed.
42 Serre. Linear Representations of Finite Groups.
43 Gillman/Jerison. Rings of Continuous Functions.
44 KENDIG. Elementary Algebraic Geometry.
45 Loève. Probability Theory I. 4th ed.
46 Loève. Probability Theory II. 4th ed.
47 Moise. Geometric Topology in Dimensions 2 and 3.
48 SACHS/WU. General Relativity for Mathematicians.
49 Gruenberg/Weir. Linear Geometry. 2nd ed.
50 Edwards. Fermat's Last Theorem.
51 KLINGENBERG. A Course in Differential Geometry.
52 Hartshorne. Algebraic Geometry.
53 Manin. A Course in Mathematical Logic.
54 Graver/Watkins. Combinatorics with Emphasis on the Theory of Graphs.
55 Brown/Pearcy. Introduction to Operator Theory I: Elements of Functional Analysis.
56 Massey. Algebraic Topology: An Introduction.
57 Crowell/Fox. Introduction to Knot Theory.
58 KOBLITZ. \( p \) -adic Numbers, \( p \) -adic Analysis, and Zeta-Functions. 2nd ed.
59 LANG. Cyclotomic Fields.
60 Arnol.D. Mathematical Methods in Classical Mechanics. 2nd ed.
Reinhold Remmert
# Classical Topics in Complex Function Theory
Translated by Leslie Kay
With 19 Illustrations
Reinhold Remmert Leslie Kay (Translator)
Mathematisches Institut Department of Mathematics
Westfälische Wilhelms-Universität Münster Virginia Polytechnic Institute
Einsteinstrasse 62 and State University
Münster D-48149 Blacksburg, VA 24061-0123
Germany USA
Editorial Board
S. Axler F.W. Gehring P.R. Halmos
Department of Department of Department of Mathematics Mathematics Mathematics
Michigan State University University of Michigan Santa Clara University East Lansing, MI 48824 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
## Mathematics Subject Classification (1991): 30-01, 32-01
Library of Congress Cataloging-in-Publication Data
Remmert, Reinhold.
[Funktionentheorie. 2. English]
Classical topics in complex function theory / Reinhold Remmert : translated by
Leslie Kay.
p. cm. - (Graduate texts in mathematics ; 172)
Translation of: Funktionentheorie II.
Includes bibliographical references and indexes.
ISBN 978-1-4419-3114-6 ISBN 978-1-4757-2956-6 (eBook)
DOI 10.1007/978-1-4757-2956-6
1. Functions of complex variables. I. Title. II. Series.
QA331.7.R4613 1997
\( {515}^{\prime }{.9} - \mathrm{{dc}}{21} \) 97-10091
Printed on acid-free paper.
(C) 1998 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1998
Softcover reprint of the hardcover 1st edition 1998
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, Springer Science+Business Media, LLC,
except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Production managed by Lesley Poliner; manufacturing supervised by Jeffrey Taub.
Photocomposed pages prepared from the author’s TEX files.
Max Koecher
in memory
## Preface
## Preface to the Second German Edition
In addition to the correction of typographical errors, the text has been materially changed in three places. The derivation of Stirling's formula in Chapter 2, \( §4 \), now follows the method of Stieltjes in a more systematic way. The proof of Picard’s little theorem in Chapter 10, \( §2 \), is carried out following an idea of H. König. Finally, in Chapter 11, \( §4 \), an inaccuracy has been corrected in the proof of Szegö's theorem.
Oberwolfach, 3 October 1994 Reinhold Remmert
## Preface to the First German Edition
Wer sich mit einer Wissenschaft bekannt machen will, darf nicht nur nach den reifen Früchten greifen
- er muß sich darum bekümmern, wie und wo sie gewachsen sind. (Whoever wants to get to know a science shouldn't just grab the ripe fruit - he must also pay attention to how and where it grew.)
- J. C. Poggendorf
Presentation of function theory with vigorous connections to historical development and related disciplines: This is also the leitmotif of this second volume. It is intended that the reader experience function theory personally and participate in the work of the creative mathematician. Of course, the scaffolding used to build cathedrals cannot always be erected afterwards; but a textbook need not follow Gauss, who said that once a good building is completed its scaffolding should no longer be seen. \( {}^{1} \) Sometimes even the framework of a smoothly plastered house should be exposed.
The edifice of function theory was built by Abel, Cauchy, Jacobi, Riemann, and Weierstrass. Many others made important and beautiful contributions; not only the work of the kings should be portrayed, but also the life of the nobles and the citizenry in the kingdoms. For this reason, the bibliographies became quite extensive. But this seems a small price to pay. "Man kann der studierenden Jugend keinen größeren Dienst erweisen als wenn man sie zweckmäßig anleitet, sich durch das Studium der Quellen mit den Fortschritten der Wissenschaft bekannt zu machen." (One can render young students no greater service than by suitably directing them to familiarize themselves with the advances of science through study of the sources.) (letter from Weierstrass to Casorati, 21 December 1868)
Unlike the first volume, this one contains numerous glimpses of the function theory of several complex variables. It should be emphasized how independent this discipline has become of the classical function theory from which it sprang.
In citing references, I endeavored - as in the first volume - to give primarily original works. Once again I ask indulgence if this was not always successful. The search for the first appearance of a new idea that quickly becomes mathematical folklore is often difficult. The Xenion is well known:
Allegire der Erste nur falsch, da schreiben ihm zwanzig
Immer den Irrthum nach, ohne den Text zu besehn. \( {}^{2} \)
The selection of material is conservative. The Weierstrass product theorem, Mittag-Leffler's theorem, the Riemann mapping theorem, and Runge's approximation theory are central. In addition to these required topics, the reader will find
- Eisenstein's proof of Euler's product formula for the sine;
- Wielandt's uniqueness theorem for the gamma function;
- an intensive discussion of Stirling's formula;
- Iss'sa's theorem;
\( {}^{1} \) Cf. W. Sartorius von Waltershausen: Gauß zum Gedächtnis, Hirzel, Leipzig 1856; reprinted by Martin Sändig oHG, Wiesbaden 1965, p. 82.
\( {}^{2} \) Just let the first one come up with a wrong reference, twenty others will copy his error without ever consulting the text. [The translator is grateful to Mr. Ingo Seidler for his help in translating this couplet.]
- Besse’s proof that all domains in \( \mathbb{C} \) are domains of holomorphy;
- Wedderburn's lemma and the ideal theory of rings of holomorphic functions;
- Estermann's proofs of the overconvergence theorem and Bloch's theorem;
- a holomorphic imbedding of the unit disc in \( {\mathbb{C}}^{3} \) ;
- Gauss's expert opinion of November 1851 on Riemann's dissertation.
An effort was made to keep the presentation concise. One worries, however:
Weiß uns der Leser auch für unsre Kürze Dank?
Wohl kaum? Denn Kürze ward durch Vielheit leider! lang. \( {}^{3} \)
\( {}^{3} \) Is the reader even grateful for our brev |
1006_(GTM172)Classical Topics in Complex Function Theory | 1 | ;
\( {}^{1} \) Cf. W. Sartorius von Waltershausen: Gauß zum Gedächtnis, Hirzel, Leipzig 1856; reprinted by Martin Sändig oHG, Wiesbaden 1965, p. 82.
\( {}^{2} \) Just let the first one come up with a wrong reference, twenty others will copy his error without ever consulting the text. [The translator is grateful to Mr. Ingo Seidler for his help in translating this couplet.]
- Besse’s proof that all domains in \( \mathbb{C} \) are domains of holomorphy;
- Wedderburn's lemma and the ideal theory of rings of holomorphic functions;
- Estermann's proofs of the overconvergence theorem and Bloch's theorem;
- a holomorphic imbedding of the unit disc in \( {\mathbb{C}}^{3} \) ;
- Gauss's expert opinion of November 1851 on Riemann's dissertation.
An effort was made to keep the presentation concise. One worries, however:
Weiß uns der Leser auch für unsre Kürze Dank?
Wohl kaum? Denn Kürze ward durch Vielheit leider! lang. \( {}^{3} \)
\( {}^{3} \) Is the reader even grateful for our brevity? Hardly? For brevity, through abundance, alas! turned long.
## Gratias ago
It is impossible here to thank by name all those who gave me valuable advice. I would like to mention Messrs. R. B. Burckel, J. Elstrodt, D. Gaier, W. Kaup, M. Koecher, K. Lamotke, K.-J. Ramspott, and P. Ullrich, who gave their critical opinions. I must also mention the Volkswagen Foundation, which supported the first work on this book through an academic stipend in the winter semester 1982-83.
Thanks are also due to Mrs. S. Terveer and Mr. K. Schlöter. They gave valuable help in the preparatory work and eliminated many flaws in the text. They both went through the last version critically and meticulously, proofread it, and compiled the indices.
Advice to the reader. Parts A, B, and C are to a large extent mutually independent. A reference 3.4.2 means Subsection 2 in Section 4 of Chapter 3. The chapter number is omitted within a chapter, and the section number within a section. Cross-references to the volume Funktionentheorie I refer to the third edition 1992; the Roman numeral I begins the reference, e.g. I.3.4.2. \( {}^{4} \) No later use will be made of material in small print; chapters, sections and subsections marked by \( * \) can be skipped on a first reading. Historical comments are usually given after the actual mathematics. Bibliographies are arranged at the end of each chapter (occasionally at the end of each section); page numbers, when given, refer to the editions listed.
Readers in search of the older literature may consult A. Gutzmer's German-language revision of G. Vivanti’s Theorie der eindeutigen Funk-tionen, Teubner 1906, in which 672 titles (through 1904) are collected.
---
\( {}^{4} \) [In this translation, references, still indicated by the Roman numeral I, are to Theory of Complex Functions (Springer, 1991), the English translation by R. B. Burckel of the second German edition of Funktionentheorie I. Trans.]
---
## Contents
Preface to the Second German Edition vii
Preface to the First German Edition vii
Acknowledgments \( \mathrm{X} \)
Advice to the reader \( \mathrm{X} \)
1 Infinite Products of Holomorphic Functions 3
§1. Infinite Products 4
1. Infinite products of numbers 4
2. Infinite products of functions 6
§2. Normal Convergence 7
1. Normal convergence 7
2. Normally convergent products of holomorphic
functions 9
3. Logarithmic differentiation 10
§3. The Sine Product \( \sin {\pi z} = {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {z}^{2}/{\nu }^{2}}\right) \) 12
1. Standard proof 12
2. Characterization of the sine
by the duplication formula 14
3. Proof of Euler's formula using Lemma 2 15
\( {4}^{ * } \) . Proof of the duplication formula for Euler’s product,
following Eisenstein 16
5. On the history of the sine product 17
## A Infinite Products and Partial Fraction Series 1
§4*. Euler Partition Products 18
1. Partitions of natural numbers and Euler products 19
2. Pentagonal number theorem. Recursion formulas for
\( p\left( n\right) \) and \( \sigma \left( n\right) \) 20
3. Series expansion of \( \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 + {q}^{\nu }z}\right) \) in powers of \( z \) . 22
4. On the history of partitions and the pentagonal
number theorem 24
§5*. Jacobi's Product Representation of the Series
\( J\left( {z, q}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\nu = - \infty }}^{\infty }{q}^{\dot{{\nu }^{2}}}{z}^{\nu } \) 25
1. Jacobi's theorem 25
2. Discussion of Jacobi's theorem 26
3. On the history of Jacobi's identity 28
Bibliography 30
2 The Gamma Function 33
§1. The Weierstrass Function \( \Delta \left( z\right) = z{e}^{\gamma z}\mathop{\prod }\limits_{{\nu > 1}}\left( {1 + z/\nu }\right) {e}^{-z/\nu } \) 36
1. The auxiliary function
\( H\left( z\right) \mathrel{\text{:=}} z\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 + z/\nu }\right) {e}^{-z/\nu } \) 36
2. The entire function \( \Delta \left( z\right) \mathrel{\text{:=}} {e}^{\gamma z}H\left( z\right) \) 37
§2. The Gamma Function 39
1. Properties of the \( \Gamma \) -function 39
2. Historical notes 41
3. The logarithmic derivative 42
4. The uniqueness problem 43
5. Multiplication formulas 45
\( {6}^{ * } \) . Hölder’s theorem 46
\( {7}^{ * } \) . The logarithm of the \( \Gamma \) -function 47
§3. Euler’s and Hankel’s Integral Representations of \( \Gamma \left( z\right) \) 49
1. Convergence of Euler's integral 49
2. Euler's theorem 51
\( {3}^{ * } \) . The equation 52
4*. Hankel's loop integral 53
§4. Stirling's Formula and Gudermann's Series 55
1. Stieltjes’s definition of the function \( \mu \left( z\right) \) 56
2. Stirling's formula 58
3. Growth of \( \left| {\Gamma \left( {x + {iy}}\right) }\right| \) for \( \left| y\right| \rightarrow \infty \) 59
\( {4}^{ * } \) . Gudermann’s series. 60
5*. Stirling's series 61
\( {6}^{ * } \) . Delicate estimates for the remainder term 63
\( {7}^{ * } \) . Binet’s integral 64
8*. Lindelöf's estimate 66
§5. The Beta Function 67
1. Proof of Euler's identity 68
2. Classical proofs of Euler's identity 69
Bibliography 70
3 Entire Functions with Prescribed Zeros 73
§1. The Weierstrass Product Theorem for \( \mathbb{C} \) 74
1. Divisors and principal divisors 74
2. Weierstrass products 75
3. Weierstrass factors 76
4. The Weierstrass product theorem 77
5. Consequences 78
6. On the history of the product theorem 79
§2. Discussion of the Product Theorem 80
1. Canonical products . 81
2. Three classical canonical products 82
3. The \( \sigma \) -function 83
4. The \( \wp \) -function 85
\( {5}^{ * } \) . An observation of Hurwitz 85
Bibliography 86
4* Holomorphic Functions with Prescribed Zeros 89
§1. The Product Theorem for Arbitrary Regions 89
1. Convergence lemma 90
2. The product theorem for special divisors 91
3. The general product theorem 92
4. Second proof of the general product theorem 92
5. Consequences 93
§2. Applications and Examples 94 1. Divisibility in the ring \( \mathcal{O}\left( G\right) \) . Greatest common divisors. 94
2. Examples of Weierstrass products 96
3. On the history of the general product theorem 97
4. Glimpses of several variables 97
§3. Bounded Functions on \( \mathbb{E} \) and Their Divisors 99
1. Generalization of Schwarz's lemma 99
2. Necessity of the Blaschke condition 100
3. Blaschke products 100
4. Bounded functions on the right half-plane 102
Appendix to Section 3: Jensen's Formula 102
Bibliography 104
5 Iss'sa's Theorem. Domains of Holomorphy 107
§1. Iss'sa's Theorem 107
1. Bers's theorem 108
2. Iss'sa's theorem 109
3. Proof of the lemma 109
4. Historical remarks on the theorems of
Bers and Iss'sa 110
\( {5}^{ * } \) . Determination of all the valuations on \( \mathcal{M}\left( G\right) \) 111
§2. Domains of Holomorphy 112
1. A construction of Goursat 113
2. Well-distributed boundary sets. First proof of the
existence theorem 115
3. Discussion of the concept of domains of holomorphy 116
4. Peripheral sets. Second proof of the existence
theorem 118
5. On the history of the concept of domains of
holomorphy 119
6. Glimpse of several variables 120
§3. Simple Examples of Domains of Holomorphy 120
1. Examples for \( \mathbb{E} \) 120
2. Lifting theorem 122
3. Cassini regions and domains of holomorphy 122
Bibliography 123
6 Functions with Prescribed Principal Parts 125
§1. Mittag-Leffler’s Theorem for \( \mathbb{C} \) 126
1. Principal part distributions 126
2. Mittag-Leffler series 127
3. Mittag-Leffler's theorem. 128
4. Consequences 128
5. Canonical Mittag-Leffler series. Examples 129
6. On the history of Mittag-Leffler’s theorem for \( \mathbb{C} \) 130
§2. Mittag-Leffler’s Theorem for Arbitrary Regions 131
1. Special principal part distributions 131
2. Mittag-Leffler's general theorem 132
3. Consequences 133
4. On the history of Mittag-Leffler's general theorem 134
5. Glimpses of several variables 135
§3*. Ideal Theory in Rings of Holomorphic Functions 135
1. Ideals in \( \mathcal{O}\left( G\right) \) that are not finitely generated 136
2. Wedderburn's lemma (representation of 1) 136
3. Linear representation of the gcd. Principal ideal
theorem 138
4. Nonvanishing ideals 138
5. Main theorem of the ideal theory of \( \mathcal{O}\left( G\right) \) 139
6. On the history of the ideal theory of holomorphic
functions 140
7. Glimpses of several variables 141
Bibliography 142
145
7 The Theorems of Montel and Vitali 147
§1. Montel's Theorem 148
1. Montel's theorem for sequences 148
2. Proof of Montel's theorem 150
3. Montel's convergence criterion 150
4. Vitali's theorem 150
\( {5}^{ * } \) . Pointwise convergent sequences of holomorphic
functions 151
§2. Normal Families 152
1. Montel's theorem for normal families 152
2. Discussion of Montel's theorem 153
3. On the history of Montel's theorem 154
\( {4}^{ * } \) . Square-integrable functions and normal families 154
§3*. Vitali's Theorem |
1006_(GTM172)Classical Topics in Complex Function Theory | 2 | ings of Holomorphic Functions 135
1. Ideals in \( \mathcal{O}\left( G\right) \) that are not finitely generated 136
2. Wedderburn's lemma (representation of 1) 136
3. Linear representation of the gcd. Principal ideal
theorem 138
4. Nonvanishing ideals 138
5. Main theorem of the ideal theory of \( \mathcal{O}\left( G\right) \) 139
6. On the history of the ideal theory of holomorphic
functions 140
7. Glimpses of several variables 141
Bibliography 142
145
7 The Theorems of Montel and Vitali 147
§1. Montel's Theorem 148
1. Montel's theorem for sequences 148
2. Proof of Montel's theorem 150
3. Montel's convergence criterion 150
4. Vitali's theorem 150
\( {5}^{ * } \) . Pointwise convergent sequences of holomorphic
functions 151
§2. Normal Families 152
1. Montel's theorem for normal families 152
2. Discussion of Montel's theorem 153
3. On the history of Montel's theorem 154
\( {4}^{ * } \) . Square-integrable functions and normal families 154
§3*. Vitali's Theorem 156
1. Convergence lemma 156
2. Vitali's theorem (final version) 157
3. On the history of Vitali's theorem 158
§4*. Applications of Vitali's theorem 159
1. Interchanging integration and differentiation 159
2. Compact convergence of the \( \Gamma \) -integral . 160
3. Müntz's theorem 161
§5. Consequences of a Theorem of Hurwitz . 162
Bibliography 164
8 The Riemann Mapping Theorem 167
§1. Integral Theorems for Homotopic Paths 168
1. Fixed-endpoint homotopic paths 168
2. Freely homotopic closed paths 169
3. Null homotopy and null homology 170
4. Simply connected domains . 171
\( {5}^{ * } \) . Reduction of the integral theorem 1 to a lemma 172
\( {6}^{ * } \) . Proof of Lemma 5* 174
§2. The Riemann Mapping Theorem 175
1. Reduction to \( Q \) -domains 175
2. Existence of holomorphic injections 177
3. Existence of expansions 177
4. Existence proof by means of an extremal principle 178
5. On the uniqueness of the mapping function 179
6. Equivalence theorem 180
§3. On the History of the Riemann Mapping Theorem 181
1. Riemann's dissertation 181
2. Early history. 183
3. From Carathéodory-Koebe to Fejér-Riesz 184
## B Mapping Theory
4. Carathéodory's final proof 184
5. Historical remarks on uniqueness and boundary
behavior. 186
6. Glimpses of several variables 187
§4. Isotropy Groups of Simply Connected Domains 188
1. Examples 188
2. The group \( {\operatorname{Aut}}_{a}G \) for simply connected domains
\( G \neq \mathbb{C} \) 189
3*. Mapping radius. Monotonicity theorem 189
Appendix to Chapter 8: Carathéodory-Koebe Theory 191
§1. Simple Properties of Expansions 191
1. Expansion lemma . 191
2. Admissible expansions. The square root method 192
3*. The crescent expansion 193
§2. The Carathéodory-Koebe Algorithm 194
1. Properties of expansion sequences 194
2. Convergence theorem 195
3. Koebe families and Koebe sequences 196
4. Summary. Quality of convergence 197
5. Historical remarks. The competition between
Carathéodory and Koebe 197
§3. The Koebe Families \( {\mathcal{K}}_{m} \) and \( {\mathcal{K}}_{\infty } \) 198
1. A lemma 198
2. The families \( {\mathcal{K}}_{m} \) and \( {\mathcal{K}}_{\infty } \) 199
Bibliography for Chapter 8 and the Appendix 201
9 Automorphisms and Finite Inner Maps 203
§1. Inner Maps and Automorphisms 203
1. Convergent sequences in \( \operatorname{Hol}G \) and \( \operatorname{Aut}G \) 204
2. Convergence theorem for sequences of
automorphisms 204
3. Bounded homogeneous domains 205
\( {4}^{ * } \) . Inner maps of \( \mathbb{H} \) and homotheties 206
§2. Iteration of Inner Maps 206
1. Elementary properties 207
2. H. Cartan's theorem 207
3. The group \( {\operatorname{Aut}}_{a}G \) for bounded domains 208
4. The closed subgroups of the circle group 209
5*. Automorphisms of domains with holes. Annulus
theorem 210
§3. Finite Holomorphic Maps 211
1. Three general properties 212
2. Finite inner maps of \( \mathbb{E} \) 212
3. Boundary lemma for annuli 213
4. Finite inner maps of annuli 215
5. Determination of all the finite maps
between annuli 216
§4*. Radó's Theorem. Mapping Degree 217
1. Closed maps. Equivalence theorem 217
2. Winding maps 218
3. Radó's theorem 219
4. Mapping degree 220
5. Glimpses 221
Bibliography 221
## C Selecta
223
10 The Theorems of Bloch, Picard, and Schottky 225
§1. Bloch's Theorem 226
1. Preparation for the proof 226
2. Proof of Bloch's theorem 227
\( {3}^{ * } \) . Improvement of the bound by the solution of an extremal problem 228
\( {4}^{ * } \) . Ahlfors’s theorem 230
5*. Landau's universal constant 232
§2. Picard's Little Theorem 233
1. Representation of functions that omit two values 233
2. Proof of Picard's little theorem 234
3. Two amusing applications 235
§3. Schottky's Theorem and Consequences 236
1. Proof of Schottky's theorem 237
2. Landau's sharpened form of
Picard's little theorem 238
3. Sharpened forms of Montel's
and Vitali's theorems 239
§4. Picard's Great Theorem 240
1. Proof of Picard's great theorem 240
2. On the history of the theorems of this chapter 240
Bibliography 241
11 Boundary Behavior of Power Series 243
§1. Convergence on the Boundary 244
1. Theorems of Fatou, M. Riesz, and Ostrowski 244
2. A lemma of M. Riesz 245
3. Proof of the theorems in 1 247
4. A criterion for nonextendibility 248
Bibliography for Section 1 249
§2. Theory of Overconvergence. Gap Theorem 249
1. Overconvergent power series 249
2. Ostrowski's overconvergence theorem 250
3. Hadamard's gap theorem 252
4. Porter's construction of overconvergent series 253
5. On the history of the gap theorem 254
6. On the history of overconvergence 255
7. Glimpses 255
Bibliography for Section 2 256
§3. A Theorem of Fatou-Hurwitz-Pólya 257
1. Hurwitz's proof 258
2. Glimpses 259
Bibliography for Section 3 259
§4. An Extension Theorem of Szegö 260
1. Preliminaries for the proof of (Sz) 260
2. A lemma 262
3. Proof of (Sz) 263
4. An application 263
5. Glimpses 265
Bibliography for Section 4 266
12 Runge Theory for Compact Sets 267
§1. Techniques 268
1. Cauchy integral formula for compact sets 269
2. Approximation by rational functions 271
3. Pole-shifting theorem 272
§2. Runge Theory for Compact Sets 273
1. Runge's approximation theorem 273
2. Consequences of Runge's little theorem 275
3. Main theorem of Runge theory for compact sets 276
§3. Applications of Runge's Little Theorem 278
1. Pointwise convergent sequences of polynomials that
do not converge compactly everywhere 278
2. Holomorphic imbedding of the unit disc in \( {\mathbb{C}}^{3} \) 281
§4. Discussion of the Cauchy Integral Formula for
Compact Sets 283
1. Final form of Theorem 1.1 284
2. Circuit theorem 285
Bibliography 287
13 Runge Theory for Regions 289
§1. Runge's Theorem for Regions 290
1. Filling in compact sets. Runge's proof of
Mittag-Leffler's theorem 291
2. Runge's approximation theorem 292
3. Main theorem of Cauchy function theory 292
4. On the theory of holes 293
5. On the history of Runge theory 294
§2. Runge Pairs 295
1. Topological characterization of Runge pairs 295
2. Runge hulls 296
3. Homological characterization of Runge hulls.
The Behnke-Stein theorem 297
4. Runge regions. 298
5*. Approximation and holomorphic extendibility 299
§3. Holomorphically Convex Hulls and Runge Pairs 300
1. Properties of the hull operator 300
2. Characterization of Runge pairs by means of
holomorphically convex hulls 302
Appendix: On the Components of Locally Compact Spaces. Šura-
Bura's Theorem . 303
1. Components 303
2. Existence of open compact sets 304
3. Filling in holes 305
4. Proof of Sura-Bura's theorem 305
Bibliography 306
14 Invariance of the Number of Holes 309
§1. Homology Theory. Separation Lemma 309
1. Homology groups. The Betti number 310
2. Induced homomorphisms. Natural properties 311
3. Separation of holes by closed paths 312
§2. Invariance of the Number of Holes. Product Theorem
for Units 313
1. On the structure of the homology groups 313
2. The number of holes and the Betti number 314
3. Normal forms of multiply connected
domains (report) 315
4. On the structure of the multiplicative
group \( \mathcal{O}{\left( G\right) }^{ \times } \) 316
5. Glimpses 318
Bibliography 318
Short Biographies 321
Symbol Index 329
Name Index 331
Subject Index 337
Part A
Infinite Products
and
Partial Fraction Series 1
Infinite Products of Holomorphic
Functions
Allgemeine Sätze über die Convergenz der unend-lichen Producte sind zum grossen Theile bekannt. (General theorems on the convergence of infinite products are for the most part well known.)
- Weierstrass, 1854
Infinite products first appeared in 1579 in the work of F. Vieta (Opera, p. 400, Leyden, 1646); he gave the formula
\[
\frac{2}{\pi } = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}}} \cdot \ldots
\]
for \( \pi \) (cf. [Z], p. 104 and p. 118). In 1655 J. Wallis discovered the famous product
\[
\frac{\pi }{2} = \frac{2 \cdot 2}{1 \cdot 3} \cdot \frac{4 \cdot 4}{3 \cdot 5} \cdot \frac{6 \cdot 6}{5 \cdot 7} \cdot \ldots \cdot \frac{{2n} \cdot {2n}}{\left( {{2n} - 1}\right) \cdot \left( {{2n} - 1}\right) } \cdot \ldots ,
\]
which appears in "Arithmetica infinitorum," Opera I, p. 468 (cf. [Z], p. 104 and p. 119). But L. Euler was the first to work systematically with infinite products and to formulate important product expansions; cf. Chapter 9 of his Introductio. The first convergence criterion is due to Cauchy, Cours d'analyse, p. 562 ff. Infinite products had found their permanent place in analysis by 1854 at the latest, through Weierstrass ([Wei], p. 172 ff.). \( {}^{1} \)
\( {}^{1} \) In 1847 Eisenstein, in his long-forgotten work [Ei], had already systematically used infinite products. He also uses conditionally convergent products (and
One goal of this chapter is the der |
1006_(GTM172)Classical Topics in Complex Function Theory | 3 |
\]
for \( \pi \) (cf. [Z], p. 104 and p. 118). In 1655 J. Wallis discovered the famous product
\[
\frac{\pi }{2} = \frac{2 \cdot 2}{1 \cdot 3} \cdot \frac{4 \cdot 4}{3 \cdot 5} \cdot \frac{6 \cdot 6}{5 \cdot 7} \cdot \ldots \cdot \frac{{2n} \cdot {2n}}{\left( {{2n} - 1}\right) \cdot \left( {{2n} - 1}\right) } \cdot \ldots ,
\]
which appears in "Arithmetica infinitorum," Opera I, p. 468 (cf. [Z], p. 104 and p. 119). But L. Euler was the first to work systematically with infinite products and to formulate important product expansions; cf. Chapter 9 of his Introductio. The first convergence criterion is due to Cauchy, Cours d'analyse, p. 562 ff. Infinite products had found their permanent place in analysis by 1854 at the latest, through Weierstrass ([Wei], p. 172 ff.). \( {}^{1} \)
\( {}^{1} \) In 1847 Eisenstein, in his long-forgotten work [Ei], had already systematically used infinite products. He also uses conditionally convergent products (and
One goal of this chapter is the derivation and discussion of Euler's product
\[
\sin {\pi z} = {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{{z}^{2}}{{\nu }^{2}}}\right)
\]
for the sine function; we give two proofs in Section 3.
Since infinite products are only rarely treated in lectures and textbooks on infinitesimal calculus, we begin by collecting, in Section 1, some basic facts about infinite products of numbers and of holomorphic functions. Normally convergent infinite products \( \prod {f}_{\nu } \) of functions are investigated in Section 2; in particular, the important theorem on logarithmic differentiation of products is proved.
## §1. Infinite Products
We first consider infinite products of sequences of complex numbers. In the second section, the essentials of the theory of compactly convergent products of functions are stated. A detailed discussion of infinite products can be found in \( \left\lbrack \mathrm{{Kn}}\right\rbrack \) .
1. Infinite products of numbers. If \( {\left( {a}_{\nu }\right) }_{\nu \geq k} \) is a sequence of complex numbers, the sequence \( {\left( \mathop{\prod }\limits_{{\nu = k}}^{n}{a}_{\nu }\right) }_{n \geq k} \) of partial products is called a(n) (infinite) product with the factors \( {a}_{\nu } \) . We write \( \mathop{\prod }\limits_{{\nu = k}}^{\infty }{a}_{\nu } \) or \( \mathop{\prod }\limits_{{\nu > k}}{a}_{\nu } \) or simply \( \prod {a}_{\nu } \) ; in general, \( k = 0 \) or \( k = 1 \) .
If we now — by analogy with series — were to call a product \( \prod {a}_{\nu } \) convergent whenever the sequence of partial products had a limit \( a \), undesirable pathologies would result: for one thing, a product would be convergent with value 0 if just one factor \( {a}_{\nu } \) were zero; for another, \( \prod {a}_{\nu } \) could be zero even if not a single factor were zero (e.g. if \( \left| {a}_{\nu }\right| \leq q < 1 \) for all \( \nu \) ). We will therefore take precautions against zero factors and convergence to zero. We introduce the partial products
\[
{p}_{m, n} \mathrel{\text{:=}} {a}_{m}{a}_{m + 1}\ldots {a}_{n} = \mathop{\prod }\limits_{{\nu = m}}^{n}{a}_{\nu },\;k \leq m \leq n,
\]
and call the product \( \prod {a}_{\nu } \) convergent if there exists an index \( m \) such that the sequence \( {\left( {p}_{m, n}\right) }_{n \geq m} \) has a limit \( {\widehat{a}}_{m} \neq 0 \) .
---
series) and carefully discusses the problems, then barely recognized, of conditional and absolute convergence; but he does not deal with questions of compact convergence. Thus logarithms of infinite products are taken without hesitation, and infinite series are casually differentiated term by term; this carelessness may perhaps explain why Weierstrass nowhere cites Eisenstein's work.
---
We then call \( a \mathrel{\text{:=}} {a}_{k}{a}_{k + 1}\ldots {a}_{m - 1}{\widehat{a}}_{m} \) the value of the product and introduce the suggestive notation
\[
\prod {a}_{\nu } \mathrel{\text{:=}} {a}_{k}{a}_{k + 1}\ldots {a}_{m - 1}{\widehat{a}}_{m} = a.
\]
The number \( a \) is independent of the index \( m \) : since \( {\widehat{a}}_{m} \neq 0 \), we have \( {a}_{n} \neq 0 \) for all \( n \geq m \) ; hence for each fixed \( l > m \) the sequence \( {\left( {p}_{l, n}\right) }_{n \geq l} \) also has a limit \( {\widehat{a}}_{l} \neq 0 \), and \( a = {a}_{k}{a}_{k + 1}\ldots {a}_{l - 1}{\widehat{a}}_{l} \) . Nonconvergent products are called divergent. The following result is immediate:
A product \( \prod {a}_{\nu } \) is convergent if and only if at most finitely many factors are zero and the sequence of partial products consisting of the nonzero elements has a limit \( \neq 0 \) .
The restrictions we have found take into account as well as possible the special role of zero. Just as for finite products, the following holds (by definition):
A convergent product \( \prod {a}_{\nu } \) is zero if and only if at least one factor is zero.
We note further:
If \( \mathop{\prod }\limits_{{\nu = 0}}^{\infty }{a}_{\nu } \) converges, then \( {\widehat{a}}_{n} \mathrel{\text{:=}} \mathop{\prod }\limits_{{\nu = n}}^{\infty }{a}_{\nu } \) exists for all \( n \in \mathbb{N} \) . Moreover, \( \lim {\widehat{a}}_{n} = 1 \) and \( \lim {a}_{n} = 1 \) .
Proof. We may assume that \( a \mathrel{\text{:=}} \prod {a}_{\nu } \neq 0 \) . Then \( {\widehat{a}}_{n} = a/{p}_{0, n - 1} \) . Since \( \lim {p}_{0, n - 1} = a \), it follows that \( \lim {\widehat{a}}_{n} = 1 \) . The equality \( \lim {a}_{n} = 1 \) holds because, for all \( n,{\widehat{a}}_{n} \neq 0 \) and \( {a}_{n} = {\widehat{a}}_{n}/{\widehat{a}}_{n + 1} \) .
Examples. a) Let \( {a}_{0} \mathrel{\text{:=}} 0,{a}_{\nu } \mathrel{\text{:=}} 1 \) for \( \nu \geq 1 \) . Then \( \prod {a}_{\nu } = 0 \) .
b) Let \( {a}_{\nu } \mathrel{\text{:=}} 1 - \frac{1}{{\nu }^{2}},\nu \geq 2 \) . Then \( {p}_{2, n} = \frac{1}{2}\left( {1 + \frac{1}{n}}\right) \) ; hence \( \mathop{\prod }\limits_{{\nu \geq 2}}{a}_{\nu } = \frac{1}{2} \) .
c) Let \( {a}_{\nu } \mathrel{\text{:=}} 1 - \frac{1}{\nu },\nu \geq 2 \) . Then \( {p}_{2, n} = \frac{1}{n} \) ; hence \( \lim {p}_{2, n} = 0 \) . The product \( \mathop{\prod }\limits_{{\nu \geq 2}}{a}_{\nu } \) is divergent (since no factor vanishes) although \( \lim {a}_{n} = 1 \) .
In 4.3.2 we will need the following generalization of c):
d) Let \( {a}_{0},{a}_{1},{a}_{2},\ldots \) be a sequence of real numbers with \( {a}_{n} \geq 0 \) and \( \sum \left( {1 - {a}_{\nu }}\right) = + \infty \) . Then \( \lim \mathop{\prod }\limits_{{\nu = 0}}^{n}{a}_{\nu } = 0 \) .
Proof. \( 0 \leq {p}_{0, n} = \mathop{\prod }\limits_{0}^{n}{a}_{\nu } \leq \exp \left\lbrack {-\mathop{\sum }\limits_{0}^{n}\left( {1 - {a}_{\nu }}\right) }\right\rbrack, n \in \mathbb{N} \), since \( t \leq {e}^{t - 1} \) for all \( t \in \mathbb{R} \) . Since \( \sum \left( {1 - {a}_{\nu }}\right) = + \infty \), it follows that \( \lim {p}_{0, n} = 0 \) .
It is not appropriate to introduce, by analogy with series, the concept of absolute convergence. If we were to call a product \( \prod {a}_{\nu } \) absolutely convergent whenever \( \prod \left| {a}_{\nu }\right| \) converged, then convergence would always imply absolute convergence - but \( \prod {\left( -1\right) }^{\nu } \) would be absolutely convergent without being convergent! The first comprehensive treatment of the convergence theory of infinite products was given in 1889 by A. Pringsheim [P]. Exercises. Show:
\[
\text{a)}\mathop{\prod }\limits_{{\nu = 2}}^{\infty }\frac{{\nu }^{3} - 1}{{\nu }^{3} + 1} = \frac{2}{3},\;\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\frac{\nu + {\left( -1\right) }^{\nu + 1}}{\nu } = 1\text{,}
\]
\[
\text{b)}\mathop{\prod }\limits_{{\nu = 2}}^{\infty }\cos \frac{\pi }{{2}^{\nu }} = \frac{2}{\pi }\text{(Vieta’s product).}
\]
2. Infinite products of functions. Let \( X \) denote a locally compact metric space. It is well known that the concepts of compact convergence and locally uniform convergence coincide for such spaces; cf. I.3.1.3. For a sequence \( {f}_{\nu } \in \) \( \mathcal{C}\left( X\right) \) of continuous functions on \( X \) with values in \( \mathbb{C} \), the (infinite) product \( \prod {f}_{\nu } \) is called compactly convergent in \( X \) if, for every compact set \( K \) in \( X \) , there is an index \( m = m\left( K\right) \) such that the sequence \( {p}_{m, n} \mathrel{\text{:=}} {f}_{m}{f}_{m + 1}\ldots {f}_{n} \) , \( n \geq m \), converges uniformly on \( K \) to a nonvanishing function \( {\widehat{f}}_{m} \) . Then, for each point \( x \in X \) ,
\[
f\left( x\right) \mathrel{\text{:=}} \prod {f}_{\nu }\left( x\right) \in \mathbb{C}
\]
exists (in the sense of Subsection 1); we call the function \( f : X \rightarrow \mathbb{C} \) the limit of the product and write
\[
f = \prod {f}_{\nu };\;\text{ then, on }K,\;f \mid K = \left( {{f}_{0} \mid K}\right) \cdot \ldots \cdot \left( {{f}_{m - 1} \mid K}\right) \cdot {\widehat{f}}_{m}.
\]
The next two statements follow immediately from the continuity theorem I.3.1.2.
a) If \( \prod {f}_{\nu } \) converges compactly to \( f \) in \( X \), then \( f \) is continuous in \( X \) and the sequence \( {f}_{\nu } \) converges compactly in \( X \) to 1 .
b) If \( \prod {f}_{\nu } \) and \( \prod {g}_{\nu } \) converge compactly in \( X \), then so does \( \prod {f}_{\nu }{g}_{\nu } \) :
\[
\prod {f}_{\nu }{g}_{\nu } = \left( {\prod {f}_{\nu }}\right) \left( {\prod {g}_{\nu }}\right)
\]
We are primarily interested in the case where \( X \) is a domain \( {}^{2} \) in \( \mathbb{C} \) and all the functions \( {f}_{\nu } \) are holomorphic. The following is clear by the Weierstrass convergence theorem (cf. I.8.4.1).
c) Let \( G \) be a domain in \( \mathbb{C} \) . Every product \( \prod {f}_{\nu } \) of functions \( {f}_{\nu } \) holomorphic in \( G \) that converges compactly in \( G \) has a limit \( f \) that is holomorphic in \( G \) .
Examples. a) The functions \( {f}_{\nu } \mathrel{\text{:=}} \left( { |
1006_(GTM172)Classical Topics in Complex Function Theory | 4 | mediately from the continuity theorem I.3.1.2.
a) If \( \prod {f}_{\nu } \) converges compactly to \( f \) in \( X \), then \( f \) is continuous in \( X \) and the sequence \( {f}_{\nu } \) converges compactly in \( X \) to 1 .
b) If \( \prod {f}_{\nu } \) and \( \prod {g}_{\nu } \) converge compactly in \( X \), then so does \( \prod {f}_{\nu }{g}_{\nu } \) :
\[
\prod {f}_{\nu }{g}_{\nu } = \left( {\prod {f}_{\nu }}\right) \left( {\prod {g}_{\nu }}\right)
\]
We are primarily interested in the case where \( X \) is a domain \( {}^{2} \) in \( \mathbb{C} \) and all the functions \( {f}_{\nu } \) are holomorphic. The following is clear by the Weierstrass convergence theorem (cf. I.8.4.1).
c) Let \( G \) be a domain in \( \mathbb{C} \) . Every product \( \prod {f}_{\nu } \) of functions \( {f}_{\nu } \) holomorphic in \( G \) that converges compactly in \( G \) has a limit \( f \) that is holomorphic in \( G \) .
Examples. a) The functions \( {f}_{\nu } \mathrel{\text{:=}} \left( {1 + \frac{2z}{{2\nu } - 1}}\right) {\left( 1 + \frac{2z}{{2\nu } + 1}\right) }^{-1},\nu \geq 1 \), are holomorphic in the unit disc \( \mathbb{E} \) . We have
\[
{p}_{2, n} = \left( {1 + \frac{2}{3}z}\right) {\left( 1 + \frac{2z}{{2n} + 1}\right) }^{-1} \in \mathcal{O}\left( \mathbb{E}\right) ;\;\text{ hence }\;\lim {p}_{2, n} = 1 + \frac{2}{3}z,
\]
\( {}^{2}\lbrack \) As defined in Funktionentheorie I, a region (“Bereich” in German) is a nonempty open subset of \( \mathbb{C} \) ; a domain ("Gebiet" in German) is a connected region. In consulting Theory of Complex Functions, the reader should be aware that there "Bereich" was translated as "domain" and "Gebiet" as "region." Trans.] and the product \( \mathop{\prod }\limits_{{\nu = 1}}^{\infty }{f}_{\nu } \) therefore converges compactly in \( \mathbb{E} \) to \( 1 + {2z} \) .
b) Let \( {f}_{\nu }\left( z\right) \equiv z \) for all \( \nu \geq 0 \) . The product \( \mathop{\prod }\limits_{{\nu = 0}}^{\infty }{f}_{\nu } \) does not converge (even pointwise) in the unit disc \( \mathbb{E} \), since the sequence \( {p}_{m, n} = {z}^{n - m + 1} \) converges to zero for every \( m \) .
We note an important sufficient
Convergence criterion. Let \( {f}_{\nu } \in \mathcal{C}\left( X\right) ,\nu \geq 0 \) . Suppose there exists an \( m \in \mathbb{N} \) such that every function \( {f}_{\nu },\nu \geq m \), has a logarithm \( \log {f}_{\nu } \in \mathcal{C}\left( X\right) \) . If \( \mathop{\sum }\limits_{{\nu > m}}\log {f}_{\nu } \) converges compactly in \( X \) to \( s \in \mathcal{C}\left( X\right) \), then \( \prod {f}_{\nu } \) converges compactly in \( X \) to \( {f}_{0}{f}_{1}\ldots {f}_{m - 1}\exp s \) .
Proof. Since the sequence \( {s}_{n} \mathrel{\text{:=}} \mathop{\sum }\limits_{{\nu = m}}^{n}\log {f}_{\nu } \) converges compactly to \( s \), the sequence \( {p}_{m, n} = \mathop{\prod }\limits_{{\nu = m}}^{n}{f}_{\nu } = \exp {s}_{n} \) converges compactly in \( X \) to \( \exp s \) . As \( \exp s \) does not vanish, the assertion follows. \( {}^{3} \)
## §2. Normal Convergence
The convergence criterion 1.2 is hardly suitable for applications, since series consisting of logarithms are generally hard to handle. Moreover, we need a criterion - by analogy with infinite series - that ensures the compact convergence of all partial products and all rearrangements. Here again, as for series, "normal convergence" proves superior to "compact convergence." We recall this concept of convergence for series, again assuming the space \( X \) to be locally compact: then \( \sum {f}_{\nu },{f}_{\nu } \in \mathcal{C}\left( X\right) \), is normally convergent in \( X \) if and only if \( \sum {\left| {f}_{\nu }\right| }_{K} < \infty \) for every compact set \( K \subset X \) (cf. I.3.3.2). Normally convergent series are compactly convergent; normal convergence is preserved under passage to partial sums and arbitrary rearrangements (cf. I.3.3.1).
The factors of a product \( \prod {f}_{\nu } \) are often written in the form \( {f}_{\nu } = 1 + {g}_{\nu } \) ; by \( {1.2}\mathrm{a} \) ), the sequence \( {g}_{\nu } \) converges compactly to zero if \( \prod {f}_{\nu } \) converges compactly.
1. Normal convergence. A product \( \prod {f}_{\nu } \) with \( {f}_{\nu } = 1 + {g}_{\nu } \in \mathcal{C}\left( X\right) \) is called normally convergent in \( X \) if the series \( \sum {g}_{\nu } \) converges normally in \( X \) . It is easy to see that
if \( \mathop{\prod }\limits_{{\nu \geq 0}}{f}_{\nu } \) converges normally in \( X \), then
- for every bijection \( \tau : \mathbb{N} \rightarrow \mathbb{N} \), the product \( \mathop{\prod }\limits_{{\nu \geq 0}}{f}_{\tau \left( \nu \right) } \) converges normally in \( X \) ;
---
\( {}^{3} \) The simple proof that the compact convergence of \( {s}_{n} \) to \( s \) implies the compact convergence of \( \exp {s}_{n} \) to \( \exp s \) can be found in I.5.4.3 (composition lemma).
---
- every subproduct \( \mathop{\prod }\limits_{{j \geq 0}}{f}_{{\nu }_{j}} \) converges normally in \( X \) ;
- the product converges compactly in \( X \) .
We will see that the concept of normal convergence is a good one. At the moment, however, it is not clear that a normally convergent product even has a limit. We immediately prove this and more:
Rearrangement theorem. Let \( \mathop{\prod }\limits_{{\nu > 0}}{f}_{\nu } \) be normally convergent in \( X \) . Then there is a function \( f : X \rightarrow \mathbb{C} \) such that for every bijection \( \tau : \mathbb{N} \rightarrow \mathbb{N} \) the rearrangement \( \mathop{\prod }\limits_{{\nu \geq 0}}{f}_{\tau \left( \nu \right) } \) of the product converges compactly to \( f \) in \( X \) .
Proof. For \( w \in \mathbb{E} \) we have \( \log \left( {1 + w}\right) = \mathop{\sum }\limits_{{\nu \geq 1}}\frac{{\left( -1\right) }^{\nu - 1}}{\nu }{w}^{\nu } \) . It follows that
\( \left| {\log \left( {1 + w}\right) }\right| \leq \left| w\right| \left( {1 + \left| w\right| + {\left| w\right| }^{2} + \cdots }\right) \) ; hence \( \left| {\log \left( {1 + w}\right) }\right| \leq 2\left| w\right| \) if \( \left| w\right| \leq 1/2 \) .
Now let \( K \subset X \) be an arbitrary compact set and let \( {g}_{n} = {f}_{n} - 1 \) . There is an \( m \in \mathbb{N} \) such that \( {\left| {g}_{n}\right| }_{K} \leq \frac{1}{2} \) for \( n \geq m \) . For all such \( n \) ,
\[
\log {f}_{n} = \sum \frac{{\left( -1\right) }^{\nu - 1}}{\nu }{g}_{n}^{\nu } \in \mathcal{C}\left( K\right) ,\;\text{ where }{\left| \log {f}_{n}\right| }_{K} \leq 2{\left| {g}_{n}\right| }_{K}
\]
We see that \( \mathop{\sum }\limits_{{\nu > m}}{\left| \log {f}_{\nu }\right| }_{K} \leq \mathop{\sum }\limits_{{\nu > m}}{\left| {g}_{\nu }\right| }_{K} < \infty \) . Hence, by the rearrangement theorem for series (cf. I.0.4.3), for every bijection \( \sigma \) of \( {\mathbb{N}}_{m} \mathrel{\text{:=}} \) \( \{ n \in \mathbb{N} : n \geq m\} \) the series \( \mathop{\sum }\limits_{{\nu > m}}\log {f}_{\sigma \left( \nu \right) } \) converges uniformly in \( K \) to \( \mathop{\sum }\limits_{{\nu > m}}\log {f}_{\nu } \) . By 1.2, it follows that for such \( \sigma \) the products \( \mathop{\prod }\limits_{{\nu \geq m}}{f}_{\sigma \left( \nu \right) } \) and \( \mathop{\prod }\limits_{{\nu > m}}{f}_{\nu } \) converge uniformly in \( K \) to the same limit function. But an arbitrary bijection \( \tau \) of \( \mathbb{N} \) (= permutation of \( \mathbb{N} \) ) differs only by finitely many transpositions (which have no effect on convergence) from a permutation \( {\sigma }^{\prime } : \mathbb{N} \rightarrow \mathbb{N} \) with \( {\sigma }^{\prime }\left( {\mathbb{N}}_{m}\right) = {\mathbb{N}}_{m} \) . Hence there exists a function \( f : X \rightarrow \mathbb{C} \) such that every product \( \mathop{\prod }\limits_{{\nu \geq 0}}{f}_{\tau \left( \nu \right) } \) converges compactly in \( X \) to \( f \) .
Corollary. Let \( f = \mathop{\prod }\limits_{{\nu \geq 0}}{f}_{\nu } \) converge normally in \( X \) . Then the following statements hold.
1) Every product \( {\widehat{f}}_{n} \mathrel{\text{:=}} \mathop{\prod }\limits_{{\nu \geq n}}{f}_{\nu } \) converges normally in \( X \), and
\[
f = {f}_{0}{f}_{1}\ldots {f}_{n - 1}{\widehat{f}}_{n}
\]
2) If \( \mathbb{N} = \mathop{\bigcup }\limits_{1}^{\infty }{N}_{\kappa } \) is a (finite or infinite) partition of \( \mathbb{N} \) into pairwise disjoint subsets \( {N}_{1},\ldots ,{N}_{\kappa },\ldots \), then every product \( \mathop{\prod }\limits_{{\nu \in {N}_{\kappa }}}{f}_{\nu } \) converges normally in \( X \) and
\[
f = \mathop{\prod }\limits_{{\kappa = 1}}^{\infty }\left( {\mathop{\prod }\limits_{{\nu \in {N}_{\kappa }}}{f}_{\nu }}\right)
\]
Products can converge compactly without being normally convergent, as is shown, for example, by \( \mathop{\prod }\limits_{{\nu > 1}}\left( {1 + {g}_{\nu }}\right) ,{g}_{\nu } \mathrel{\text{:=}} {\left( -1\right) }^{\nu - 1}/\nu \) . It is always true that \( \left( {1 + {g}_{{2\nu } - 1}}\right) \left( {1 + {g}_{2\nu }}\right) = 1 \) ; hence \( {p}_{1, n} = 1 \) for even \( n \) and \( {p}_{1, n} = 1 + \frac{1}{n} \) for odd \( n \) . The product \( \mathop{\prod }\limits_{{\nu > 1}}\left( {1 + {g}_{\nu }}\right) \) thus converges compactly in \( \mathbb{C} \) to 1 . In this example the subproduct \( {\bar{\Pi }}_{\nu \geq 1}\left( {1 + {g}_{{2\nu } - 1}}\right) \) is not convergent!
All later applications (sine product, Jacobi’s triple product, Weierstrass’s factorial, general Weierstrass products) will involve normally convergent products.
Exercises. 1) Prove that if the products \( \prod {f}_{\nu } \) and \( \prod {\widetilde{f}}_{\nu } \) converge normally in \( X \) , then the product \( \prod \left( {{f}_{\nu }{\widetilde{f}}_{\nu }}\right) \) also converges normally in \( X \) .
2) Show that the following products converge normally in the unit disc \( \mathbb{E} \), and prove the identities
\[
\mathop{\prod }\limits_{{\nu \geq 0}}\left( {1 + {z}^{{2} |
1006_(GTM172)Classical Topics in Complex Function Theory | 5 | } - 1}}\right) \left( {1 + {g}_{2\nu }}\right) = 1 \) ; hence \( {p}_{1, n} = 1 \) for even \( n \) and \( {p}_{1, n} = 1 + \frac{1}{n} \) for odd \( n \) . The product \( \mathop{\prod }\limits_{{\nu > 1}}\left( {1 + {g}_{\nu }}\right) \) thus converges compactly in \( \mathbb{C} \) to 1 . In this example the subproduct \( {\bar{\Pi }}_{\nu \geq 1}\left( {1 + {g}_{{2\nu } - 1}}\right) \) is not convergent!
All later applications (sine product, Jacobi’s triple product, Weierstrass’s factorial, general Weierstrass products) will involve normally convergent products.
Exercises. 1) Prove that if the products \( \prod {f}_{\nu } \) and \( \prod {\widetilde{f}}_{\nu } \) converge normally in \( X \) , then the product \( \prod \left( {{f}_{\nu }{\widetilde{f}}_{\nu }}\right) \) also converges normally in \( X \) .
2) Show that the following products converge normally in the unit disc \( \mathbb{E} \), and prove the identities
\[
\mathop{\prod }\limits_{{\nu \geq 0}}\left( {1 + {z}^{{2}^{\nu }}}\right) = \frac{1}{1 - z},\;\mathop{\prod }\limits_{{\nu \geq 1}}\left\lbrack {\left( {1 + {z}^{\nu }}\right) \left( {1 - {z}^{{2\nu } - 1}}\right) }\right\rbrack = 1.
\]
2. Normally convergent products of holomorphic functions. The zero set \( Z\left( f\right) \) of any function \( f \neq 0 \) holomorphic in \( G \) is locally finite in \( G;{}^{4} \) hence \( Z\left( f\right) \) is at most countably infinite (see I.8.1.3).
For finitely many functions \( {f}_{0},{f}_{1},\ldots ,{f}_{n} \in \mathcal{O}\left( G\right) ,{f}_{\nu } \neq 0 \) ,
\[
Z\left( {{f}_{0}{f}_{1}\ldots {f}_{n}}\right) = \mathop{\bigcup }\limits_{0}^{n}Z\left( {f}_{\nu }\right) \;\text{ and }\;{o}_{c}\left( {{f}_{0}{f}_{1}\ldots {f}_{n}}\right) = \mathop{\sum }\limits_{0}^{n}{o}_{c}\left( {f}_{\nu }\right) ,\;c \in G,
\]
where \( {o}_{c}\left( f\right) \) denotes the order of the zero of \( f \) at \( c \) (I.8.1.4). For infinite products, we have the following result.
Proposition. Let \( f = \prod {f}_{\nu },{f}_{\nu } \neq 0 \), be a normally convergent product in \( G \) of functions holomorphic in \( G \) . Then
\[
f \neq 0,\;Z\left( f\right) = \bigcup Z\left( {f}_{\nu }\right) ,\;{o}_{c}\left( f\right) = \sum {o}_{c}\left( {f}_{\nu }\right) \;\text{ for all }c \in G.
\]
Proof. Let \( c \in G \) be fixed. Since \( f\left( c\right) = \prod {f}_{\nu }\left( c\right) \) converges, there exists an index \( n \) such that \( {f}_{\nu }\left( c\right) \neq 0 \) for all \( \nu \geq n \) . By Corollary 1,1), \( f = {f}_{0}{f}_{1}\ldots {f}_{n - 1}{\widehat{f}}_{n} \), where \( {\widehat{f}}_{n} \mathrel{\text{:=}} \mathop{\prod }\limits_{{\nu \geq n}}{f}_{\nu } \in \mathcal{O}\left( G\right) \) by the Weierstrass convergence theorem. It follows that
\[
{o}_{c}\left( f\right) = \mathop{\sum }\limits_{0}^{{n - 1}}{o}_{c}\left( {f}_{\nu }\right) + {o}_{c}\left( {\widehat{f}}_{n}\right) ,\;\text{ with }\;{o}_{c}\left( {\widehat{f}}_{n}\right) = 0\;\left( {\text{ since }{\widehat{f}}_{n}\left( c\right) \neq 0}\right) .
\]
\( {}^{4} \) Let \( G \) be an open subset of \( \mathbb{C} \) . A subset of \( G \) is locally finite in \( G \) if it intersects every compact set in \( G \) in only a finite number of points. Equivalently, a subset of \( G \) is locally finite in \( G \) if it is discrete and closed in \( G \) .
This proves the addition rule for infinite products. In particular, \( Z\left( f\right) = \) \( \cup Z\left( {f}_{\nu }\right) \) . Since each \( {f}_{\nu } \neq 0 \), all the sets \( Z\left( {f}_{\nu }\right) \) and hence also their countable union are countable; it follows that \( f \neq 0 \) .
Remark. The proposition is true even if the convergence of the product in \( G \) is only compact. The proof remains valid word for word, since it is easy to see that for every \( n \) the tail end \( {\widehat{f}}_{n} = \mathop{\prod }\limits_{{\nu \geq n}}{f}_{\nu } \) converges compactly in \( G \) .
We will need the following result in the next section.
If \( f = \prod {f}_{\nu },{f}_{\nu } \in \mathcal{O}\left( G\right) \), is normally convergent in \( G \), then the sequence \( {\widehat{f}}_{n} = \mathop{\prod }\limits_{{\nu \geq n}}{f}_{\nu } \in \mathcal{O}\left( G\right) \) converges compactly in \( G \) to 1 .
Proof. Let \( {\widehat{f}}_{m} \neq 0 \) . Then \( A \mathrel{\text{:=}} Z\left( {\widehat{f}}_{m}\right) \) is locally finite in \( G \) . All the partial products \( {p}_{m, n - 1} \in \mathcal{O}\left( G\right), n > m \), are nonvanishing in \( G \smallsetminus A \) and
\[
{\widehat{f}}_{n}\left( z\right) = {\widehat{f}}_{m}\left( z\right) \cdot \left( \frac{1}{{p}_{m, n - 1}\left( z\right) }\right) \;\text{ for all }\;z \in G \smallsetminus A.
\]
Now the sequence \( 1/{p}_{m, n - 1} \) converges compactly in \( G \smallsetminus A \) to \( 1/{\widehat{f}}_{m} \) . Hence, by the sharpened version of the Weierstrass convergence theorem (see I.8.5.4), this sequence also converges compactly in \( G \) to 1 .
Exercise. Show that \( f = \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\cos \left( {z/{2\nu }}\right) \) converges normally in \( \mathbb{C} \) . Determine \( Z\left( f\right) \) . Show that for each \( k \in \mathbb{N} \smallsetminus \{ 0\} \) there exists a zero of order \( k \) of \( f \) and that
\[
\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\cos \frac{z}{2\nu } = \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {\frac{{2\nu } - 1}{z}\sin \frac{z}{{2\nu } - 1}}\right) .
\]
3. Logarithmic differentiation. The logarithmic derivative of a meromorphic function \( h \in \mathcal{M}\left( G\right), h \neq 0 \), is by definition the function \( {h}^{\prime }/h \in \) \( \mathcal{M}\left( G\right) \) (see also I.9.3.1, where the case of nonvanishing holomorphic functions is discussed). For finite products \( h = {h}_{1}{h}_{2}\ldots {h}_{m},{h}_{\mu } \in \mathcal{M}\left( G\right) \), we have the
Addition formula: \( \frac{{h}^{\prime }}{h} = \frac{{h}_{1}^{\prime }}{{h}_{1}} + \frac{{h}_{2}^{\prime }}{{h}_{2}} + \cdots + \frac{{h}_{m}^{\prime }}{{h}_{m}} \) .
This formula carries over to infinite products of holomorphic functions.
Differentiation theorem. Let \( f = \prod {f}_{\nu } \) be a product of holomorphic functions that converges normally in \( G \) . Then \( \sum {f}_{\nu }^{\prime }/{f}_{\nu } \) is a series of meromorphic functions that converges normally in \( \bar{G} \), and
\[
\frac{{f}^{\prime }}{f} = \sum \frac{{f}_{\nu }^{\prime }}{{f}_{\nu }} \in \mathcal{M}\left( G\right)
\]
Proof. 1) For all \( n \in \mathbb{N} \) (by Corollary 1,1)),
\[
f = {f}_{0}{f}_{1}\ldots {f}_{n - 1}{\widehat{f}}_{n},\text{ with }{\widehat{f}}_{n} \mathrel{\text{:=}} \mathop{\prod }\limits_{{\nu \geq n}}{f}_{\nu };\text{ hence }\frac{{f}^{\prime }}{f} = \mathop{\sum }\limits_{{\nu = 1}}^{{n - 1}}\frac{{f}_{\nu }^{\prime }}{{f}_{\nu }} + \frac{{\widehat{f}}_{n}^{\prime }}{{\widehat{f}}_{n}}.
\]
Since the sequence \( {\widehat{f}}_{n} \) converges compactly in \( G \) to 1 (cf. 2), the derivatives \( {\widehat{f}}_{n}^{\prime } \) converge compactly in \( G \) to 0 by Weierstrass. For every disc \( B \) with \( \bar{B} \subset G \) there is thus an \( m \in \mathbb{N} \) such that all \( {f}_{n}, n \geq m \), are nonvanishing in \( B \) and the sequence \( {\widehat{f}}_{n}^{\prime }/{\widehat{f}}_{n} \in \mathcal{O}\left( B\right), n \geq m \), converges compactly in \( B \) to zero. This shows that \( \sum {f}_{\nu }^{\prime }/{f}_{\nu } \) converges compactly in \( G \) to \( {f}^{\prime }/f \) .
2) We now show that \( \sum {f}_{\nu }^{\prime }/{f}_{\nu } \) converges normally in \( G \) . Let \( {g}_{\nu } \mathrel{\text{:=}} {f}_{\nu } - 1 \) . We must assign an index \( m \) to every compact set \( K \) in \( G \) so that every pole set \( P\left( {{f}_{\nu }^{\prime }/{f}_{\nu }}\right) ,\nu \geq m \), is disjoint from \( K \) and
\( \left( *\right) \)
\[
\mathop{\sum }\limits_{{\nu \geq m}}{\left| \frac{{f}_{\nu }^{\prime }}{{f}_{\nu }}\right| }_{K} = \mathop{\sum }\limits_{{\nu \geq m}}{\left| \frac{{g}_{\nu }^{\prime }}{{f}_{\nu }}\right| }_{K} < \infty \;\text{ (cf. I. 11.1.1). }
\]
We choose \( m \) so large that all the sets \( Z\left( {f}_{\nu }\right) \cap K,\nu \geq m \), are empty and \( \mathop{\min }\limits_{{z \in K}}\left| {{f}_{\nu }\left( z\right) }\right| \geq \frac{1}{2} \) for all \( \nu \geq m \) (this is possible, since the sequence \( {f}_{\nu } \) converges compactly to 1). Now, by the Cauchy estimates for derivatives, there exist a compact set \( L \supset K \) in \( G \) and a constant \( M > 0 \) such that \( {\left| {g}_{\nu }^{\prime }\right| }_{K} \leq \) \( {\left. M{\left| {g}_{\nu }\right| }_{L}\text{ for all }\nu \text{ (cf. I.8.3.1). Thus }{\left| {g}_{\nu }^{\prime }/{f}_{\nu }\right| }_{K} \leq {\left| {g}_{\nu }^{\prime }\right| }_{K} \cdot \left( \mathop{\min }\limits_{{z \in K}}\left| {f}_{\nu }\left( z\right) \right| \right. \right| }^{-1} \leq \) \( {2M}{\left| {g}_{\nu }\right| }_{L} \) for \( \nu \geq m \) . Since \( \sum {\left| {g}_{\nu }\right| }_{L} < \infty \) by hypothesis, \( \left( *\right) \) follows.
The differentiation theorem is an important tool for concrete computations; for example, we use it in the next subsection to derive Euler's product for the sine, and we give another application in 2.2.3. The theorem holds verbatim if the word "normal" is replaced by "compact." (Prove this.)
The differentiation theorem can be used to prove:
If \( f \) is holomorphic at the origin, then \( f \) can be represented uniquely in a disc \( B \) about 0 as a product
\[
f\left( z\right) = b{z}^{k}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 + {b}_{\nu }{z}^{\nu }}\right) ,\;b,{b}_{\nu } \in \mathbb{C},\;k \in \mathbb{N},
\]
which converges normally in \( B \) to \( f \) .
This theorem was proved in 1929 by J. F. Ritt [R]. It is not claimed that the product converges in the largest disc about 0 in which \( f \) is holomorphic. There seem to be no compell |
1006_(GTM172)Classical Topics in Complex Function Theory | 6 | ) for \( \nu \geq m \) . Since \( \sum {\left| {g}_{\nu }\right| }_{L} < \infty \) by hypothesis, \( \left( *\right) \) follows.
The differentiation theorem is an important tool for concrete computations; for example, we use it in the next subsection to derive Euler's product for the sine, and we give another application in 2.2.3. The theorem holds verbatim if the word "normal" is replaced by "compact." (Prove this.)
The differentiation theorem can be used to prove:
If \( f \) is holomorphic at the origin, then \( f \) can be represented uniquely in a disc \( B \) about 0 as a product
\[
f\left( z\right) = b{z}^{k}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 + {b}_{\nu }{z}^{\nu }}\right) ,\;b,{b}_{\nu } \in \mathbb{C},\;k \in \mathbb{N},
\]
which converges normally in \( B \) to \( f \) .
This theorem was proved in 1929 by J. F. Ritt [R]. It is not claimed that the product converges in the largest disc about 0 in which \( f \) is holomorphic. There seem to be no compelling applications of this product expansion, which is a multiplicative analogue of the Taylor series.
§3. The Sine Product \( \sin {\pi z} = {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {z}^{2}/{\nu }^{2}}\right) \)
The product \( \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {z}^{2}/{\nu }^{2}}\right) \) is normally convergent in \( \mathbb{C} \), since \( \mathop{\sum }\limits_{{\nu = 1}}^{\infty }{z}^{2}/{\nu }^{2} \) converges normally in \( \mathbb{C} \) . In 1734 Euler discovered that
(1)
\[
\sin {\pi z} = {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{{z}^{2}}{{\nu }^{2}}}\right) ,\;z \in \mathbb{C}.
\]
We give two proofs of this formula.
1. Standard proof (using logarithmic differentiation and the partial fraction decomposition for the cotangent). Setting \( {f}_{\nu } \mathrel{\text{:=}} 1 - {z}^{2}/{\nu }^{2} \) and \( f\left( z\right) \mathrel{\text{:=}} \) \( {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }{f}_{\nu } \) gives
\[
{f}_{\nu }^{\prime }/{f}_{\nu } = \frac{2z}{{z}^{2} - {\nu }^{2}},\;\text{ and thus }\;{f}^{\prime }\left( z\right) /f\left( z\right) = \frac{1}{z} + \mathop{\sum }\limits_{{\nu = 1}}^{\infty }\frac{2z}{{z}^{2} - {\nu }^{2}}.
\]
Here the right-hand side is the function \( \pi \cot {\pi z} \) (cf. I.11.2.1). As this is also the logarithmic derivative of \( \sin {\pi z} \), we have \( {}^{5}f\left( z\right) = c\sin {\pi z} \) with \( c \in {\mathbb{C}}^{ \times } \) . Since \( \mathop{\lim }\limits_{{z \rightarrow 0}}\frac{f\left( z\right) }{\pi z} = 1 = \mathop{\lim }\limits_{{z \rightarrow 0}}\frac{\sin {\pi z}}{\pi z} \), it follows that \( c = 1 \) .
Substituting special values for \( z \) in (1) yields interesting (and uninteresting) formulas. Setting \( z \mathrel{\text{:=}} \frac{1}{2} \) gives the product formula
\[
\frac{\pi }{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \ldots = \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\frac{2\nu }{{2\nu } - 1} \cdot \frac{2\nu }{{2\nu } + 1}\;\text{ (Wallis,1655). }
\]
For \( z \mathrel{\text{:=}} 1 \), one obtains the trivial equality \( \frac{1}{2} = \mathop{\prod }\limits_{{\nu = 2}}^{\infty }\left( {1 - \frac{1}{{\nu }^{2}}}\right) \) (cf. Example 1.1, b); on the other hand, setting \( z \mathrel{\text{:=}} i \) and using the identity \( \sin {\pi i} = \frac{i}{2}\left( {{e}^{\pi } - {e}^{-\pi }}\right) \) give the bizarre formula
\[
\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 + \frac{1}{{\nu }^{2}}}\right) = \frac{{e}^{\pi } - {e}^{-\pi }}{2\pi }
\]
Using the identity \( \sin z\cos z = \frac{1}{2}\sin {2z} \) and Corollary 2.1, one obtains
\[
\cos {\pi z}\sin {\pi z} = {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {\left( \frac{2z}{\nu }\right) }^{2}}\right)
\]
\[
= {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {\left( \frac{2z}{2\nu }\right) }^{2}}\right) \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {\left( \frac{2z}{{2\nu } - 1}\right) }^{2}}\right)
\]
\( {}^{5} \) Let \( f \neq 0, g \neq 0 \) be two meromorphic functions on a domain \( G \) which have the same logarithmic derivative. Then \( f = {cg} \), with \( c \in {\mathbb{C}}^{ \times } \) . To prove this, note that \( f/g \in \mathcal{M}\left( G\right) \) and \( {\left( f/g\right) }^{\prime } \equiv 0 \) .
§3. The Sine Product \( \sin {\pi z} = {\pi z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {z}^{2}/{\nu }^{2}}\right) \)
and hence Euler's product representation for the cosine:
\[
\cos {\pi z} = \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{4{z}^{2}}{{\left( 2\nu - 1\right) }^{2}}}\right) ,\;z \in \mathbb{C}.
\]
In 1734-35, with his sine product, Euler could in principle compute all the numbers \( \zeta \left( {2n}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\nu = 1}}^{\infty }{\nu }^{-{2n}}, n = 1,2,\ldots \) (cf. also I.11.3.2). Thus it follows immediately, for example, that \( \zeta \left( 2\right) = \frac{{\pi }^{2}}{6} \) : Since \( {f}_{n}\left( z\right) \mathrel{\text{:=}} \mathop{\prod }\limits_{{\nu = 1}}^{n}\left( {1 - {z}^{2}/{\nu }^{2}}\right) = \) \( 1 - \left( {\mathop{\sum }\limits_{{\nu = 1}}^{n}{\nu }^{-2}}\right) {z}^{2} + \cdots \) tends compactly to \( f\left( z\right) \mathrel{\text{:=}} \left( {\sin {\pi z}}\right) /\left( {\pi z}\right) = 1 - \frac{{\pi }^{2}{z}^{2}}{6} + \cdots \) , it follows that \( \frac{1}{2}{f}_{n}^{\prime \prime }\left( 0\right) = - \mathop{\sum }\limits_{{\nu = 1}}^{n}{\nu }^{-2} \) converges to \( \frac{1}{2}{f}^{\prime \prime }\left( 0\right) = - \frac{1}{6}{\pi }^{2} \) .
Wallis's formula permits an elementary calculation of the Gaussian error integral \( {\int }_{0}^{\infty }{e}^{-{x}^{2}}{dx} \) . For \( {I}_{n} \mathrel{\text{:=}} {\int }_{0}^{\infty }{x}^{n}{e}^{-{x}^{2}}{dx} \), we have
\[
2{I}_{n} = \left( {n - 1}\right) {I}_{n - 2},\;n \geq 2\;\text{ (integration by parts!). }
\]
Since \( {I}_{1} = \frac{1}{2} \), an induction argument gives
(o)
\[
{2}^{k}{I}_{2k} = 1 \cdot 3 \cdot 5 \cdot \ldots \cdot \left( {{2k} - 1}\right) {I}_{0},\;2{I}_{{2k} + 1} = k!,\;k \in \mathbb{N}.
\]
Since \( {I}_{n + 1} + {2t}{I}_{n} + {t}^{2}{I}_{n - 1} = {\int }_{0}^{\infty }{x}^{n - 1}{\left( x + t\right) }^{2}{e}^{-{x}^{2}}{dx} \) for all \( t \in \mathbb{R} \), it follows that
\[
{I}_{n}^{2} < {I}_{n - 1}{I}_{n + 1};\;\text{ hence }\;2{I}_{n}^{2} < n{I}_{n - 1}^{2}.
\]
With ( \( \circ \) ) we now obtain
\[
\frac{{\left( k!\right) }^{2}}{{4k} + 2} = \frac{2}{{2k} + 1}{I}_{{2k} + 1}^{2} < {I}_{2k}^{2} < {I}_{{2k} - 1}{I}_{{2k} + 1} = \frac{{\left( k!\right) }^{2}}{4k}.
\]
This can also be written
\[
{I}_{2k}^{2} = \frac{{\left( k!\right) }^{2}}{{4k} + 2}\left( {1 + {\varepsilon }_{k}}\right) ,\;\text{ with }\;0 < {\varepsilon }_{k} < \frac{1}{2k}.
\]
Using (o) to substitute \( {I}_{0} \) into this yields
\[
2{I}_{0}^{2} = \frac{{\left\lbrack 2 \cdot 4 \cdot 6 \cdot \ldots \cdot \left( 2k\right) \right\rbrack }^{2}}{{\left\lbrack 1 \cdot 3 \cdot 5 \cdot \ldots \cdot \left( 2k - 1\right) \right\rbrack }^{2}\left( {{2k} + 1}\right) }\left( {1 + {\varepsilon }_{k}}\right) .
\]
From \( \lim {\varepsilon }_{k} = 0 \) and Wallis’s formula, it follows that \( 2{I}_{0}^{2} = \frac{1}{2}\pi \) and hence that \( {\int }_{0}^{\infty }{e}^{-{x}^{2}}{dx} = \frac{1}{2}\sqrt{\pi }. \)
This derivation was given by T.-J. Stieltjes: Note sur l’intégrale \( {\int }_{0}^{\infty }{e}^{-{u}^{2}}{du} \) , Nouv. Ann. Math. 9, 3rd ser., 479-480 (1890); Euvres complètes 2, 2nd ed., Springer, 1993, 263-264.
Exercises. Prove:
1) \( \lim \frac{2 \cdot 4 \cdot 6 \cdot \ldots \cdot {2n}}{3 \cdot 5 \cdot 7 \cdot \ldots \cdot \left( {{2n} + 1}\right) }\sqrt{n} = \frac{1}{2}\sqrt{\pi } \) ;
2) \( \frac{1}{4}\pi = \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{1}{{\left( 2\nu + 1\right) }^{2}}}\right) \) ;
3) \( {e}^{az} - {e}^{bz} = \left( {a - b}\right) z{e}^{\frac{1}{2}\left( {a + b}\right) z}\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 + \frac{{\left( a - b\right) }^{2}{z}^{2}}{4{\nu }^{2}{\pi }^{2}}}\right) \) ;
\[
\text{4)}\cos \left( {\frac{1}{4}{\pi z}}\right) - \sin \left( {\frac{1}{4}{\pi z}}\right) = \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 + \frac{{\left( -1\right) }^{n}z}{{2n} - 1}}\right) \text{.}
\]
2. Characterization of the sine by the duplication formula. We characterize the sine function by properties that are easy to verify for the product \( z\prod \left( {1 - {z}^{2}/{\nu }^{2}}\right) \) . The equality \( \sin {2z} = 2\sin z\cos z \) is a
Duplication formula: \( \sin {2\pi z} = 2\sin {\pi z}\sin \pi \left( {z + \frac{1}{2}}\right) ,\;z \in \mathbb{C} \) .
In order to use it in characterizing the sine, we first prove a lemma.
Lemma (Herglotz, multiplicative form). \( {}^{6} \) Let \( G \subset \mathbb{C} \) be a domain that contains an interval \( \lbrack 0, r), r > 1 \) . Suppose that \( g \in \mathcal{O}\left( G\right) \) has no zeros in \( \lbrack 0, r) \) and satisfies a multiplicative duplication formula
\[
\text{(*)}g\left( {2z}\right) = {cg}\left( z\right) g\left( {z + \frac{1}{2}}\right) \;\text{when}\;z, z + \frac{1}{2},{2z} \in \lbrack 0, r)\;\text{(with}c \in {\mathbb{C}}^{ \times }\text{).}
\]
Then \( g\left( z\right) = a{e}^{bz} \) with \( 1 = {ac}{e}^{\frac{1}{2}b} \) .
Proof. The function \( h \mathrel{\text{:=}} {g}^{\prime }/g \in \mathcal{M}\left( G\right) \) is holomorphic throughout \( \lbrack 0, r) \) , and \( {2h}\left( {2z}\right) = 2{g}^{\prime }\left( {2z}\right) /g\left( {2z}\right) = h\left( z\right) + h\left( {z + \frac{1}{2}}\right) \) whenever \( z, z + \frac{1}{2},{2z} \in \lbrack 0, r) \) . By Herglotz’s lemma (additive form), \( h \) is constant. \( {}^{6} \) It follows that \( {g}^{\prime } = {bg} \) with \( b \in \mathbb{C} \) ; hence \( g\left( z\right) = a{e}^{bz} \) . By \( \left( *\right) \), ace \( {}^{\frac{1}{2}b} = 1 \) .
The |
1006_(GTM172)Classical Topics in Complex Function Theory | 7 | \in \mathcal{O}\left( G\right) \) has no zeros in \( \lbrack 0, r) \) and satisfies a multiplicative duplication formula
\[
\text{(*)}g\left( {2z}\right) = {cg}\left( z\right) g\left( {z + \frac{1}{2}}\right) \;\text{when}\;z, z + \frac{1}{2},{2z} \in \lbrack 0, r)\;\text{(with}c \in {\mathbb{C}}^{ \times }\text{).}
\]
Then \( g\left( z\right) = a{e}^{bz} \) with \( 1 = {ac}{e}^{\frac{1}{2}b} \) .
Proof. The function \( h \mathrel{\text{:=}} {g}^{\prime }/g \in \mathcal{M}\left( G\right) \) is holomorphic throughout \( \lbrack 0, r) \) , and \( {2h}\left( {2z}\right) = 2{g}^{\prime }\left( {2z}\right) /g\left( {2z}\right) = h\left( z\right) + h\left( {z + \frac{1}{2}}\right) \) whenever \( z, z + \frac{1}{2},{2z} \in \lbrack 0, r) \) . By Herglotz’s lemma (additive form), \( h \) is constant. \( {}^{6} \) It follows that \( {g}^{\prime } = {bg} \) with \( b \in \mathbb{C} \) ; hence \( g\left( z\right) = a{e}^{bz} \) . By \( \left( *\right) \), ace \( {}^{\frac{1}{2}b} = 1 \) .
The next theorem now follows quickly.
Theorem. Let \( f \) be an odd entire function that vanishes in \( \left\lbrack {0,1}\right\rbrack \) only at 0 and 1, and vanishes to first order there. Suppose that it satisfies the
Duplication formula: \( f\left( {2z}\right) = {cf}\left( z\right) f\left( {z + \frac{1}{2}}\right) ,\;z \in \mathbb{C},\; \) where \( \;c \in {\mathbb{C}}^{ \times }. \)
Then \( f\left( z\right) = 2{c}^{-1}\sin {\pi z} \) .
Proof. The function \( g\left( z\right) \mathrel{\text{:=}} f\left( z\right) /\sin {\pi z} \) is holomorphic and nowhere zero in a domain \( G \supset \lbrack 0, r), r > 1 \) ; we have \( g\left( {2z}\right) = \frac{1}{2}{cg}\left( z\right) g\left( {z + \frac{1}{2}}\right) \) . By Herglotz, \( f\left( z\right) = a{e}^{bz}\sin {\pi z} \) with \( {ac}{e}^{\frac{1}{2}b} = 2 \) . Since \( f\left( {-z}\right) = f\left( z\right) \), it also follows that \( b = 0 \) .
\( {}^{6} \) We recall the following lemma, discussed in I.11.2.2:
Herglotz’s lemma (additive form). Let \( \lbrack 0, r) \subset G \) with \( r > 1 \) . Let \( h \in \mathcal{O}\left( G\right) \) and assume that the additive duplication formula \( {2h}\left( {2z}\right) = h\left( z\right) + h\left( {z + \frac{1}{2}}\right) \) holds when \( z, z + \frac{1}{2},{2z} \in \lbrack 0, r) \) . Then \( h \) is constant.
Proof. Let \( t \in \left( {1, r}\right) \) and \( M \mathrel{\text{:=}} \max \left\{ {\left| {{h}^{\prime }\left( z\right) }\right| : z \in \lbrack 0, t)}\right\} \) . Since \( 4{h}^{\prime }\left( {2z}\right) = {h}^{\prime }\left( z\right) + \) \( {h}^{\prime }\left( {z + \frac{1}{2}}\right) \) and \( \frac{1}{2}z \) and \( \frac{1}{2}\left( {z + 1}\right) \) always lie in \( \left\lbrack {0, t}\right\rbrack \) whenever \( z \) does, it follows that \( {4M} \leq {2M} \), and hence that \( M = 0 \) . By the identity theorem, \( {h}^{\prime } = 0 \) ; thus \( h = \) const.
We also use the duplication formula for the sine to derive an integral that will be needed in the appendix to 4.3 for the proof of Jensen's formula:
(1)
\[
{\int }_{0}^{1}\log \sin {\pi tdt} = - \log 2
\]
Proof. Assuming for the moment that the integral exists, we have
(o)
\[
{\int }_{0}^{\frac{1}{2}}\log \sin {2\pi tdt} = \frac{1}{2}\log 2 + {\int }_{0}^{\frac{1}{2}}\log \sin {\pi tdt} + {\int }_{0}^{\frac{1}{2}}\log \sin \pi \left( {t + \frac{1}{2}}\right) {dt}.
\]
Setting \( \tau \mathrel{\text{:=}} {2t} \) on the left-hand side and \( \tau \mathrel{\text{:=}} t + \frac{1}{2} \) in the integral on the extreme right immediately yields (1). The second integral on the right in (o) exists whenever the first one does (set \( t + \frac{1}{2} = 1 - \tau \) ). The first integral exists since \( g\left( t\right) \mathrel{\text{:=}} {t}^{-1}\sin {\pi t} \) is continuous and nonvanishing in \( \left\lbrack {0,\frac{1}{2}}\right\rbrack {.}^{7} \)
3. Proof of Euler's formula using Lemma 2. The function
\[
s\left( z\right) \mathrel{\text{:=}} z \cdot \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - {z}^{2}/{\nu }^{2}}\right)
\]
is entire and odd and has zeros precisely at the points of \( \mathbb{Z} \), and these are first-order zeros. Since \( {s}^{\prime }\left( 0\right) = \mathop{\lim }\limits_{{z \rightarrow 0}}s\left( z\right) /z = 1 \), Theorem 2 implies that \( \sin {\pi z} = {\pi s}\left( z\right) \) whenever \( s \) satisfies a duplication formula. This can be verified immediately. Since \( s \) converges normally, it follows from Corollary 2.1 that
\( \left( +\right) \)
\[
s\left( {2z}\right) = {2z} \cdot \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{{\left( 2z\right) }^{2}}{{\left( 2\nu \right) }^{2}}}\right) \cdot \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{4{z}^{2}}{{\left( 2\nu - 1\right) }^{2}}}\right)
\]
\[
= {2s}\left( z\right) \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{4{z}^{2}}{{\left( 2\nu - 1\right) }^{2}}}\right) .
\]
A computation (!) gives
\[
\left( {1 - \frac{1}{4{\nu }^{2}}}\right) \left( {1 - \frac{4{z}^{2}}{{\left( 2\nu - 1\right) }^{2}}}\right) = \frac{1 + {2z}/\left( {{2\nu } - 1}\right) }{1 + {2z}/\left( {{2\nu } + 1}\right) }\left( {1 - \frac{{\left( 2z + 1\right) }^{2}}{4{\nu }^{2}}}\right) ,\;\nu \geq 1.
\]
If we take Example a) of 1.2 into account, this yields
\[
\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{1}{4{\nu }^{2}}}\right) \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{4{z}^{2}}{{\left( 2\nu - 1\right) }^{2}}}\right) = \left( {1 + {2z}}\right) \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{{\left( 2z + 1\right) }^{2}}{4{\nu }^{2}}}\right)
\]
\[
= {2s}\left( {z + \frac{1}{2}}\right) \text{.}
\]
\( {}^{7} \) Let \( f\left( t\right) = {t}^{-n}g\left( t\right), t \in \mathbb{N} \), where \( g \) is continuous and nonvanishing in \( \left\lbrack {0, r}\right\rbrack \) , \( r > 0 \) . Then \( {\int }_{0}^{r}\log f\left( t\right) {dt} \) exists. This is clear since \( {\int }_{0}^{r}\log {tdt} \) exists \( (x\log x - x \) is an antiderivative, and \( \left. {\mathop{\lim }\limits_{{\delta \searrow 0}}\delta \log \delta = 0}\right) \) . Thus \( \left( +\right) \) is a duplication formula: \( s\left( {2z}\right) = 4{a}^{-1}s\left( z\right) s\left( {z + \frac{1}{2}}\right) \), where \( a \mathrel{\text{:=}} \) \( \prod \left( {1 - 1/4{\nu }^{2}}\right) \neq 0. \)
This multiplicative proof dates back to the American mathematician E. H. Moore; a number of computations are carried out in his 1894 paper [M]. The reader should note the close relationship with Schottky's proof of the equation
\[
\pi \cot {\pi z} = \frac{1}{z} + \mathop{\sum }\limits_{{\nu = - \infty }}^{\infty }\left( {\frac{1}{z + \nu } - \frac{1}{\nu }}\right)
\]
in I.11.2.1; Moore probably did not know Schottky’s 1892 paper.
\( {4}^{ * } \) . Proof of the duplication formula for Euler’s product, following Eisenstein. Long before Moore, Eisenstein had proved the duplication formula for \( s\left( z\right) \) in passing. In \( {1847} \) ([Ei], p. \( {461}\mathrm{{ff}} \) .), he considered the apparently complicated product
\[
E\left( {w, z}\right) \mathrel{\text{:=}} \mathop{\prod }\limits_{{\nu = - \infty }}^{\infty }\left( {1 + \frac{z}{\nu + w}}\right) = \left( {1 + \frac{z}{w}}\right) \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{\nu = - n}}^{n}\left( {1 + \frac{z}{\nu + w}}\right)
\]
of two variables \( \left( {w, z}\right) \in \left( {\mathbb{C} \smallsetminus \mathbb{Z}}\right) \times \mathbb{C} \) ; here \( \mathop{\prod }\limits_{e} = \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{\nu = - n}}^{n} \) denotes the Eisenstein multiplication (by analogy with the Eisenstein summation \( \mathop{\sum }\limits_{e} \), which we introduced in I.11.2). Moreover, \( \mathop{\prod }\limits^{\prime } \) indicates that the factor with index 0 is omitted. The Eisenstein product \( E\left( {w, z}\right) \) is normally convergent in the \( \left( {w, z}\right) \) - space \( \left( {\mathbb{C} \smallsetminus \mathbb{Z}}\right) \times \mathbb{C} \), since
\[
\mathop{\prod }\limits_{{\nu = - n}}^{n}\left( {1 + \frac{z}{\nu + w}}\right) = \mathop{\prod }\limits_{{\nu = 1}}^{n}\left( {1 - \frac{{z}^{2} + {2wz}}{{\nu }^{2} - {w}^{2}}}\right)
\]
and \( \mathop{\sum }\limits_{{\nu = 1}}^{\infty }1/\left( {{w}^{2} - {\nu }^{2}}\right) \) converges normally in \( \mathbb{C} \smallsetminus \mathbb{Z} \) (cf. I.11.1.3). The function \( E\left( {z, w}\right) \) is therefore continuous in \( \left( {\mathbb{C} \smallsetminus \mathbb{Z}}\right) \times \mathbb{C} \) and, for fixed \( w \), holomorphic in each \( z \in \mathbb{C} \) . Computations can be carried out elegantly with \( E\left( {z, w}\right) \), and the following is immediate.
Duplication formula. \( E\left( {{2w},{2z}}\right) = E\left( {w, z}\right) E\left( {w + \frac{1}{2}, z}\right) \) .
Proof.
\[
E\left( {{2w},{2z}}\right) = \mathop{\prod }\limits_{{\nu = - \infty }}^{\infty }\left( {1 + \frac{2z}{{2\nu } + {2w}}}\right) \cdot \mathop{\prod }\limits_{{\nu = - \infty }}^{\infty }\left( {1 + \frac{2z}{{2\nu } + 1 + {2w}}}\right)
\]
\[
= E\left( {w, z}\right) E\left( {w + \frac{1}{2}, z}\right) \text{.}
\]
口
Eisenstein used the (trivial, but astonishing!) formula
\( \left( *\right) \)
\[
1 + \frac{z}{\nu + w} = \left( {1 + \frac{w + z}{\nu }}\right) /\left( {1 + \frac{w}{\nu }}\right) \;\text{ (Eisenstein’s trick) }
\]
to reduce his "double product" to Euler's product:
\[
E\left( {w, z}\right) = \frac{s\left( {w + z}\right) }{s\left( w\right) },\;\text{ where }\;s\left( z\right) = z\mathop{\prod }\limits_{{\nu \geq 1}}\left( {1 - \frac{{z}^{2}}{{\nu }^{2}}}\right) .
\]
Proof.
\[
E\left( {w, z}\right) = \frac{w + z}{w}\mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{\nu = - n}}^{n}\left( {1 + \frac{w + z}{\nu }}\right) /\mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{\nu = - n}}^{n}\left( {1 |
1006_(GTM172)Classical Topics in Complex Function Theory | 8 | }}^{\infty }\left( {1 + \frac{2z}{{2\nu } + {2w}}}\right) \cdot \mathop{\prod }\limits_{{\nu = - \infty }}^{\infty }\left( {1 + \frac{2z}{{2\nu } + 1 + {2w}}}\right)
\]
\[
= E\left( {w, z}\right) E\left( {w + \frac{1}{2}, z}\right) \text{.}
\]
口
Eisenstein used the (trivial, but astonishing!) formula
\( \left( *\right) \)
\[
1 + \frac{z}{\nu + w} = \left( {1 + \frac{w + z}{\nu }}\right) /\left( {1 + \frac{w}{\nu }}\right) \;\text{ (Eisenstein’s trick) }
\]
to reduce his "double product" to Euler's product:
\[
E\left( {w, z}\right) = \frac{s\left( {w + z}\right) }{s\left( w\right) },\;\text{ where }\;s\left( z\right) = z\mathop{\prod }\limits_{{\nu \geq 1}}\left( {1 - \frac{{z}^{2}}{{\nu }^{2}}}\right) .
\]
Proof.
\[
E\left( {w, z}\right) = \frac{w + z}{w}\mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{\nu = - n}}^{n}\left( {1 + \frac{w + z}{\nu }}\right) /\mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{\nu = - n}}^{n}\left( {1 + \frac{w}{\nu }}\right)
\]
\[
= \left( {w + z}\right) \mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{{\left( w + z\right) }^{2}}{{\nu }^{2}}}\right) /\left( {w\mathop{\prod }\limits_{{\nu = 1}}^{\infty }\left( {1 - \frac{{w}^{2}}{{\nu }^{2}}}\right) }\right)
\]
\[
= \frac{s\left( {w + z}\right) }{s\left( w\right) }
\]
The duplication formula for \( s\left( z\right) \) is now contained in the equation
\[
\frac{s\left( {{2w} + {2z}}\right) }{s\left( {2w}\right) } = E\left( {{2w},{2z}}\right) = E\left( {w, z}\right) E\left( {w + \frac{1}{2}, z}\right) = \frac{s\left( {w + z}\right) }{s\left( w\right) } \cdot \frac{s\left( {w + \frac{1}{2} + z}\right) }{s\left( {w + \frac{1}{2}}\right) }.
\]
Since \( s \) is continuous and \( \mathop{\lim }\limits_{{w \rightarrow 0}}\frac{s\left( {2w}\right) }{s\left( w\right) } = 2 \), it follows that
\[
s\left( {2z}\right) = \mathop{\lim }\limits_{{w \rightarrow 0}}\frac{s\left( {2w}\right) }{s\left( w\right) }s\left( {w + z}\right) \frac{s\left( {w + \frac{1}{2} + z}\right) }{s\left( {w + \frac{1}{2}}\right) } = {2s}{\left( \frac{1}{2}\right) }^{-1}s\left( z\right) s\left( {z + \frac{1}{2}}\right) .
\]
The elegance of Eisenstein's reasoning is made possible by the second variable \( w \) . Eisenstein also notes (loc. cit.) that \( E \) is periodic in \( w : E(w + \) \( 1, z) = E\left( {w, z}\right) \) (proved by substituting \( \nu + 1 \) for \( \nu \) ); he uses \( E \) and \( s \) to prove the quadratic reciprocity law; the duplication formula appears there at the bottom of p. 462. Eisenstein calls the identity \( E\left( {w, z}\right) = s\left( {w + z}\right) /s\left( w\right) \) the fundamental formula and writes it as follows (p. 402; the interpretation is left to the reader):
\[
\mathop{\prod }\limits_{{m \in \mathbb{Z}}}\left( {1 - \frac{z}{{\alpha m} + \beta }}\right) = \frac{\sin \pi \left( {\beta - z}\right) /\alpha }{\sin {\pi \beta }/\alpha },\;\alpha ,\beta \in \mathbb{C},\;\beta /\alpha \notin \mathbb{Z}.
\]
5. On the history of the sine product. Euler discovered the cosine and sine products in 1734-35 and published them in the famous paper "De Summis Serierum Reciprocarum" ([Eu], I-14, pp. 73-86); the formula
\[
1 - \frac{{s}^{2}}{1 \cdot 2 \cdot 3} + \frac{{s}^{4}}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} - \frac{{s}^{6}}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \cdots
\]
\[
= \left( {1 - \frac{{s}^{2}}{{p}^{2}}}\right) \left( {1 - \frac{{s}^{2}}{4{p}^{2}}}\right) \left( {1 - \frac{{s}^{2}}{9{p}^{2}}}\right) \left( {1 - \frac{{s}^{2}}{{16}{p}^{2}}}\right) \ldots
\]
(with \( p \mathrel{\text{:=}} \pi \) ) appears on p. 84. As justification Euler asserts that the zeros of the series are \( p, - p,{2p}, - {2p},{3p}, - {3p} \), etc., and that the series is therefore (by analogy with polynomials) divisible by \( 1 - \frac{s}{p},\;1 + \frac{s}{p},\;1 - \frac{s}{2p},\;1 + \frac{s}{2p}\; \) etc.!
In a letter to Euler dated 2 April 1737, Joh. Bernóulli emphasizes that this reasoning would be legitimate only if one knew that the function \( \sin z \) had no zeros in \( \mathbb{C} \) other than \( {n\pi }, n \in \mathbb{Z} \) : "demonstrandum esset nullam contineri radicem impossibilem" ([C], vol. 2, p.16); D. and N. Bernoulli made further criticisms; cf. [Weil], pp. 264-265. These objections, acknowledged to some extent by Euler, were among the factors giving incentive to his discovery of the formula \( {e}^{iz} = \cos z + i\sin z \) ; from this Euler, in 1743, derived his product formula, which then gives him all the zeros of \( \cos z \) and \( \sin z \) as a byproduct.
Euler argues as follows: since \( \lim {\left( 1 + z/n\right) }^{n} = {e}^{z} \) and \( \sin z = \left( {{e}^{iz} - {e}^{-{iz}}}\right) /{2i} \) ,
\[
\sin z = \frac{1}{2i}\lim {p}_{n}\left( \frac{iz}{n}\right) ,\;\text{ where }\;{p}_{n}\left( w\right) \mathrel{\text{:=}} {\left( 1 + w\right) }^{n} - {\left( 1 - w\right) }^{n}.
\]
For every even index \( n = {2m} \), it follows that
\( \left( *\right) \)
\[
{p}_{n}\left( w\right) = {2nw}\left( {1 + w + \cdots + {w}^{n - 2}}\right) .
\]
The roots \( \omega \) of \( {p}_{n} \) are given by \( \left( {1 + \omega }\right) = \zeta \left( {1 - \omega }\right) \), where \( \zeta = \exp \left( {{2\nu \pi i}/n}\right) \) is any \( n \) th root of unity; hence \( {p}_{2m} \), as an odd polynomial of degree \( n - 1 \), has the \( n - 1 \) distinct zeros \( 0, \pm {\omega }_{1},\ldots , \pm {\omega }_{m - 1} \), where
\[
{\omega }_{\nu } = \frac{\exp \left( {{2\nu \pi i}/n}\right) - 1}{\exp \left( {{2\nu \pi i}/n}\right) + 1} = i\tan \frac{\nu \pi }{n},\;\nu = 1,\ldots, m - 1.
\]
The factorization
\[
{p}_{2m}\left( w\right) = {2nw}\mathop{\prod }\limits_{{\nu = 1}}^{{m - 1}}\left( {1 - \frac{w}{{\omega }_{\nu }}}\right) \left( {1 + \frac{w}{{\omega }_{\nu }}}\right) = {2nw}\mathop{\prod }\limits_{{\nu = 1}}^{{m - 1}}\left( {1 + {w}^{2}{\cot }^{2}\frac{\nu \pi }{n}}\right)
\]
then follows from \( \left( *\right) \) . Thus
\[
\sin z = z\mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{\nu = 1}}^{{\frac{1}{2}n - 1}}\left( {1 - {z}^{2}{\left( \frac{1}{n}\cot \frac{\nu \pi }{n}\right) }^{2}}\right) .
\]
Since \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {\frac{1}{n}\cot \frac{\nu \pi }{n}}\right) = \frac{1}{\nu \pi } \), interchanging the limits yields the product formula. This last step can, of course, be rigorously justified (cf., for example, [V], p. 42 and p. 56). An even simpler derivation of the sine product, based on the same fundamental idea, is given in [Nu], 5.4.3.
## §4*. Euler Partition Products
Euler intensively studied the product
\[
Q\left( {z, q}\right) \mathrel{\text{:=}} \mathop{\prod }\limits_{{\nu \geq 1}}\left( {1 + {q}^{\nu }z}\right) = \left( {1 + {qz}}\right) \left( {1 + {q}^{2}z}\right) \left( {1 + {q}^{3}z}\right) \cdots
\]
as well as the sine product. \( Q\left( {z, q}\right) \) converges normally in \( \mathbb{C} \) for every \( q \in \mathbb{E} \) since \( \sum {\left| q\right| }^{\nu } < \infty \) ; the product is therefore an entire function in \( z \), which |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 0 |
# Reinhard Diestel Graph Theory
## Electronic Edition 2005
(C) Springer-Verlag Heidelberg, New York 1997, 2000, 2005
This is an electronic version of the third (2005) edition of the above Springer book, from their series Graduate Texts in Mathematics, vol. 173. The cross-references in the text and in the margins are active links: click on them to be taken to the appropriate page.
The printed edition of this book can be ordered via
http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
where also errata, reviews etc. are posted.
Substantial discounts and free copies for lecturers are available for course adoptions; see here.
## Preface
Almost two decades have passed since the appearance of those graph theory texts that still set the agenda for most introductory courses taught today. The canon created by those books has helped to identify some main fields of study and research, and will doubtless continue to influence the development of the discipline for some time to come.
Yet much has happened in those 20 years, in graph theory no less than elsewhere: deep new theorems have been found, seemingly disparate methods and results have become interrelated, entire new branches have arisen. To name just a few such developments, one may think of how the new notion of list colouring has bridged the gulf between invariants such as average degree and chromatic number, how probabilistic methods and the regularity lemma have pervaded extremal graph theory and Ramsey theory, or how the entirely new field of graph minors and tree-decompositions has brought standard methods of surface topology to bear on long-standing algorithmic graph problems.
Clearly, then, the time has come for a reappraisal: what are, today, the essential areas, methods and results that should form the centre of an introductory graph theory course aiming to equip its audience for the most likely developments ahead?
I have tried in this book to offer material for such a course. In view of the increasing complexity and maturity of the subject, I have broken with the tradition of attempting to cover both theory and applications: this book offers an introduction to the theory of graphs as part of (pure) mathematics; it contains neither explicit algorithms nor 'real world' applications. My hope is that the potential for depth gained by this restriction in scope will serve students of computer science as much as their peers in mathematics: assuming that they prefer algorithms but will benefit from an encounter with pure mathematics of some kind, it seems an ideal opportunity to look for this close to where their heart lies!
In the selection and presentation of material, I have tried to accommodate two conflicting goals. On the one hand, I believe that an introductory text should be lean and concentrate on the essential, so as to offer guidance to those new to the field. As a graduate text, moreover, it should get to the heart of the matter quickly: after all, the idea is to convey at least an impression of the depth and methods of the subject. On the other hand, it has been my particular concern to write with sufficient detail to make the text enjoyable and easy to read: guiding questions and ideas will be discussed explicitly, and all proofs presented will be rigorous and complete.
A typical chapter, therefore, begins with a brief discussion of what are the guiding questions in the area it covers, continues with a succinct account of its classic results (often with simplified proofs), and then presents one or two deeper theorems that bring out the full flavour of that area. The proofs of these latter results are typically preceded by (or interspersed with) an informal account of their main ideas, but are then presented formally at the same level of detail as their simpler counterparts. I soon noticed that, as a consequence, some of those proofs came out rather longer in print than seemed fair to their often beautifully simple conception. I would hope, however, that even for the professional reader the relatively detailed account of those proofs will at least help to minimize reading time...
If desired, this text can be used for a lecture course with little or no further preparation. The simplest way to do this would be to follow the order of presentation, chapter by chapter: apart from two clearly marked exceptions, any results used in the proof of others precede them in the text.
Alternatively, a lecturer may wish to divide the material into an easy basic course for one semester, and a more challenging follow-up course for another. To help with the preparation of courses deviating from the order of presentation, I have listed in the margin next to each proof the reference numbers of those results that are used in that proof. These references are given in round brackets: for example, a reference (4.1.2) in the margin next to the proof of Theorem 4.3.2 indicates that Lemma 4.1.2 will be used in this proof. Correspondingly, in the margin next to Lemma 4.1.2 there is a reference [4.3.2] (in square brackets) informing the reader that this lemma will be used in the proof of Theorem 4.3.2. Note that this system applies between different sections only (of the same or of different chapters): the sections themselves are written as units and best read in their order of presentation.
The mathematical prerequisites for this book, as for most graph theory texts, are minimal: a first grounding in linear algebra is assumed for Chapter 1.9 and once in Chapter 5.5, some basic topological concepts about the Euclidean plane and 3-space are used in Chapter 4, and a previous first encounter with elementary probability will help with Chapter 11. (Even here, all that is assumed formally is the knowledge of basic definitions: the few probabilistic tools used are developed in the text.) There are two areas of graph theory which I find both fascinating and important, especially from the perspective of pure mathematics adopted here, but which are not covered in this book: these are algebraic graph theory and infinite graphs.
At the end of each chapter, there is a section with exercises and another with bibliographical and historical notes. Many of the exercises were chosen to complement the main narrative of the text: they illustrate new concepts, show how a new invariant relates to earlier ones, or indicate ways in which a result stated in the text is best possible. Particularly easy exercises are identified by the superscript \( {}^{ - } \), the more challenging ones carry a \( {}^{ + } \) . The notes are intended to guide the reader on to further reading, in particular to any monographs or survey articles on the theme of that chapter. They also offer some historical and other remarks on the material presented in the text.
Ends of proofs are marked by the symbol \( ▱ \) . Where this symbol is found directly below a formal assertion, it means that the proof should be clear after what has been said - a claim waiting to be verified! There are also some deeper theorems which are stated, without proof, as background information: these can be identified by the absence of both proof and \( ▱ \) .
Almost every book contains errors, and this one will hardly be an exception. I shall try to post on the Web any corrections that become necessary. The relevant site may change in time, but will always be accessible via the following two addresses:
http://www.springer-ny.com/supplements/diestel/
http://www.springer.de/catalog/html-files/deutsch/math/3540609180.html
Please let me know about any errors you find.
Little in a textbook is truly original: even the style of writing and of presentation will invariably be influenced by examples. The book that no doubt influenced me most is the classic GTM graph theory text by Bollobás: it was in the course recorded by this text that I learnt my first graph theory as a student. Anyone who knows this book well will feel its influence here, despite all differences in contents and presentation.
I should like to thank all who gave so generously of their time, knowledge and advice in connection with this book. I have benefited particularly from the help of N. Alon, G. Brightwell, R. Gillett, R. Halin, M. Hintz, A. Huck, I. Leader, T. Luczak, W. Mader, V. Rödl, A.D. Scott, P.D. Seymour, G. Simonyi, M. Škoviera, R. Thomas, C. Thomassen and P. Valtr. I am particularly grateful also to Tommy R. Jensen, who taught me much about colouring and all I know about \( k \) -flows, and who invested immense amounts of diligence and energy in his proofreading of the preliminary German version of this book.
## About the second edition
Naturally, I am delighted at having to write this addendum so soon after this book came out in the summer of 1997. It is particularly gratifying to hear that people are gradually adopting it not only for their personal use but more and more also as a course text; this, after all, was my aim when I wrote it, and my excuse for agonizing more over presentation than I might otherwise have done.
There are two major changes. The last chapter on graph minors now gives a complete proof of one of the major results of the Robertson-Seymour theory, their theorem that excluding a graph as a minor bounds the tree-width if and only if that graph is planar. This short proof did not exist when I wrote the first edition, which is why I then included a short proof of the next best thing, the analogous result for path-width. That theorem has now been dropped from Chapter 12. Another addition in this chapter is that the tree-width duality theorem, Theorem 12.3.9, now comes with a (short) proof too.
The second major change is the addition of a complete set of hints for the exercises. These are largely Tommy Jensen's work, and I am grateful for the time he donated to this project. The aim of these hints is to help those who use the book to study graph theory on their own, but not to spoil the fun. The exercises, including hints, co |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 1 | ing more over presentation than I might otherwise have done.
There are two major changes. The last chapter on graph minors now gives a complete proof of one of the major results of the Robertson-Seymour theory, their theorem that excluding a graph as a minor bounds the tree-width if and only if that graph is planar. This short proof did not exist when I wrote the first edition, which is why I then included a short proof of the next best thing, the analogous result for path-width. That theorem has now been dropped from Chapter 12. Another addition in this chapter is that the tree-width duality theorem, Theorem 12.3.9, now comes with a (short) proof too.
The second major change is the addition of a complete set of hints for the exercises. These are largely Tommy Jensen's work, and I am grateful for the time he donated to this project. The aim of these hints is to help those who use the book to study graph theory on their own, but not to spoil the fun. The exercises, including hints, continue to be intended for classroom use.
Apart from these two changes, there are a few additions. The most noticable of these are the formal introduction of depth-first search trees in Section 1.5 (which has led to some simplifications in later proofs) and an ingenious new proof of Menger's theorem due to Böhme, Göring and Harant (which has not otherwise been published).
Finally, there is a host of small simplifications and clarifications of arguments that I noticed as I taught from the book, or which were pointed out to me by others. To all these I offer my special thanks.
The Web site for the book has followed me to
http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
I expect this address to be stable for some time.
Once more, my thanks go to all who contributed to this second edition by commenting on the first - and I look forward to further comments!
## About the third edition
There is no denying that this book has grown. Is it still as 'lean and concentrating on the essential' as I said it should be when I wrote the preface to the first edition, now almost eight years ago?
I believe that it is, perhaps now more than ever. So why the increase in volume? Part of the answer is that I have continued to pursue the original dual aim of offering two different things between one pair of covers:
- a reliable first introduction to graph theory that can be used either for personal study or as a course text;
- a graduate text that offers some depth in selected areas.
For each of these aims, some material has been added. Some of this covers new topics, which can be included or skipped as desired. An example at the introductory level is the new section on packing and covering with the Erdős-Pósa theorem, or the inclusion of the stable marriage theorem in the matching chapter. An example at the graduate level is the Robertson-Seymour structure theorem for graphs without a given minor: a result that takes a few lines to state, but one which is increasingly relied on in the literature, so that an easily accessible reference seems desirable. Another addition, also in the chapter on graph minors, is a new proof of the 'Kuratowski theorem for higher surfaces' - a proof which illustrates the interplay between graph minor theory and surface topology better than was previously possible. The proof is complemented by an appendix on surfaces, which supplies the required background and also sheds some more light on the proof of the graph minor theorem.
Changes that affect previously existing material are rare, except for countless local improvements intended to consolidate and polish rather than change. I am aware that, as this book is increasingly adopted as a course text, there is a certain desire for stability. Many of these local improvements are the result of generous feedback I got from colleagues using the book in this way, and I am very grateful for their help and advice.
There are also some local additions. Most of these developed from my own notes, pencilled in the margin as I prepared to teach from the book. They typically complement an important but technical proof, when I felt that its essential ideas might get overlooked in the formal write-up. For example, the proof of the Erdős-Stone theorem now has an informal post-mortem that looks at how exactly the regularity lemma comes to be applied in it. Unlike the formal proof, the discussion starts out from the main idea, and finally arrives at how the parameters to be declared at the start of the formal proof must be specified. Similarly, there is now a discussion pointing to some ideas in the proof of the perfect graph theorem. However, in all these cases the formal proofs have been left essentially untouched.
The only substantial change to existing material is that the old Theorem 8.1.1 (that \( c{r}^{2}n \) edges force a \( T{K}^{r} \) ) seems to have lost its nice (and long) proof. Previously, this proof had served as a welcome opportunity to explain some methods in sparse extremal graph theory. These methods have migrated to the connectivity chapter, where they now live under the roof of the new proof by Thomas and Wollan that \( {8kn} \) edges make a \( {2k} \) -connected graph \( k \) -linked. So they are still there, leaner than ever before, and just presenting themselves under a new guise. As a consequence of this change, the two earlier chapters on dense and sparse extremal graph theory could be reunited, to form a new chapter appropriately named as Extremal Graph Theory.
Finally, there is an entirely new chapter, on infinite graphs. When graph theory first emerged as a mathematical discipline, finite and infinite graphs were usually treated on a par. This has changed in recent years, which I see as a regrettable loss: infinite graphs continue to provide a natural and frequently used bridge to other fields of mathematics, and they hold some special fascination of their own. One aspect of this is that proofs often have to be more constructive and algorithmic in nature than their finite counterparts. The infinite version of Menger's theorem in Section 8.4 is a typical example: it offers algorithmic insights into connectivity problems in networks that are invisible to the slick inductive proofs of the finite theorem given in Chapter 3.3.
Once more, my thanks go to all the readers and colleagues whose comments helped to improve the book. I am particularly grateful to Imre Leader for his judicious comments on the whole of the infinite chapter; to my graph theory seminar, in particular to Lilian Matthiesen and Philipp Sprüssel, for giving the chapter a test run and solving all its exercises (of which eighty survived their scrutiny); to Angelos Georgakopoulos for much proofreading elsewhere; to Melanie Win Myint for recompiling the index and extending it substantially; and to Tim Stelldinger for nursing the whale on page 366 until it was strong enough to carry its baby dinosaur.
## Contents
vii
1. The Basics 1
1.1 Graphs* 2
1.2 The degree of a vertex* 5
1.3 Paths and cycles* 6
1.4 Connectivity* 10
1.5 Trees and forests* 13
1.6 Bipartite graphs* 17
1.7 Contraction and minors* 18
1.8 Euler tours* 22
1.9 Some linear algebra 23
1.10 Other notions of graphs 28
Exercises 30
Notes 32
2. Matching, Covering and Packing 33
2.1 Matching in bipartite graphs* 34
2.2 Matching in general graphs \( {}^{\left( *\right) } \) . 39
2.3 Packing and covering 44
2.4 Tree-packing and arboricity 46
2.5 Path covers 49
Exercises 51
Notes 53
---
* Sections marked by an asterisk are recommended for a first course.
Of sections marked \( {}^{\left( *\right) } \), the beginning is recommended for a first course.
---
3. Connectivity 55
3.1 2-Connected graphs and subgraphs* 55
3.2 The structure of 3-connected graphs \( {}^{\left( *\right) } \) . 57
3.3 Menger's theorem* 62
3.4 Mader's theorem 67
3.5 Linking pairs of vertices \( {}^{\left( *\right) } \) . 69
Exercises 78
Notes 80
4. Planar Graphs 83
4.1 Topological prerequisites* 84
4.2 Plane graphs* 86
4.3 Drawings 92
4.4 Planar graphs: Kuratowski's theorem* 96
4.5 Algebraic planarity criteria 101
4.6 Plane duality 103
Exercises 106
Notes 109
5. Colouring 111
5.1 Colouring maps and planar graphs* 112
5.2 Colouring vertices* 114
5.3 Colouring edges* 119
5.4 List colouring 121
5.5 Perfect graphs 126
Exercises 133
Notes 136
6. Flows 139
6.1 Circulations \( {}^{\left( *\right) } \) . 140
6.2 Flows in networks* 141
6.3 Group-valued flows 144
\( {6.4k} \) -Flows for small \( k \) 149
6.5 Flow-colouring duality 152
6.6 Tutte's flow conjectures 156
Exercises 160
Notes 161
7. Extremal Graph Theory 163
7.1 Subgraphs* 164
7.2 Minors(*) 169
7.3 Hadwiger's conjecture* 172
7.4 Szemerédi's regularity lemma 175
7.5 Applying the regularity lemma 183
Exercises 189
Notes 192
8. Infinite Graphs 195
8.1 Basic notions, facts and techniques* 196
8.2 Paths, trees, and ends \( {}^{\left( *\right) } \) . 204
8.3 Homogeneous and universal graphs* 212
8.4 Connectivity and matching 216
8.5 The topological end space 226
Exercises 237
Notes 244
9. Ramsey Theory for Graphs 251
9.1 Ramsey's original theorems* 252
9.2 Ramsey numbers \( {}^{\left( *\right) } \) . 255
9.3 Induced Ramsey theorems 258
9.4 Ramsey properties and connectivity \( {}^{\left( *\right) } \) . 268
Exercises 271
Notes 272
10. Hamilton Cycles 275
10.1 Simple sufficient conditions* 275
10.2 Hamilton cycles and degree sequences* 278
10.3 Hamilton cycles in the square of a graph 281
Exercises 289
Notes 290
11. Random Graphs 293
11.1 The notion of a random graph* 294
11.2 The probabilistic method* 299
11.3 Properties of almost all graphs* 302
11.4 Threshold functions and second moments 306
Exercises 312
Notes 313
12. Minors, Trees and WQO 315
12.1 Well-quasi-ordering* 316
12.2 The graph minor theorem for trees* 317
12.3 Tree-decompositions 319
12.4 Tree-width and forbidden minors 327
12.5 The graph mino |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 2 | Homogeneous and universal graphs* 212
8.4 Connectivity and matching 216
8.5 The topological end space 226
Exercises 237
Notes 244
9. Ramsey Theory for Graphs 251
9.1 Ramsey's original theorems* 252
9.2 Ramsey numbers \( {}^{\left( *\right) } \) . 255
9.3 Induced Ramsey theorems 258
9.4 Ramsey properties and connectivity \( {}^{\left( *\right) } \) . 268
Exercises 271
Notes 272
10. Hamilton Cycles 275
10.1 Simple sufficient conditions* 275
10.2 Hamilton cycles and degree sequences* 278
10.3 Hamilton cycles in the square of a graph 281
Exercises 289
Notes 290
11. Random Graphs 293
11.1 The notion of a random graph* 294
11.2 The probabilistic method* 299
11.3 Properties of almost all graphs* 302
11.4 Threshold functions and second moments 306
Exercises 312
Notes 313
12. Minors, Trees and WQO 315
12.1 Well-quasi-ordering* 316
12.2 The graph minor theorem for trees* 317
12.3 Tree-decompositions 319
12.4 Tree-width and forbidden minors 327
12.5 The graph minor theorem \( {}^{\left( *\right) } \) . 341
Exercises 350
Notes 354
A. Infinite sets 357
B. Surfaces 361
Hints for all the exercises. . 369
Index . 393
Symbol index . 409
1
## The Basics
This chapter gives a gentle yet concise introduction to most of the terminology used later in the book. Fortunately, much of standard graph theoretic terminology is so intuitive that it is easy to remember; the few terms better understood in their proper setting will be introduced later, when their time has come.
Section 1.1 offers a brief but self-contained summary of the most basic definitions in graph theory, those centred round the notion of a graph. Most readers will have met these definitions before, or will have them explained to them as they begin to read this book. For this reason, Section 1.1 does not dwell on these definitions more than clarity requires: its main purpose is to collect the most basic terms in one place, for easy reference later.
From Section 1.2 onwards, all new definitions will be brought to life almost immediately by a number of simple yet fundamental propositions. Often, these will relate the newly defined terms to one another: the question of how the value of one invariant influences that of another underlies much of graph theory, and it will be good to become familiar with this line of thinking early.
By \( \mathbb{N} \) we denote the set of natural numbers, including zero. The set \( \mathbb{Z}/n\mathbb{Z} \) of integers modulo \( n \) is denoted by \( {\mathbb{Z}}_{n} \) ; its elements are written \( {\mathbb{Z}}_{n} \) as \( \bar{i} \mathrel{\text{:=}} i + n\mathbb{Z} \) . For a real number \( x \) we denote by \( \lfloor x\rfloor \) the greatest integer \( \leq x \), and by \( \lceil x\rceil \) the least integer \( \geq x \) . Logarithms written as \( \lfloor x\rfloor ,\lceil x\rceil \) ’log’ are taken at base 2; the natural logarithm will be denoted by ’ln’. \( \log ,\ln \) A set \( \mathcal{A} = \left\{ {{A}_{1},\ldots ,{A}_{k}}\right\} \) of disjoint subsets of a set \( A \) is a partition partition of \( A \) if the union \( \bigcup \mathcal{A} \) of all the sets \( {A}_{i} \in A \) is \( A \) and \( {A}_{i} \neq \varnothing \) for every \( i \) . UA Another partition \( \left\{ {{A}_{1}^{\prime },\ldots ,{A}_{\ell }^{\prime }}\right\} \) of \( A \) refines the partition \( \mathcal{A} \) if each \( {A}_{i}^{\prime } \) is contained in some \( {A}_{j} \) . By \( {\left\lbrack A\right\rbrack }^{k} \) we denote the set of all \( k \) -element subsets
\( {\left\lbrack A\right\rbrack }^{k} \)
of \( A \) . Sets with \( k \) elements will be called \( k \) -sets; subsets with \( k \) elements are \( k \) -subsets. \( k \) -set
## 1.1 Graphs
graph A graph is a pair \( G = \left( {V, E}\right) \) of sets such that \( E \subseteq {\left\lbrack V\right\rbrack }^{2} \) ; thus, the elements of \( E \) are 2-element subsets of \( V \) . To avoid notational ambiguities, we shall always assume tacitly that \( V \cap E = \varnothing \) . The elements of \( V \) are the
---
vertex edge
---
vertices (or nodes, or points) of the graph \( G \), the elements of \( E \) are its edges (or lines). The usual way to picture a graph is by drawing a dot for each vertex and joining two of these dots by a line if the corresponding two vertices form an edge. Just how these dots and lines are drawn is considered irrelevant: all that matters is the information of which pairs of vertices form an edge and which do not.
Fig. 1.1.1. The graph on \( V = \{ 1,\ldots ,7\} \) with edge set
\[
E = \{ \{ 1,2\} ,\{ 1,5\} ,\{ 2,5\} ,\{ 3,4\} ,\{ 5,7\} \}
\]
on A graph with vertex set \( V \) is said to be a graph on \( V \) . The vertex
\( V\left( G\right), E\left( G\right) \) set of a graph \( G \) is referred to as \( V\left( G\right) \), its edge set as \( E\left( G\right) \) . These conventions are independent of any actual names of these two sets: the vertex set \( W \) of a graph \( H = \left( {W, F}\right) \) is still referred to as \( V\left( H\right) \), not as \( W\left( H\right) \) . We shall not always distinguish strictly between a graph and its vertex or edge set. For example, we may speak of a vertex \( v \in G \) (rather than \( v \in V\left( G\right) \) ), an edge \( e \in G \), and so on.
order The number of vertices of a graph \( G \) is its order, written as \( \left| G\right| \) ; its \( \left| G\right| ,\parallel G\parallel \) number of edges is denoted by \( \parallel G\parallel \) . Graphs are finite, infinite, countable and so on according to their order. Except in Chapter 8, our graphs will be finite unless otherwise stated.
O trivial For the empty graph \( \left( {\varnothing ,\varnothing }\right) \) we simply write \( \varnothing \) . A graph of order 0 or 1 graph is called trivial. Sometimes, e.g. to start an induction, trivial graphs can be useful; at other times they form silly counterexamples and become a nuisance. To avoid cluttering the text with non-triviality conditions, we shall mostly treat the trivial graphs, and particularly the empty graph \( \varnothing \) , with generous disregard.
incident A vertex \( v \) is incident with an edge \( e \) if \( v \in e \) ; then \( e \) is an edge at \( v \) . ends The two vertices incident with an edge are its endvertices or ends, and an edge joins its ends. An edge \( \{ x, y\} \) is usually written as \( {xy} \) (or \( {yx} \) ). If \( x \in X \) and \( y \in Y \), then \( {xy} \) is an \( X - Y \) edge. The set of all \( X - Y \) edges \( E\left( {X, Y}\right) \) in a set \( E \) is denoted by \( E\left( {X, Y}\right) \) ; instead of \( E\left( {\{ x\}, Y}\right) \) and \( E\left( {X,\{ y\} }\right) \) we simply write \( E\left( {x, Y}\right) \) and \( E\left( {X, y}\right) \) . The set of all the edges in \( E \) at a \( E\left( v\right) \) vertex \( v \) is denoted by \( E\left( v\right) \) .
Two vertices \( x, y \) of \( G \) are adjacent, or neighbours, if \( {xy} \) is an edge
---
adjacent neighbour complete \( {K}^{n} \)
---
of \( G \) . Two edges \( e \neq f \) are adjacent if they have an end in common. If all the vertices of \( G \) are pairwise adjacent, then \( G \) is complete. A complete graph on \( n \) vertices is a \( {K}^{n} \) ; a \( {K}^{3} \) is called a triangle.
Pairwise non-adjacent vertices or edges are called independent.
More formally, a set of vertices or of edges is independent (or stable) inde- pendent if no two of its elements are adjacent.
Let \( G = \left( {V, E}\right) \) and \( {G}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \) be two graphs. We call \( G \) and \( {G}^{\prime } \) isomorphic, and write \( G \simeq {G}^{\prime } \), if there exists a bijection \( \varphi : V \rightarrow {V}^{\prime } \) with \( {xy} \in E \Leftrightarrow \varphi \left( x\right) \varphi \left( y\right) \in {E}^{\prime } \) for all \( x, y \in V \) . Such a map \( \varphi \) is called an isomorphism; if \( G = {G}^{\prime } \), it is called an automorphism. We do not isomor-
phism
normally distinguish between isomorphic graphs. Thus, we usually write \( G = {G}^{\prime } \) rather than \( G \simeq {G}^{\prime } \), speak of the complete graph on 17 vertices, and so on.
A class of graphs that is closed under isomorphism is called a graph property. For example, 'containing a triangle' is a graph property: if
property
\( G \) contains three pairwise adjacent vertices then so does every graph isomorphic to \( G \) . A map taking graphs as arguments is called a graph invariant if it assigns equal values to isomorphic graphs. The number invariant of vertices and the number of edges of a graph are two simple graph invariants; the greatest number of pairwise adjacent vertices is another.
Fig. 1.1.2. Union, difference and intersection; the vertices 2,3,4 induce (or span) a triangle in \( G \cup {G}^{\prime } \) but not in \( G \)
We set \( G \cup {G}^{\prime } \mathrel{\text{:=}} \left( {V \cup {V}^{\prime }, E \cup {E}^{\prime }}\right) \) and \( G \cap {G}^{\prime } \mathrel{\text{:=}} \left( {V \cap {V}^{\prime }, E \cap {E}^{\prime }}\right) \) . \( G \cap {G}^{\prime } \) If \( G \cap {G}^{\prime } = \varnothing \), then \( G \) and \( {G}^{\prime } \) are disjoint. If \( {V}^{\prime } \subseteq V \) and \( {E}^{\prime } \subseteq E \), then subgraph \( {G}^{\prime } \) is a subgraph of \( G \) (and \( G \) a supergraph of \( {G}^{\prime } \) ), written as \( {G}^{\prime } \subseteq G \) . \( {G}^{\prime } \subseteq G \) Less formally, we say that \( G \) contains \( {G}^{\prime } \) . If \( {G}^{\prime } \subseteq G \) and \( {G}^{\prime } \neq G \), then \( {G}^{\prime } \) is a proper subgraph of \( G \) .
If \( {G}^{\prime } \subseteq G \) and \( {G}^{\prime } \) contains all the edges \( {xy} \i |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 3 | rsection; the vertices 2,3,4 induce (or span) a triangle in \( G \cup {G}^{\prime } \) but not in \( G \)
We set \( G \cup {G}^{\prime } \mathrel{\text{:=}} \left( {V \cup {V}^{\prime }, E \cup {E}^{\prime }}\right) \) and \( G \cap {G}^{\prime } \mathrel{\text{:=}} \left( {V \cap {V}^{\prime }, E \cap {E}^{\prime }}\right) \) . \( G \cap {G}^{\prime } \) If \( G \cap {G}^{\prime } = \varnothing \), then \( G \) and \( {G}^{\prime } \) are disjoint. If \( {V}^{\prime } \subseteq V \) and \( {E}^{\prime } \subseteq E \), then subgraph \( {G}^{\prime } \) is a subgraph of \( G \) (and \( G \) a supergraph of \( {G}^{\prime } \) ), written as \( {G}^{\prime } \subseteq G \) . \( {G}^{\prime } \subseteq G \) Less formally, we say that \( G \) contains \( {G}^{\prime } \) . If \( {G}^{\prime } \subseteq G \) and \( {G}^{\prime } \neq G \), then \( {G}^{\prime } \) is a proper subgraph of \( G \) .
If \( {G}^{\prime } \subseteq G \) and \( {G}^{\prime } \) contains all the edges \( {xy} \in E \) with \( x, y \in {V}^{\prime } \), then
\( {G}^{\prime } \) is an induced subgraph of \( G \) ; we say that \( {V}^{\prime } \) induces or spans \( {G}^{\prime } \) in \( G \) , induced subgraph
Fig. 1.1.3. A graph \( G \) with subgraphs \( {G}^{\prime } \) and \( {G}^{\prime \prime } \) :
\( {G}^{\prime } \) is an induced subgraph of \( G \), but \( {G}^{\prime \prime } \) is not
\( G\left\lbrack U\right\rbrack \) and write \( {G}^{\prime } = : G\left\lbrack {V}^{\prime }\right\rbrack \) . Thus if \( U \subseteq V \) is any set of vertices, then \( G\left\lbrack U\right\rbrack \) denotes the graph on \( U \) whose edges are precisely the edges of \( G \) with both ends in \( U \) . If \( H \) is a subgraph of \( G \), not necessarily induced, we spanning abbreviate \( G\left\lbrack {V\left( H\right) }\right\rbrack \) to \( G\left\lbrack H\right\rbrack \) . Finally, \( {G}^{\prime } \subseteq G \) is a spanning subgraph of \( G \) if \( {V}^{\prime } \) spans all of \( G \), i.e. if \( {V}^{\prime } = V \) .
If \( U \) is any set of vertices (usually of \( G \) ), we write \( G - U \) for \( G\left\lbrack {V \smallsetminus U}\right\rbrack \) . In other words, \( G - U \) is obtained from \( G \) by deleting all the vertices in \( U \cap V \) and their incident edges. If \( U = \{ v\} \) is a singleton, we write \( G - v \) rather than \( G - \{ v\} \) . Instead of \( G - V\left( {G}^{\prime }\right) \) we simply \( + \) write \( G - {G}^{\prime } \) . For a subset \( F \) of \( {\left\lbrack V\right\rbrack }^{2} \) we write \( G - F \mathrel{\text{:=}} \left( {V, E \smallsetminus F}\right) \) and \( G + F \mathrel{\text{:=}} \left( {V, E \cup F}\right) \) ; as above, \( G - \{ e\} \) and \( G + \{ e\} \) are abbreviated to edge- \( G - e \) and \( G + e \) . We call \( G \) edge-maximal with a given graph property maximal if \( G \) itself has the property but no graph \( G + {xy} \) does, for non-adjacent vertices \( x, y \in G \) .
minimal More generally, when we call a graph minimal or maximal with some maximal property but have not specified any particular ordering, we are referring to the subgraph relation. When we speak of minimal or maximal sets of vertices or edges, the reference is simply to set inclusion.
\( G * {G}^{\prime } \) If \( G \) and \( {G}^{\prime } \) are disjoint, we denote by \( G * {G}^{\prime } \) the graph obtained from \( G \cup {G}^{\prime } \) by joining all the vertices of \( G \) to all the vertices of \( {G}^{\prime } \) . For comple- example, \( {K}^{2} * {K}^{3} = {K}^{5} \) . The complement \( \bar{G} \) of \( G \) is the graph on \( V \) ment \( \bar{G} \) with edge set \( {\left\lbrack V\right\rbrack }^{2} \smallsetminus E \) . The line graph \( L\left( G\right) \) of \( G \) is the graph on \( E \) in line graph which \( x, y \in E \) are adjacent as vertices if and only if they are adjacent \( L\left( G\right) \) as edges in \( G \) .
Fig. 1.1.4. A graph isomorphic to its complement
## 1.2 The degree of a vertex
Let \( G = \left( {V, E}\right) \) be a (non-empty) graph. The set of neighbours of a vertex \( v \) in \( G \) is denoted by \( {N}_{G}\left( v\right) \), or briefly by \( N\left( v\right) {.}^{1} \) More generally
\( N\left( v\right) \)
for \( U \subseteq V \), the neighbours in \( V \smallsetminus U \) of vertices in \( U \) are called neighbours of \( U \) ; their set is denoted by \( N\left( U\right) \) .
The degree (or valency) \( {d}_{G}\left( v\right) = d\left( v\right) \) of a vertex \( v \) is the number
degree \( d\left( v\right) \)
\( \left| {E\left( v\right) }\right| \) of edges at \( v \) ; by our definition of a graph, \( {}^{2} \) this is equal to the number of neighbours of \( v \) . A vertex of degree 0 is isolated. The number isolated \( \delta \left( G\right) \mathrel{\text{:=}} \min \{ d\left( v\right) \mid v \in V\} \) is the minimum degree of \( G \), the number
\( \delta \left( G\right) \)
\( \Delta \left( G\right) \mathrel{\text{:=}} \max \{ d\left( v\right) \mid v \in V\} \) its maximum degree. If all the vertices
\( \Delta \left( G\right) \)
of \( G \) have the same degree \( k \), then \( G \) is \( k \) -regular, or simply regular. A regular 3-regular graph is called cubic. cubic
The number
\[
d\left( G\right) \mathrel{\text{:=}} \frac{1}{\left| V\right| }\mathop{\sum }\limits_{{v \in V}}d\left( v\right)
\]
\( d\left( G\right) \)
is the average degree of \( G \) . Clearly, average degree
\[
\delta \left( G\right) \leq d\left( G\right) \leq \Delta \left( G\right)
\]
The average degree quantifies globally what is measured locally by the vertex degrees: the number of edges of \( G \) per vertex. Sometimes it will be convenient to express this ratio directly, as \( \varepsilon \left( G\right) \mathrel{\text{:=}} \left| E\right| /\left| V\right| \) .
\( \varepsilon \left( G\right) \)
The quantities \( d \) and \( \varepsilon \) are, of course, intimately related. Indeed, if we sum up all the vertex degrees in \( G \), we count every edge exactly twice: once from each of its ends. Thus
\[
\left| E\right| = \frac{1}{2}\mathop{\sum }\limits_{{v \in V}}d\left( v\right) = \frac{1}{2}d\left( G\right) \cdot \left| V\right|
\]
and therefore
\[
\varepsilon \left( G\right) = \frac{1}{2}d\left( G\right) .
\]
Proposition 1.2.1. The number of vertices of odd degree in a graph is
\( \left\lbrack {10.3.3}\right\rbrack \)
always even.
Proof. A graph on \( V \) has \( \frac{1}{2}\mathop{\sum }\limits_{{v \in V}}d\left( v\right) \) edges, so \( \sum d\left( v\right) \) is an even number.
---
\( {}^{1} \) Here, as elsewhere, we drop the index referring to the underlying graph if the reference is clear.
2 but not for multigraphs; see Section 1.10
---
If a graph has large minimum degree, i.e. everywhere, locally, many edges per vertex, it also has many edges per vertex globally: \( \varepsilon \left( G\right) = \) \( \frac{1}{2}d\left( G\right) \geq \frac{1}{2}\delta \left( G\right) \) . Conversely, of course, its average degree may be large even when its minimum degree is small. However, the vertices of large degree cannot be scattered completely among vertices of small degree: as the next proposition shows, every graph \( G \) has a subgraph whose average degree is no less than the average degree of \( G \), and whose minimum degree is more than half its average degree:
Proposition 1.2.2. Every graph \( G \) with at least one edge has a subgraph \( H \) with \( \delta \left( H\right) > \varepsilon \left( H\right) \geq \varepsilon \left( G\right) \) .
Proof. To construct \( H \) from \( G \), let us try to delete vertices of small degree one by one, until only vertices of large degree remain. Up to which degree \( d\left( v\right) \) can we afford to delete a vertex \( v \), without lowering \( \varepsilon \) ? Clearly, up to \( d\left( v\right) = \varepsilon \) : then the number of vertices decreases by 1 and the number of edges by at most \( \varepsilon \), so the overall ratio \( \varepsilon \) of edges to vertices will not decrease.
Formally, we construct a sequence \( G = {G}_{0} \supseteq {G}_{1} \supseteq \ldots \) of induced subgraphs of \( G \) as follows. If \( {G}_{i} \) has a vertex \( {v}_{i} \) of degree \( d\left( {v}_{i}\right) \leq \varepsilon \left( {G}_{i}\right) \) , we let \( {G}_{i + 1} \mathrel{\text{:=}} {G}_{i} - {v}_{i} \) ; if not, we terminate our sequence and set \( H \mathrel{\text{:=}} {G}_{i} \) . By the choices of \( {v}_{i} \) we have \( \varepsilon \left( {G}_{i + 1}\right) \geq \varepsilon \left( {G}_{i}\right) \) for all \( i \), and hence \( \varepsilon \left( H\right) \geq \varepsilon \left( G\right) \) .
What else can we say about the graph \( H \) ? Since \( \varepsilon \left( {K}^{1}\right) = 0 < \varepsilon \left( G\right) \) , none of the graphs in our sequence is trivial, so in particular \( H \neq \varnothing \) . The fact that \( H \) has no vertex suitable for deletion thus implies \( \delta \left( H\right) > \varepsilon \left( H\right) \) , as claimed.
## 1.3 Paths and cycles
path A path is a non-empty graph \( P = \left( {V, E}\right) \) of the form
\[
V = \left\{ {{x}_{0},{x}_{1},\ldots ,{x}_{k}}\right\} \;E = \left\{ {{x}_{0}{x}_{1},{x}_{1}{x}_{2},\ldots ,{x}_{k - 1}{x}_{k}}\right\} ,
\]
where the \( {x}_{i} \) are all distinct. The vertices \( {x}_{0} \) and \( {x}_{k} \) are linked by \( P \) and are called its ends; the vertices \( {x}_{1},\ldots ,{x}_{k - 1} \) are the inner vertices of \( P \) .
---
length \( {P}^{k} \)
---
The number of edges of a path is its length, and the path of length \( k \) is denoted by \( {P}^{k} \) . Note that \( k \) is allowed to be zero; thus, \( {P}^{0} = {K}^{1} \) .
We often refer to a path by the natural sequence of its v |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 4 | arepsilon \left( G\right) \) , none of the graphs in our sequence is trivial, so in particular \( H \neq \varnothing \) . The fact that \( H \) has no vertex suitable for deletion thus implies \( \delta \left( H\right) > \varepsilon \left( H\right) \) , as claimed.
## 1.3 Paths and cycles
path A path is a non-empty graph \( P = \left( {V, E}\right) \) of the form
\[
V = \left\{ {{x}_{0},{x}_{1},\ldots ,{x}_{k}}\right\} \;E = \left\{ {{x}_{0}{x}_{1},{x}_{1}{x}_{2},\ldots ,{x}_{k - 1}{x}_{k}}\right\} ,
\]
where the \( {x}_{i} \) are all distinct. The vertices \( {x}_{0} \) and \( {x}_{k} \) are linked by \( P \) and are called its ends; the vertices \( {x}_{1},\ldots ,{x}_{k - 1} \) are the inner vertices of \( P \) .
---
length \( {P}^{k} \)
---
The number of edges of a path is its length, and the path of length \( k \) is denoted by \( {P}^{k} \) . Note that \( k \) is allowed to be zero; thus, \( {P}^{0} = {K}^{1} \) .
We often refer to a path by the natural sequence of its vertices, \( {}^{3} \) writing, say, \( P = {x}_{0}{x}_{1}\ldots {x}_{k} \) and calling \( P \) a path from \( {x}_{0} \) to \( {x}_{k} \) (as well as between \( {x}_{0} \) and \( {x}_{k} \) ).
---
3 More precisely, by one of the two natural sequences: \( {x}_{0}\ldots {x}_{k} \) and \( {x}_{k}\ldots {x}_{0} \) denote the same path. Still, it often helps to fix one of these two orderings of \( V\left( P\right) \) notationally: we may then speak of things like the ’first’ vertex on \( P \) with a certain property, etc.
---
Fig. 1.3.1. A path \( P = {P}^{6} \) in \( G \)
For \( 0 \leq i \leq j \leq k \) we write \( {xPy},\overset{ \circ }{P} \)
\[
P{x}_{i} \mathrel{\text{:=}} {x}_{0}\ldots {x}_{i}
\]
\[
{x}_{i}P \mathrel{\text{:=}} {x}_{i}\ldots {x}_{k}
\]
\[
{x}_{i}P{x}_{j} \mathrel{\text{:=}} {x}_{i}\ldots {x}_{j}
\]
and
\[
\overset{ \circ }{P} \mathrel{\text{:=}} {x}_{1}\ldots {x}_{k - 1}
\]
\[
P{\overset{ \circ }{x}}_{i} \mathrel{\text{:=}} {x}_{0}\ldots {x}_{i - 1}
\]
\[
{\overset{ \circ }{x}}_{i}P \mathrel{\text{:=}} {x}_{i + 1}\ldots {x}_{k}
\]
\[
{\overset{ \circ }{x}}_{i}P{\overset{ \circ }{x}}_{j} \mathrel{\text{:=}} {x}_{i + 1}\ldots {x}_{j - 1}
\]
for the appropriate subpaths of \( P \) . We use similar intuitive notation for the concatenation of paths; for example, if the union \( {Px} \cup {xQy} \cup {yR} \) of
three paths is again a path, we may simply denote it by \( {PxQyR} \) . \( {PxQyR} \)
Fig. 1.3.2. Paths \( P, Q \) and \( {xPyQz} \)
Given sets \( A, B \) of vertices, we call \( P = {x}_{0}\ldots {x}_{k} \) an \( A - B \) path if \( A - B \) path \( V\left( P\right) \cap A = \left\{ {x}_{0}\right\} \) and \( V\left( P\right) \cap B = \left\{ {x}_{k}\right\} \) . As before, we write \( a - B \) path rather than \( \{ a\} - B \) path, etc. Two or more paths are independent inde- pendent if none of them contains an inner vertex of another. Two \( a - b \) paths, for instance, are independent if and only if \( a \) and \( b \) are their only common vertices.
Given a graph \( H \), we call \( P \) an \( H \) -path if \( P \) is non-trivial and meets
\( H \) -path
\( H \) exactly in its ends. In particular, the edge of any \( H \) -path of length 1 is never an edge of \( H \) .
If \( P = {x}_{0}\ldots {x}_{k - 1} \) is a path and \( k \geq 3 \), then the graph \( C \mathrel{\text{:=}} \) \( P + {x}_{k - 1}{x}_{0} \) is called a cycle. As with paths, we often denote a cycle cycle by its (cyclic) sequence of vertices; the above cycle \( C \) might be written
length as \( {x}_{0}\ldots {x}_{k - 1}{x}_{0} \) . The length of a cycle is its number of edges (or vertices); \( {C}^{k} \) the cycle of length \( k \) is called a \( k \) -cycle and denoted by \( {C}^{k} \) .
girth \( g\left( G\right) \) The minimum length of a cycle (contained) in a graph \( G \) is the girth circum- \( g\left( G\right) \) of \( G \) ; the maximum length of a cycle in \( G \) is its circumference. (If ference \( G \) does not contain a cycle, we set the former to \( \infty \), the latter to zero.) chord An edge which joins two vertices of a cycle but is not itself an edge of the cycle is a chord of that cycle. Thus, an induced cycle in \( G \), a cycle in induced \( G \) forming an induced subgraph, is one that has no chords (Fig. 1.3.3). cycle
Fig. 1.3.3. A cycle \( {C}^{8} \) with chord \( {xy} \), and induced cycles \( {C}^{6},{C}^{4} \)
If a graph has large minimum degree, it contains long paths and cycles (see also Exercise 7):
\( \left\lbrack \begin{array}{l} {1.4.3} \\ {3.5.1} \end{array}\right\rbrack \) Proposition 1.3.1. Every graph \( G \) contains a path of length \( \delta \left( G\right) \) and a cycle of length at least \( \delta \left( G\right) + 1 \) (provided that \( \delta \left( G\right) \geq 2 \) ).
Proof. Let \( {x}_{0}\ldots {x}_{k} \) be a longest path in \( G \) . Then all the neighbours of \( {x}_{k} \) lie on this path (Fig. 1.3.4). Hence \( k \geq d\left( {x}_{k}\right) \geq \delta \left( G\right) \) . If \( i < k \) is minimal with \( {x}_{i}{x}_{k} \in E\left( G\right) \), then \( {x}_{i}\ldots {x}_{k}{x}_{i} \) is a cycle of length at least \( \delta \left( G\right) + 1 \) .
Fig. 1.3.4. A longest path \( {x}_{0}\ldots {x}_{k} \), and the neighbours of \( {x}_{k} \)
Minimum degree and girth, on the other hand, are not related (unless we fix the number of vertices): as we shall see in Chapter 11, there are graphs combining arbitrarily large minimum degree with arbitrarily
---
distance \( d\left( {x, y}\right) \)
---
large girth.
The distance \( {d}_{G}\left( {x, y}\right) \) in \( G \) of two vertices \( x, y \) is the length of a shortest \( x - y \) path in \( G \) ; if no such path exists, we set \( d\left( {x, y}\right) \mathrel{\text{:=}} \infty \) . The
---
diameter \( \operatorname{diam}G \)
---
greatest distance between any two vertices in \( G \) is the diameter of \( G \) , denoted by \( \operatorname{diam}G \) . Diameter and girth are, of course, related:
Proposition 1.3.2. Every graph \( G \) containing a cycle satisfies \( g\left( G\right) \leq \) \( 2\operatorname{diam}G + 1 \) .
Proof. Let \( C \) be a shortest cycle in \( G \) . If \( g\left( G\right) \geq 2\operatorname{diam}G + 2 \), then \( C \) has two vertices whose distance in \( C \) is at least \( \operatorname{diam}G + 1 \) . In \( G \) , these vertices have a lesser distance; any shortest path \( P \) between them is therefore not a subgraph of \( C \) . Thus, \( P \) contains a \( C \) -path \( {xPy} \) . Together with the shorter of the two \( x - y \) paths in \( C \), this path \( {xPy} \) forms a shorter cycle than \( C \), a contradiction.
A vertex is central in \( G \) if its greatest distance from any other vertex central is as small as possible. This distance is the radius of \( G \), denoted by \( \operatorname{rad}G \) . Thus, formally, \( \operatorname{rad}G = \mathop{\min }\limits_{{x \in V\left( G\right) }}\mathop{\max }\limits_{{y \in V\left( G\right) }}{d}_{G}\left( {x, y}\right) \) . As one easily radius \( \operatorname{rad}G \) checks (exercise), we have
\[
\operatorname{rad}G \leq \operatorname{diam}G \leq 2\operatorname{rad}G.
\]
Diameter and radius are not related to minimum, average or maximum degree if we say nothing about the order of the graph. However, graphs of large diameter and minimum degree are clearly large (much larger than forced by each of the two parameters alone; see Exercise 8), and graphs of small diameter and maximum degree must be small:
Proposition 1.3.3. A graph \( G \) of radius at most \( k \) and maximum degree at most \( d \geq 3 \) has fewer than \( \frac{d}{d - 2}{\left( d - 1\right) }^{k} \) vertices.
Proof. Let \( z \) be a central vertex in \( G \), and let \( {D}_{i} \) denote the set of vertices of \( G \) at distance \( i \) from \( z \) . Then \( V\left( G\right) = \mathop{\bigcup }\limits_{{i = 0}}^{k}{D}_{i} \) . Clearly \( \left| {D}_{0}\right| = 1 \) and \( \left| {D}_{1}\right| \leq d \) . For \( i \geq 1 \) we have \( \left| {D}_{i + 1}\right| \leq \left( {d - 1}\right) \left| {D}_{i}\right| \), because every vertex in \( {D}_{i + 1} \) is a neighbour of a vertex in \( {D}_{i} \), and each vertex in \( {D}_{i} \) has at most \( d - 1 \) neighbours in \( {D}_{i + 1} \) (since it has another neighbour in \( \left. {D}_{i - 1}\right) \) . Thus \( \left| {D}_{i + 1}\right| \leq d{\left( d - 1\right) }^{i} \) for all \( i < k \) by induction, giving
\[
\left| G\right| \leq 1 + d\mathop{\sum }\limits_{{i = 0}}^{{k - 1}}{\left( d - 1\right) }^{i} = 1 + \frac{d}{d - 2}\left( {{\left( d - 1\right) }^{k} - 1}\right) < \frac{d}{d - 2}{\left( d - 1\right) }^{k}.
\]
Similarly, we can bound the order of \( G \) from below by assuming that both its minimum degree and girth are large. For \( d \in \mathbb{R} \) and \( g \in \mathbb{N} \) let
\[
{n}_{0}\left( {d, g}\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} 1 + d\mathop{\sum }\limits_{{i = 0}}^{{r - 1}}{\left( d - 1\right) }^{i} & \text{ if }g = : {2r} + 1\text{ is odd; } \\ 2\mathop{\sum }\limits_{{i = 0}}^{{r - 1}}{\left( d - 1\right) }^{i} & \text{ if }g = : {2r}\text{ is even. } \end{array}\right.
\]
It is not difficult to prove that a graph of minimum degree \( \delta \) and girth \( g \) has at least \( {n}_{0}\left( {\delta, g}\right) \) vertices (Exercise 6). Interestingly, one can obtain the same bound for its average degree:
Theorem 1.3.4. (Alon, Hoory & Linial 2002)
Let \( G \) be a graph. If \( d\left( G\right) \geq d \geq 2 \) and \( g\left( G\right) \geq g \in \ |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 5 | + \frac{d}{d - 2}\left( {{\left( d - 1\right) }^{k} - 1}\right) < \frac{d}{d - 2}{\left( d - 1\right) }^{k}.
\]
Similarly, we can bound the order of \( G \) from below by assuming that both its minimum degree and girth are large. For \( d \in \mathbb{R} \) and \( g \in \mathbb{N} \) let
\[
{n}_{0}\left( {d, g}\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} 1 + d\mathop{\sum }\limits_{{i = 0}}^{{r - 1}}{\left( d - 1\right) }^{i} & \text{ if }g = : {2r} + 1\text{ is odd; } \\ 2\mathop{\sum }\limits_{{i = 0}}^{{r - 1}}{\left( d - 1\right) }^{i} & \text{ if }g = : {2r}\text{ is even. } \end{array}\right.
\]
It is not difficult to prove that a graph of minimum degree \( \delta \) and girth \( g \) has at least \( {n}_{0}\left( {\delta, g}\right) \) vertices (Exercise 6). Interestingly, one can obtain the same bound for its average degree:
Theorem 1.3.4. (Alon, Hoory & Linial 2002)
Let \( G \) be a graph. If \( d\left( G\right) \geq d \geq 2 \) and \( g\left( G\right) \geq g \in \mathbb{N} \) then \( \left| G\right| \geq {n}_{0}\left( {d, g}\right) \) .
One aspect of Theorem 1.3.4 is that it guarantees the existence of a short cycle compared with \( \left| G\right| \) . Using just the easy minimum degree version of Exercise 6, we get the following rather general bound:
2.3.1] Corollary 1.3.5. If \( \delta \left( G\right) \geq 3 \) then \( g\left( G\right) < 2\log \left| G\right| \) .
Proof. If \( g \mathrel{\text{:=}} g\left( G\right) \) is even then
\[
{n}_{0}\left( {3, g}\right) = 2\frac{{2}^{g/2} - 1}{2 - 1} = {2}^{g/2} + \left( {{2}^{g/2} - 2}\right) > {2}^{g/2},
\]
while if \( g \) is odd then
\[
{n}_{0}\left( {3, g}\right) = 1 + 3\frac{{2}^{\left( {g - 1}\right) /2} - 1}{2 - 1} = \frac{3}{\sqrt{2}}{2}^{g/2} - 2 > {2}^{g/2}.
\]
As \( \left| G\right| \geq {n}_{0}\left( {3, g}\right) \), the result follows.
walk A walk (of length \( k \) ) in a graph \( G \) is a non-empty alternating sequence \( {v}_{0}{e}_{0}{v}_{1}{e}_{1}\ldots {e}_{k - 1}{v}_{k} \) of vertices and edges in \( G \) such that \( {e}_{i} = \) \( \left\{ {{v}_{i},{v}_{i + 1}}\right\} \) for all \( i < k \) . If \( {v}_{0} = {v}_{k} \), the walk is closed. If the vertices in a walk are all distinct, it defines an obvious path in \( G \) . In general, every walk between two vertices contains \( {}^{4} \) a path between these vertices (proof?).
## 1.4 Connectivity
connected A non-empty graph \( G \) is called connected if any two of its vertices are linked by a path in \( G \) . If \( U \subseteq V\left( G\right) \) and \( G\left\lbrack U\right\rbrack \) is connected, we also call \( U \) itself connected (in \( G \) ). Instead of ’not connected’ we usually say 'disconnected'.
\( \left\lbrack {1.5.2}\right\rbrack \)
Proposition 1.4.1. The vertices of a connected graph \( G \) can always be enumerated, say as \( {v}_{1},\ldots ,{v}_{n} \), so that \( {G}_{i} \mathrel{\text{:=}} G\left\lbrack {{v}_{1},\ldots ,{v}_{i}}\right\rbrack \) is connected for every \( i \) .
Proof. Pick any vertex as \( {v}_{1} \), and assume inductively that \( {v}_{1},\ldots ,{v}_{i} \) have been chosen for some \( i < \left| G\right| \) . Now pick a vertex \( v \in G - {G}_{i} \) . As \( G \) is connected, it contains a \( v - {v}_{1} \) path \( P \) . Choose as \( {v}_{i + 1} \) the last vertex of \( P \) in \( G - {G}_{i} \) ; then \( {v}_{i + 1} \) has a neighbour in \( {G}_{i} \) . The connectedness of every \( {G}_{i} \) follows by induction on \( i \) .
---
4 We shall often use terms defined for graphs also for walks, as long as their meaning is obvious.
---
Let \( G = \left( {V, E}\right) \) be a graph. A maximal connected subgraph of \( G \)
is called a component of \( G \) . Note that a component, being connected, is component always non-empty; the empty graph, therefore, has no components.
Fig. 1.4.1. A graph with three components, and a minimal spanning connected subgraph in each component
If \( A, B \subseteq V \) and \( X \subseteq V \cup E \) are such that every \( A - B \) path in \( G \)
contains a vertex or an edge from \( X \), we say that \( X \) separates the sets \( A \) separate and \( B \) in \( G \) . Note that this implies \( A \cap B \subseteq X \) . More generally we say that \( X \) separates \( G \) if \( G - X \) is disconnected, that is, if \( X \) separates in \( G \) some two vertices that are not in \( X \) . A separating set of vertices is a
separator. Separating sets of edges have no generic name, but some such separator sets do; see Section 1.9 for the definition of cuts and bonds. A vertex cutvertex which separates two other vertices of the same component is a cutvertex,
and an edge separating its ends is a bridge. Thus, the bridges in a graph bridge are precisely those edges that do not lie on any cycle.
Fig. 1.4.2. A graph with cutvertices \( v, x, y, w \) and bridge \( e = {xy} \)
The unordered pair \( \{ A, B\} \) is a separation of \( G \) if \( A \cup B = V \) and \( G \)
separation
has no edge between \( A \smallsetminus B \) and \( B \smallsetminus A \) . Clearly, the latter is equivalent to saying that \( A \cap B \) separates \( A \) from \( B \) . If both \( A \smallsetminus B \) and \( B \smallsetminus A \) are non-empty, the separation is proper. The number \( \left| {A \cap B}\right| \) is the order of the separation \( \{ A, B\} \) .
\( G \) is called \( k \) -connected (for \( k \in \mathbb{N} \) ) if \( \left| G\right| > k \) and \( G - X \) is connected
\( k \) -connected
for every set \( X \subseteq V \) with \( \left| X\right| < k \) . In other words, no two vertices of \( G \) are separated by fewer than \( k \) other vertices. Every (non-empty) graph is 0-connected, and the 1-connected graphs are precisely the non-trivial connected graphs. The greatest integer \( k \) such that \( G \) is \( k \) -connected
is the connectivity \( \kappa \left( G\right) \) of \( G \) . Thus, \( \kappa \left( G\right) = 0 \) if and only if \( G \) is connectivity
\( \kappa \left( G\right) \)
disconnected or a \( {K}^{1} \), and \( \kappa \left( {K}^{n}\right) = n - 1 \) for all \( n \geq 1 \) .
---
\( \ell \) -edge-
connected
edge-
connectivity
\( \lambda \left( G\right) \)
---
If \( \left| G\right| > 1 \) and \( G - F \) is connected for every set \( F \subseteq E \) of fewer than \( \ell \) edges, then \( G \) is called \( \ell \) -edge-connected. The greatest integer \( \ell \) such that \( G \) is \( \ell \) -edge-connected is the edge-connectivity \( \lambda \left( G\right) \) of \( G \) . In particular, we have \( \lambda \left( G\right) = 0 \) if \( G \) is disconnected.
Fig. 1.4.3. The octahedron \( G \) (left) with \( \kappa \left( G\right) = \lambda \left( G\right) = 4 \) ,
and a graph \( H \) with \( \kappa \left( H\right) = 2 \) but \( \lambda \left( H\right) = 4 \)
## Proposition 1.4.2. If \( G \) is non-trivial then \( \kappa \left( G\right) \leq \lambda \left( G\right) \leq \delta \left( G\right) \) .
Proof. The second inequality follows from the fact that all the edges incident with a fixed vertex separate \( G \) . To prove the first, let \( F \) be any minimal subset of \( E \) such that \( G - F \) is disconnected. We show that \( \kappa \left( G\right) \leq \left| F\right| \) .
Suppose first that \( G \) has a vertex \( v \) that is not incident with an edge in \( F \) . Let \( C \) be the component of \( G - F \) containing \( v \) . Then the vertices of \( C \) that are incident with an edge in \( F \) separate \( v \) from \( G - C \) . Since no edge in \( F \) has both ends in \( C \) (by the minimality of \( F \) ), there are at most \( \left| F\right| \) such vertices, giving \( \kappa \left( G\right) \leq \left| F\right| \) as desired.
Suppose now that every vertex is incident with an edge in \( F \) . Let \( v \) be any vertex, and let \( C \) be the component of \( G - F \) containing \( v \) . Then the neighbours \( w \) of \( v \) with \( {vw} \notin F \) lie in \( C \) and are incident with distinct edges in \( F \), giving \( {d}_{G}\left( v\right) \leq \left| F\right| \) . As \( {N}_{G}\left( v\right) \) separates \( v \) from all the other vertices in \( G \), this yields \( \kappa \left( G\right) \leq \left| F\right| \) -unless there are no other vertices, i.e. unless \( \{ v\} \cup N\left( v\right) = V \) . But \( v \) was an arbitrary vertex. So we may assume that \( G \) is complete, giving \( \kappa \left( G\right) = \lambda \left( G\right) = \left| G\right| - 1 \) .
By Proposition 1.4.2, high connectivity requires a large minimum degree. Conversely, large minimum degree does not ensure high connectivity, not even high edge-connectivity (examples?). It does, however, imply the existence of a highly connected subgraph:
## Theorem 1.4.3. (Mader 1972)
Let \( 0 \neq k \in \mathbb{N} \) . Every graph \( G \) with \( d\left( G\right) \geq {4k} \) has a \( \left( {k + 1}\right) \) -connected subgraph \( H \) such that \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - k \) .
Proof. Put \( \gamma \mathrel{\text{:=}} \varepsilon \left( G\right) \left( { \geq {2k}}\right) \), and consider the subgraphs \( {G}^{\prime } \subseteq G \) such that
\[
\left| {G}^{\prime }\right| \geq {2k}\;\text{ and }\;\begin{Vmatrix}{G}^{\prime }\end{Vmatrix} > \gamma \left( {\left| {G}^{\prime }\right| - k}\right) .
\]
\( \left( *\right) \)
Such graphs \( {G}^{\prime } \) exist since \( G \) is one; let \( H \) be one of smallest order. \( H \)
No graph \( {G}^{\prime } \) as in \( \left( *\right) \) can have order exactly \( {2k} \), since this would imply that \( \begin{Vmatrix}{G}^{\prime }\end{Vmatrix} > {\gamma k} \geq 2{k}^{2} > \l |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 6 | ). It does, however, imply the existence of a highly connected subgraph:
## Theorem 1.4.3. (Mader 1972)
Let \( 0 \neq k \in \mathbb{N} \) . Every graph \( G \) with \( d\left( G\right) \geq {4k} \) has a \( \left( {k + 1}\right) \) -connected subgraph \( H \) such that \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - k \) .
Proof. Put \( \gamma \mathrel{\text{:=}} \varepsilon \left( G\right) \left( { \geq {2k}}\right) \), and consider the subgraphs \( {G}^{\prime } \subseteq G \) such that
\[
\left| {G}^{\prime }\right| \geq {2k}\;\text{ and }\;\begin{Vmatrix}{G}^{\prime }\end{Vmatrix} > \gamma \left( {\left| {G}^{\prime }\right| - k}\right) .
\]
\( \left( *\right) \)
Such graphs \( {G}^{\prime } \) exist since \( G \) is one; let \( H \) be one of smallest order. \( H \)
No graph \( {G}^{\prime } \) as in \( \left( *\right) \) can have order exactly \( {2k} \), since this would imply that \( \begin{Vmatrix}{G}^{\prime }\end{Vmatrix} > {\gamma k} \geq 2{k}^{2} > \left( \begin{matrix} \left| {G}^{\prime }\right| \\ 2 \end{matrix}\right) \) . The minimality of \( H \) therefore implies that \( \delta \left( H\right) > \gamma \) : otherwise we could delete a vertex of degree at most \( \gamma \) and obtain a graph \( {G}^{\prime } \subsetneq H \) still satisfying \( \left( *\right) \) . In particular, we have \( \left| H\right| \geq \gamma \) . Dividing the inequality of \( \parallel H\parallel > \gamma \left| H\right| - {\gamma k} \) from (*) by \( \left| H\right| \) therefore yields \( \varepsilon \left( H\right) > \gamma - k \), as desired.
It remains to show that \( H \) is \( \left( {k + 1}\right) \) -connected. If not, then \( H \) has a proper separation \( \left\{ {{U}_{1},{U}_{2}}\right\} \) of order at most \( k \) ; put \( H\left\lbrack {U}_{i}\right\rbrack = : {H}_{i} \) . \( {H}_{1},{H}_{2} \) Since any vertex \( v \in {U}_{1} \smallsetminus {U}_{2} \) has all its \( d\left( v\right) \geq \delta \left( H\right) > \gamma \) neighbours from \( H \) in \( {H}_{1} \), we have \( \left| {H}_{1}\right| \geq \gamma \geq {2k} \) . Similarly, \( \left| {H}_{2}\right| \geq {2k} \) . As by the minimality of \( H \) neither \( {H}_{1} \) nor \( {H}_{2} \) satisfies \( \left( *\right) \), we further have
\[
\begin{Vmatrix}{H}_{i}\end{Vmatrix} \leq \gamma \left( {\left| {H}_{i}\right| - k}\right)
\]
for \( i = 1,2 \) . But then
\[
\parallel H\parallel \leq \begin{Vmatrix}{H}_{1}\end{Vmatrix} + \begin{Vmatrix}{H}_{2}\end{Vmatrix}
\]
\[
\leq \gamma \left( {\left| {H}_{1}\right| + \left| {H}_{2}\right| - {2k}}\right)
\]
\[
\leq \gamma \left( {\left| H\right| - k}\right) \;\left( {\text{ as }\left| {{H}_{1} \cap {H}_{2}}\right| \leq k}\right)
\]
which contradicts \( \left( *\right) \) for \( H \) .
## 1.5 Trees and forests
An acyclic graph, one not containing any cycles, is called a forest. A con- forest nected forest is called a tree. (Thus, a forest is a graph whose components tree are trees.) The vertices of degree 1 in a tree are its leaves. \( {}^{5} \) Every non- leaf trivial tree has a leaf consider, for example, the ends of a longest path. This little fact often comes in handy, especially in induction proofs about trees: if we remove a leaf from a tree, what remains is still a tree.
---
5 ... except that the root of a tree (see below) is never called a leaf, even if it has degree 1.
---
Fig. 1.5.1. A tree
\( \left\lbrack \begin{array}{l} {1.6.1} \\ {1.9.6} \\ \left\lbrack {4.2.9}\right\rbrack \end{array}\right\rbrack \) Theorem 1.5.1. The following assertions are equivalent for a graph \( T \) :
(i) \( T \) is a tree;
(ii) Any two vertices of \( T \) are linked by a unique path in \( T \) ;
(iii) \( T \) is minimally connected, i.e. \( T \) is connected but \( T - e \) is disconnected for every edge \( e \in T \) ;
(iv) \( T \) is maximally acyclic, i.e. \( T \) contains no cycle but \( T + {xy} \) does, for any two non-adjacent vertices \( x, y \in T \) .
The proof of Theorem 1.5.1 is straightforward, and a good exercise for anyone not yet familiar with all the notions it relates. Extending our \( {xTy} \) notation for paths from Section 1.3, we write \( {xTy} \) for the unique path in a tree \( T \) between two vertices \( x, y \) (see (ii) above).
A frequently used application of Theorem 1.5.1 is that every connected graph contains a spanning tree: by the equivalence of (i) and (iii), any minimal connected spanning subgraph will be a tree. Figure 1.4.1 shows a spanning tree in each of the three components of the graph depicted.
Corollary 1.5.2. The vertices of a tree can always be enumerated, say as \( {v}_{1},\ldots ,{v}_{n} \), so that every \( {v}_{i} \) with \( i \geq 2 \) has a unique neighbour in \( \left\{ {{v}_{1},\ldots ,{v}_{i - 1}}\right\} \)
(1.4.1)
Proof. Use the enumeration from Proposition 1.4.1.
\( \begin{array}{l} \left\lbrack {1.9.6}\right\rbrack \\ \left\lbrack {2.4.1}\right\rbrack \\ \left\lbrack {2.4.4}\right\rbrack \\ \left\lbrack {4.2.9}\right\rbrack \end{array} \) Corollary 1.5.3. A connected graph with \( n \) vertices is a tree if and only if it has \( n - 1 \) edges. Proof. Induction on \( i \) shows that the subgraph spanned by the first \( i \) vertices in Corollary 1.5.2 has \( i - 1 \) edges; for \( i = n \) this proves the forward implication. Conversely, let \( G \) be any connected graph with \( n \) vertices and \( n - 1 \) edges. Let \( {G}^{\prime } \) be a spanning tree in \( G \) . Since \( {G}^{\prime } \) has \( n - 1 \) edges by the first implication, it follows that \( G = {G}^{\prime } \) .
Corollary 1.5.4. If \( T \) is a tree and \( G \) is any graph with \( \delta \left( G\right) \geq \left| T\right| - 1 \) ,
\( \left\lbrack \begin{array}{l} {9.2.1} \\ {9.2.3} \end{array}\right\rbrack \)
then \( T \subseteq G \), i.e. \( G \) has a subgraph isomorphic to \( T \) .
Proof. Find a copy of \( T \) in \( G \) inductively along its vertex enumeration from Corollary 1.5.2.
Sometimes it is convenient to consider one vertex of a tree as special; such a vertex is then called the root of this tree. A tree \( T \) with a fixed root root \( r \) is a rooted tree. Writing \( x \leq y \) for \( x \in {rTy} \) then defines a partial
ordering on \( V\left( T\right) \), the tree-order associated with \( T \) and \( r \) . We shall tree-order think of this ordering as expressing ’height’: if \( x < y \) we say that \( x \) lies
below \( y \) in \( T \), we call up/above down/below
\[
\lceil y\rceil \mathrel{\text{:=}} \{ x \mid x \leq y\} \text{ and }\lfloor x\rfloor \mathrel{\text{:=}} \{ y \mid y \geq x\}
\]
\( \lceil t\rceil ,\lfloor t\rfloor \)
the down-closure of \( y \) and the up-closure of \( x \), and so on. Note that the
---
down-closure up-closure
---
root \( r \) is the least element in this partial order, the leaves of \( T \) are its maximal elements, the ends of any edge of \( T \) are comparable, and the down-closure of every vertex is a chain, a set of pairwise comparable chain elements. (Proofs?) The vertices at distance \( k \) from \( r \) have height \( k \) and height form the \( k \) th level of \( T \) . level
A rooted tree \( T \) contained in a graph \( G \) is called normal in \( G \) if normal tree the ends of every \( T \) -path in \( G \) are comparable in the tree-order of \( T \) . If \( T \) spans \( G \), this amounts to requiring that two vertices of \( T \) must be comparable whenever they are adjacent in \( G \) ; see Figure 1.5.2.
Fig. 1.5.2. A normal spanning tree with root \( r \)
A normal tree \( T \) in \( G \) can be a powerful tool for examining the structure of \( G \), because \( G \) reflects the separation properties of \( T \) :
Lemma 1.5.5. Let \( T \) be a normal tree in \( G \) .
(i) Any two vertices \( x, y \in T \) are separated in \( G \) by the set \( \lceil x\rceil \cap \lceil y\rceil \) .
(ii) If \( S \subseteq V\left( T\right) = V\left( G\right) \) and \( S \) is down-closed, then the components of \( G - S \) are spanned by the sets \( \lfloor x\rfloor \) with \( x \) minimal in \( T - S \) .
Proof. (i) Let \( P \) be any \( x - y \) path in \( G \) . Since \( T \) is normal, the vertices of \( P \) in \( T \) form a sequence \( x = {t}_{1},\ldots ,{t}_{n} = y \) for which \( {t}_{i} \) and \( {t}_{i + 1} \) are always comparable in the tree oder of \( T \) . Consider a minimal such sequence of vertices in \( P \cap T \) . In this sequence we cannot have \( {t}_{i - 1} < {t}_{i} > {t}_{i + 1} \) for any \( i \), since \( {t}_{i - 1} \) and \( {t}_{i + 1} \) would then be comparable and deleting \( {t}_{i} \) would yield a smaller such sequence. So
\[
x = {t}_{1} > \ldots > {t}_{k} < \ldots < {t}_{n} = y
\]
for some \( k \in \{ 1,\ldots, n\} \) . As \( {t}_{k} \in \lceil x\rceil \cap \lceil y\rceil \cap V\left( P\right) \), the result follows.
(ii) Since \( S \) is down-closed, the upper neighbours in \( T \) of any vertex of \( G - S \) are again in \( G - S \) (and clearly in the same component), so the components \( C \) of \( G - S \) are up-closed. As \( S \) is down-closed, minimal vertices of \( C \) are also minimal in \( G - S \) . By (i), this means that \( C \) has only one minimal vertex \( x \) and equals its up-closure \( \lfloor x\rfloor \) .
Normal spanning trees are also called depth-first search trees, because of the way they arise in computer searches on graphs (Exercise 19). This fact is often used to prove their existence. The following inductive proof, however, is simpler and illuminates nicely how normal trees capture the structure of their host graphs.
Proposition 1.5.6. Every connected graph contains a normal spanning tree, with any specified vertex as its root.
Proof. Let \( G \) be a connected graph and \( r \in G \) |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 7 | eil x\rceil \cap \lceil y\rceil \cap V\left( P\right) \), the result follows.
(ii) Since \( S \) is down-closed, the upper neighbours in \( T \) of any vertex of \( G - S \) are again in \( G - S \) (and clearly in the same component), so the components \( C \) of \( G - S \) are up-closed. As \( S \) is down-closed, minimal vertices of \( C \) are also minimal in \( G - S \) . By (i), this means that \( C \) has only one minimal vertex \( x \) and equals its up-closure \( \lfloor x\rfloor \) .
Normal spanning trees are also called depth-first search trees, because of the way they arise in computer searches on graphs (Exercise 19). This fact is often used to prove their existence. The following inductive proof, however, is simpler and illuminates nicely how normal trees capture the structure of their host graphs.
Proposition 1.5.6. Every connected graph contains a normal spanning tree, with any specified vertex as its root.
Proof. Let \( G \) be a connected graph and \( r \in G \) any specified vertex. Let \( T \) be a maximal normal tree with root \( r \) in \( G \) ; we show that \( V\left( T\right) = V\left( G\right) \) .
Suppose not, and let \( C \) be a component of \( G - T \) . As \( T \) is normal, \( N\left( C\right) \) is a chain in \( T \) . Let \( x \) be its greatest element, and let \( y \in C \) be adjacent to \( x \) . Let \( {T}^{\prime } \) be the tree obtained from \( T \) by joining \( y \) to \( x \) ; the tree-order of \( {T}^{\prime } \) then extends that of \( T \) . We shall derive a contradiction by showing that \( {T}^{\prime } \) is also normal in \( G \) .
Let \( P \) be a \( {T}^{\prime } \) -path in \( G \) . If the ends of \( P \) both lie in \( T \), then they are comparable in the tree-order of \( T \) (and hence in that of \( {T}^{\prime } \) ), because then \( P \) is also a \( T \) -path and \( T \) is normal in \( G \) by assumption. If not, then \( y \) is one end of \( P \), so \( P \) lies in \( C \) except for its other end \( z \), which lies in \( N\left( C\right) \) . Then \( z \leq x \), by the choice of \( x \) . For our proof that \( y \) and \( z \) are comparable it thus suffices to show that \( x < y \), i.e. that \( x \in r{T}^{\prime }y \) . This, however, is clear since \( y \) is a leaf of \( {T}^{\prime } \) with neighbour \( x \) .
## 1.6 Bipartite graphs
Let \( r \geq 2 \) be an integer. A graph \( G = \left( {V, E}\right) \) is called \( r \) -partite if \( r \) -partite \( V \) admits a partition into \( r \) classes such that every edge has its ends in different classes: vertices in the same partition class must not be adjacent. Instead of '2-partite' one usually says bipartite. bipartite
Fig. 1.6.1. Two 3-partite graphs
An \( r \) -partite graph in which every two vertices from different par-
tition classes are adjacent is called complete; the complete \( r \) -partite complete \( r \) -partite graphs for all \( r \) together are the complete multipartite graphs. The
complete \( r \) -partite graph \( \overline{{K}^{{n}_{1}}} * \ldots * \overline{{K}^{{n}_{r}}} \) is denoted by \( {K}_{{n}_{1},\ldots ,{n}_{r}} \) ; if \( {K}_{{n}_{1},\ldots ,{n}_{r}} \)
\( {n}_{1} = \ldots = {n}_{r} = : s \), we abbreviate this to \( {K}_{s}^{r} \) . Thus, \( {K}_{s}^{r} \) is the complete \( {K}_{s}^{r} \) \( r \) -partite graph in which every partition class contains exactly \( s \) vertices. \( {}^{6} \) (Figure 1.6.1 shows the example of the octahedron \( {K}_{2}^{3} \) ; compare its drawing with that in Figure 1.4.3.) Graphs of the form \( {K}_{1, n} \) are called stars; the vertex in the singleton partition class of this \( {K}_{1, n} \) is the star star's centre. centre
Fig. 1.6.2. Three drawings of the bipartite graph \( {K}_{3,3} = {K}_{3}^{2} \)
Clearly, a bipartite graph cannot contain an odd cycle, a cycle of odd odd cycle length. In fact, the bipartite graphs are characterized by this property:
Proposition 1.6.1. A graph is bipartite if and only if it contains no
\( \left\lbrack \begin{array}{l} {5.3.1} \\ {6.4.2} \end{array}\right\rbrack \)
odd cycle.
---
6 Note that we obtain a \( {K}_{s}^{r} \) if we replace each vertex of a \( {K}^{r} \) by an independent \( s \) -set; our notation of \( {K}_{s}^{r} \) is intended to hint at this connection.
---
(1.5.1) Proof. Let \( G = \left( {V, E}\right) \) be a graph without odd cycles; we show that \( G \) is bipartite. Clearly a graph is bipartite if all its components are bipartite or trivial, so we may assume that \( G \) is connected. Let \( T \) be a spanning tree in \( G \), pick a root \( r \in T \), and denote the associated tree-order on \( V \) by \( { \leq }_{T} \) . For each \( v \in V \), the unique path \( {rTv} \) has odd or even length. This defines a bipartition of \( V \) ; we show that \( G \) is bipartite with this partition.
Fig. 1.6.3. The cycle \( {C}_{e} \) in \( T + e \)
Let \( e = {xy} \) be an edge of \( G \) . If \( e \in T \), with \( x{ < }_{T}y \) say, then \( {rTy} = {rTxy} \) and so \( x \) and \( y \) lie in different partition classes. If \( e \notin T \) then \( {C}_{e} \mathrel{\text{:=}} {xTy} + e \) is a cycle (Fig. 1.6.3), and by the case treated already the vertices along \( {xTy} \) alternate between the two classes. Since \( {C}_{e} \) is even by assumption, \( x \) and \( y \) again lie in different classes.
## 1.7 Contraction and minors
In Section 1.1 we saw two fundamental containment relations between graphs: the 'subgraph' relation, and the 'induced subgraph' relation. In this section we meet two more: the 'minor' relation, and the 'topological minor' relation.
\( G/e \) Let \( e = {xy} \) be an edge of a graph \( G = \left( {V, E}\right) \) . By \( G/e \) we denote the
contraction graph obtained from \( G \) by contracting the edge \( e \) into a new vertex \( {v}_{e} \) , which becomes adjacent to all the former neighbours of \( x \) and of \( y \) . Formally, \( G/e \) is a graph \( \left( {{V}^{\prime },{E}^{\prime }}\right) \) with vertex set \( {V}^{\prime } \mathrel{\text{:=}} \left( {V\smallsetminus \{ x, y\} }\right) \cup \left\{ {v}_{e}\right\} \) \( {v}_{e} \) (where \( {v}_{e} \) is the ’new’ vertex, i.e. \( {v}_{e} \notin V \cup E \) ) and edge set
\[
{E}^{\prime } \mathrel{\text{:=}} \{ {vw} \in E \mid \{ v, w\} \cap \{ x, y\} = \varnothing \}
\]
\[
\cup \left\{ {{v}_{e}w \mid {xw} \in E\smallsetminus \{ e\} \text{ or }{yw} \in E\smallsetminus \{ e\} }\right\} .
\]
Fig. 1.7.1. Contracting the edge \( e = {xy} \)
More generally, if \( X \) is another graph and \( \left\{ {{V}_{x} \mid x \in V\left( X\right) }\right\} \) is a partition of \( V \) into connected subsets such that, for any two vertices \( x, y \in X \), there is a \( {V}_{x} - {V}_{y} \) edge in \( G \) if and only if \( {xy} \in E\left( X\right) \), we call
\( G \) an \( {MX} \) and write \( {}^{7}G = {MX} \) (Fig. 1.7.2). The sets \( {V}_{x} \) are the branch \( {MX} \)
sets of this \( {MX} \) . Intuitively, we obtain \( X \) from \( G \) by contracting every branch sets branch set to a single vertex and deleting any 'parallel edges' or 'loops' that may arise. In infinite graphs, branch sets are allowed to be infinite. For example, the graph shown in Figure 8.1.1 is an \( {MX} \) with \( X \) an infinite star.
Fig. 1.7.2. \( Y \supseteq G = {MX} \), so \( X \) is a minor of \( Y \)
If \( {V}_{x} = U \subseteq V \) is one of the branch sets above and every other branch set consists just of a single vertex, we also write \( G/U \) for the graph \( X \) and \( {v}_{U} \) for the vertex \( x \in X \) to which \( U \) contracts, and think of the rest of \( X \) as an induced subgraph of \( G \) . The contraction of a single edge \( u{u}^{\prime } \) defined earlier can then be viewed as the special case of \( U = \left\{ {u,{u}^{\prime }}\right\} \) .
Proposition 1.7.1. \( G \) is an \( {MX} \) if and only if \( X \) can be obtained from \( G \) by a series of edge contractions, i.e. if and only if there are graphs \( {G}_{0},\ldots ,{G}_{n} \) and edges \( {e}_{i} \in {G}_{i} \) such that \( {G}_{0} = G,{G}_{n} \simeq X \), and \( {G}_{i + 1} = {G}_{i}/{e}_{i} \) for all \( i < n \) .
Proof. Induction on \( \left| G\right| - \left| X\right| \) .
---
7 Thus formally, the expression \( {MX} \) -where \( M \) stands for ’minor’; see below-refers to a whole class of graphs, and \( G = {MX} \) means (with slight abuse of notation) that \( G \) belongs to this class.
---
If \( G = {MX} \) is a subgraph of another graph \( Y \), we call \( X \) a minor of \( Y \)
minor; \( \preccurlyeq \) and write \( X \preccurlyeq Y \) . Note that every subgraph of a graph is also its minor; in particular, every graph is its own minor. By Proposition 1.7.1, any minor of a graph can be obtained from it by first deleting some vertices and edges, and then contracting some further edges. Conversely, any graph obtained from another by repeated deletions and contractions (in any order) is its minor: this is clear for one deletion or contraction, and follows for several from the transitivity of the minor relation (Proposition 1.7.3).
If we replace the edges of \( X \) with independent paths between their
---
subdivision \( {TX} \)
topological minor
---
ends (so that none of these paths has an inner vertex on another path or in \( X \) ), we call the graph \( G \) obtained a subdivision of \( X \) and write \( G = {TX}.{}^{8} \) If \( G = {TX} \) is the |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 8 | class.
---
If \( G = {MX} \) is a subgraph of another graph \( Y \), we call \( X \) a minor of \( Y \)
minor; \( \preccurlyeq \) and write \( X \preccurlyeq Y \) . Note that every subgraph of a graph is also its minor; in particular, every graph is its own minor. By Proposition 1.7.1, any minor of a graph can be obtained from it by first deleting some vertices and edges, and then contracting some further edges. Conversely, any graph obtained from another by repeated deletions and contractions (in any order) is its minor: this is clear for one deletion or contraction, and follows for several from the transitivity of the minor relation (Proposition 1.7.3).
If we replace the edges of \( X \) with independent paths between their
---
subdivision \( {TX} \)
topological minor
---
ends (so that none of these paths has an inner vertex on another path or in \( X \) ), we call the graph \( G \) obtained a subdivision of \( X \) and write \( G = {TX}.{}^{8} \) If \( G = {TX} \) is the subgraph of another graph \( Y \), then \( X \) is a topological minor of \( Y \) (Fig. 1.7.3).
Fig. 1.7.3. \( Y \supseteq G = {TX} \), so \( X \) is a topological minor of \( Y \)
If \( G = {TX} \), we view \( V\left( X\right) \) as a subset of \( V\left( G\right) \) and call these vertices
branch the branch vertices of \( G \) ; the other vertices of \( G \) are its subdividing vertices vertices. Thus, all subdividing vertices have degree 2, while the branch vertices retain their degree from \( X \) .
## Proposition 1.7.2.
(i) Every \( {TX} \) is also an \( {MX} \) (Fig. 1.7.4); thus, every topological minor of a graph is also its (ordinary) minor.
(ii) If \( \Delta \left( X\right) \leq 3 \), then every \( {MX} \) contains a \( {TX} \) ; thus, every minor with maximum degree at most 3 of a graph is also its topological minor.
\( \left\lbrack {12.4.1}\right\rbrack \)
Proposition 1.7.3. The minor relation \( \preccurlyeq \) and the topological-minor relation are partial orderings on the class of finite graphs, i.e. they are reflexive, antisymmetric and transitive.
---
8 So again \( {TX} \) denotes an entire class of graphs: all those which, viewed as a topological space in the obvious way, are homeomorphic to \( X \) . The \( T \) in \( {TX} \) stands for 'topological'.
---
Fig. 1.7.4. A subdivision of \( {K}^{4} \) viewed as an \( M{K}^{4} \)
Now that we have met all the standard relations between graphs, we can also define what it means to embed one graph in another. Basi-
cally, an embedding of \( G \) in \( H \) is an injective map \( \varphi : V\left( G\right) \rightarrow V\left( H\right) \) that embedding preserves the kind of structure we are interested in. Thus, \( \varphi \) embeds \( G \) in \( H \) ’as a subgraph’ if it preserves the adjacency of vertices, and ’as an induced subgraph' if it preserves both adjacency and non-adjacency. If \( \varphi \) is defined on \( E\left( G\right) \) as well as on \( V\left( G\right) \) and maps the edges \( {xy} \) of \( G \) to independent paths in \( H \) between \( \varphi \left( x\right) \) and \( \varphi \left( y\right) \), it embeds \( G \) in \( H \) ’as a topological minor’. Similarly, an embedding \( \varphi \) of \( G \) in \( H \) ’as a minor’ would be a map from \( V\left( G\right) \) to disjoint connected vertex sets in \( H \) (rather than to single vertices) such that \( H \) has an edge between the sets \( \varphi \left( x\right) \) and \( \varphi \left( y\right) \) whenever \( {xy} \) is an edge of \( G \) . Further variants are possible; depending on the context, one may wish to define embeddings 'as a spanning subgraph', 'as an induced minor', and so on in the obivous way.
## 1.8 Euler tours
Any mathematician who happens to find himself in the East Prussian city of Königsberg (and in the 18th century) will lose no time to follow the great Leonhard Euler's example and inquire about a round trip through the old city that traverses each of the bridges shown in Figure 1.8.1 exactly once.
Thus inspired, \( {}^{9} \) let us call a closed walk in a graph an Euler tour if it traverses every edge of the graph exactly once. A graph is Eulerian if Eulerian it admits an Euler tour.
Theorem 1.8.1. (Euler 1736)
\( \left\lbrack \begin{array}{l} {2.1.5} \\ {10.3.3} \end{array}\right\rbrack \)
A connected graph is Eulerian if and only if every vertex has even degree.
Proof. The degree condition is clearly necessary: a vertex appearing \( k \) times in an Euler tour (or \( k + 1 \) times, if it is the starting and finishing vertex and as such counted twice) must have degree \( {2k} \) .
---
9 Anyone to whom such inspiration seems far-fetched, even after contemplating Figure 1.8.2, may seek consolation in the multigraph of Figure 1.10.1.
---
Fig. 1.8.1. The bridges of Königsberg (anno 1736)
Conversely, let \( G \) be a connected graph with all degrees even, and let
\[
W = {v}_{0}{e}_{0}\ldots {e}_{\ell - 1}{v}_{\ell }
\]
be a longest walk in \( G \) using no edge more than once. Since \( W \) cannot be extended, it already contains all the edges at \( {v}_{\ell } \) . By assumption, the number of such edges is even. Hence \( {v}_{\ell } = {v}_{0} \), so \( W \) is a closed walk.
Suppose \( W \) is not an Euler tour. Then \( G \) has an edge \( e \) outside \( W \) but incident with a vertex of \( W \), say \( e = u{v}_{i} \) . (Here we use the connectedness of \( G \), as in the proof of Proposition 1.4.1.) Then the walk
\[
{ue}{v}_{i}{e}_{i}\ldots {e}_{\ell - 1}{v}_{\ell }{e}_{0}\ldots {e}_{i - 1}{v}_{i}
\]
is longer than \( W \), a contradiction.
Fig. 1.8.2. A graph formalizing the bridge problem
## 1.9 Some linear algebra
Let \( G = \left( {V, E}\right) \) be a graph with \( n \) vertices and \( m \) edges, say \( V = \) \( G = \left( {V, E}\right) \) \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) and \( E = \left\{ {{e}_{1},\ldots ,{e}_{m}}\right\} \) . The vertex space \( \mathcal{V}\left( G\right) \) of \( G \) is the vector space over the 2-element field \( {\mathbb{F}}_{2} = \{ 0,1\} \) of all functions \( V \rightarrow {\mathbb{F}}_{2} \) .
---
vertex
space \( \mathcal{V}\left( G\right) \)
---
Every element of \( \mathcal{V}\left( G\right) \) corresponds naturally to a subset of \( V \), the set of those vertices to which it assigns a 1, and every subset of \( V \) is uniquely represented in \( \mathcal{V}\left( G\right) \) by its indicator function. We may thus think of \( \mathcal{V}\left( G\right) \) as the power set of \( V \) made into a vector space: the sum \( U + {U}^{\prime } \) of two vertex sets \( U,{U}^{\prime } \subseteq V \) is their symmetric difference (why?), and \( U = - U \) for all \( U \subseteq V \) . The zero in \( \mathcal{V}\left( G\right) \), viewed in this way, is the empty (vertex) set \( \varnothing \) . Since \( \left\{ {\left\{ {v}_{1}\right\} ,\ldots ,\left\{ {v}_{n}\right\} }\right\} \) is a basis of \( \mathcal{V}\left( G\right) \), its standard basis, we have \( \dim \mathcal{V}\left( G\right) = n \) .
In the same way as above, the functions \( E \rightarrow {\mathbb{F}}_{2} \) form the edge
space \( \mathcal{E}\left( G\right) \) of \( G \) : its elements are the subsets of \( E \), vector addition edge space
\( \mathcal{E}\left( G\right) \)
amounts to symmetric difference, \( \varnothing \subseteq E \) is the zero, and \( F = - F \) for all \( F \subseteq E \) . As before, \( \left\{ {\left\{ {e}_{1}\right\} ,\ldots ,\left\{ {e}_{m}\right\} }\right\} \) is the standard basis of \( \mathcal{E}\left( G\right) \) , standard basis and \( \dim \mathcal{E}\left( G\right) = m \) .
Since the edges of a graph carry its essential structure, we shall mostly be concerned with the edge space. Given two edge sets \( F,{F}^{\prime } \in \) \( \mathcal{E}\left( G\right) \) and their coefficients \( {\lambda }_{1},\ldots ,{\lambda }_{m} \) and \( {\lambda }_{1}^{\prime },\ldots ,{\lambda }_{m}^{\prime } \) with respect to the standard basis, we write
\[
\left\langle {F,{F}^{\prime }}\right\rangle \mathrel{\text{:=}} {\lambda }_{1}{\lambda }_{1}^{\prime } + \ldots + {\lambda }_{m}{\lambda }_{m}^{\prime } \in {\mathbb{F}}_{2}.
\]
\( \left\langle {F,{F}^{\prime }}\right\rangle \)
Note that \( \left\langle {F,{F}^{\prime }}\right\rangle = 0 \) may hold even when \( F = {F}^{\prime } \neq \varnothing \) : indeed, \( \left\langle {F,{F}^{\prime }}\right\rangle = 0 \) if and only if \( F \) and \( {F}^{\prime } \) have an even number of edges in common. Given a subspace \( \mathcal{F} \) of \( \mathcal{E}\left( G\right) \), we write
\[
{\mathcal{F}}^{ \bot } \mathrel{\text{:=}} \{ D \in \mathcal{E}\left( G\right) \mid \langle F, D\rangle = 0\text{ for all }F \in \mathcal{F}\} .
\]
\( {\mathcal{F}}^{ \bot } \)
This is again a subspace of \( \mathcal{E}\left( G\right) \) (the space of all vectors solving a certain set of linear equations-which?), and we have
\[
\dim \mathcal{F} + \dim {\mathcal{F}}^{ \bot } = m.
\]
(†)
The cycle space \( \mathcal{C} = \mathcal{C}\left( G\right) \) is the subspace of \( \mathcal{E}\left( G\right) \) spanned by all
---
cycle space
\( \mathcal{C}\left( G\right) \)
---
the cycles in \( G \) -more precisely, by their edge sets. \( {}^{10} \) The dimension of \( \mathcal{C}\left( G\right) \) is sometimes called the cyclomatic number of \( G \) .
Proposition 1.9.1. The induced cycles in \( G \) generate its entire cycle
\( \left\lbrack {\;\left\lbrack {\;\left\lbrack {\;\left\lbrack {3.2.3}\right\rbrack \;}\right\rbrack \;}\right\rbrack }\right\rbrack \)
space. |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 9 | t) \), we write
\[
{\mathcal{F}}^{ \bot } \mathrel{\text{:=}} \{ D \in \mathcal{E}\left( G\right) \mid \langle F, D\rangle = 0\text{ for all }F \in \mathcal{F}\} .
\]
\( {\mathcal{F}}^{ \bot } \)
This is again a subspace of \( \mathcal{E}\left( G\right) \) (the space of all vectors solving a certain set of linear equations-which?), and we have
\[
\dim \mathcal{F} + \dim {\mathcal{F}}^{ \bot } = m.
\]
(†)
The cycle space \( \mathcal{C} = \mathcal{C}\left( G\right) \) is the subspace of \( \mathcal{E}\left( G\right) \) spanned by all
---
cycle space
\( \mathcal{C}\left( G\right) \)
---
the cycles in \( G \) -more precisely, by their edge sets. \( {}^{10} \) The dimension of \( \mathcal{C}\left( G\right) \) is sometimes called the cyclomatic number of \( G \) .
Proposition 1.9.1. The induced cycles in \( G \) generate its entire cycle
\( \left\lbrack {\;\left\lbrack {\;\left\lbrack {\;\left\lbrack {3.2.3}\right\rbrack \;}\right\rbrack \;}\right\rbrack }\right\rbrack \)
space.
---
10 For simplicity, we shall not always distinguish between the edge sets \( F \in \mathcal{E}\left( G\right) \) and the subgraphs \( \left( {V, F}\right) \) they induce in \( G \) . When we wish to be more precise, such as in Chapter 8.5, we shall use the word 'circuit' for the edge set of a cycle.
---
Proof. By definition of \( \mathcal{C}\left( G\right) \) it suffices to show that the induced cycles in \( G \) generate every cycle \( C \subseteq G \) with a chord \( e \) . This follows at once by induction on \( \left| C\right| \) : the two cycles in \( C + e \) that have \( e \) but no other edge in common are shorter than \( C \), and their symmetric difference is precisely \( C \) .
The elements of \( \mathcal{C} \) are easily recognized by the degrees of the subgraphs they form. Moreover, to generate the cycle space from cycles we only need disjoint unions rather than arbitrary symmetric differences:
\( \left\lbrack {4.5.1}\right\rbrack \)
Proposition 1.9.2. The following assertions are equivalent for edge sets \( F \subseteq E \) :
(i) \( F \in \mathcal{C}\left( G\right) \) ;
(ii) \( F \) is a disjoint union of (edge sets of) cycles in \( G \) ;
(iii) All vertex degrees of the graph \( \left( {V, F}\right) \) are even.
Proof. Since cycles have even degrees and taking symmetric differences preserves this,(i) \( \rightarrow \) (iii) follows by induction on the number of cycles used to generate \( F \) . The implication (iii) \( \rightarrow \) (ii) follows by induction on \( \left| F\right| \) : if \( F \neq \varnothing \) then \( \left( {V, F}\right) \) contains a cycle \( C \), whose edges we delete for the induction step. The implication (ii) \( \rightarrow \) (i) is immediate from the definition of \( \mathcal{C}\left( G\right) \) .
If \( \left\{ {{V}_{1},{V}_{2}}\right\} \) is a partition of \( V \), the set \( E\left( {{V}_{1},{V}_{2}}\right) \) of all the edges of \( G \) crossing this partition is called a cut (or cocycle). Recall that for \( {V}_{1} = \{ v\} \) this cut is denoted by \( E\left( v\right) \) .
\( \left\lbrack {4.6.3}\right\rbrack \)
Proposition 1.9.3. Together with \( \varnothing \), the cuts in \( G \) form a subspace \( {\mathcal{C}}^{ * } \) of \( \mathcal{E}\left( G\right) \) . This space is generated by cuts of the form \( E\left( v\right) \) .
Proof. Let \( {\mathcal{C}}^{ * } \) denote the set of all cuts in \( G \), together with \( \varnothing \) . To prove that \( {\mathcal{C}}^{ * } \) is a subspace, we show that for all \( D,{D}^{\prime } \in {\mathcal{C}}^{ * } \) also \( D + {D}^{\prime } \) \( \left( { = D - {D}^{\prime }}\right) \) lies in \( {\mathcal{C}}^{ * } \) . Since \( D + D = \varnothing \in {\mathcal{C}}^{ * } \) and \( D + \varnothing = D \in {\mathcal{C}}^{ * } \) , we may assume that \( D \) and \( {D}^{\prime } \) are distinct and non-empty. Let \( \left\{ {{V}_{1},{V}_{2}}\right\} \) and \( \left\{ {{V}_{1}^{\prime },{V}_{2}^{\prime }}\right\} \) be the corresponding partitions of \( V \) . Then \( D + {D}^{\prime } \) consists of all the edges that cross one of these partitions but not the other (Fig. 1.9.1). But these are precisely the edges between \( \left( {{V}_{1} \cap {V}_{1}^{\prime }}\right) \cup \left( {{V}_{2} \cap {V}_{2}^{\prime }}\right) \) and \( \left( {{V}_{1} \cap {V}_{2}^{\prime }}\right) \cup \left( {{V}_{2} \cap {V}_{1}^{\prime }}\right) \), and by \( D \neq {D}^{\prime } \) these two sets form another partition of \( V \) . Hence \( D + {D}^{\prime } \in {\mathcal{C}}^{ * } \), and \( {\mathcal{C}}^{ * } \) is indeed a subspace of \( \mathcal{E}\left( G\right) \) .
Our second assertion, that the cuts \( E\left( v\right) \) generate all of \( {\mathcal{C}}^{ * } \), follows from the fact that every edge \( {xy} \in G \) lies in exactly two such cuts (in \( E\left( x\right) \) and in \( E\left( y\right) \) ); thus every partition \( \left\{ {{V}_{1},{V}_{2}}\right\} \) of \( V \) satisfies \( E\left( {{V}_{1},{V}_{2}}\right) = \) \( \mathop{\sum }\limits_{{v \in {V}_{1}}}E\left( v\right) \)
Fig. 1.9.1. Cut edges in \( D + {D}^{\prime } \)
The subspace \( {\mathcal{C}}^{ * } = : {\mathcal{C}}^{ * }\left( G\right) \) of \( \mathcal{E}\left( G\right) \) from Proposition 1.9.3 is the cut
space of \( G \) . It is not difficult to find among the cuts \( E\left( v\right) \) an explicit cut space \( {\mathcal{C}}^{ * }\left( G\right) \) basis for \( {\mathcal{C}}^{ * }\left( G\right) \), and thus to determine its dimension (Exercise 27).
A minimal non-empty cut in \( G \) is a bond. Thus, bonds are for \( {\mathcal{C}}^{ * } \) bond what cycles are for \( \mathcal{C} \) : the minimal non-empty elements. Note that the ’non-empty’ condition bites only if \( G \) is disconnected. If \( G \) is connected, its bonds are just its minimal cuts, and these are easy to recognize: clearly, a cut in a connected graph is minimal if and only if both sides of the corresponding vertex partition induce connected subgraphs. If \( G \) is disconnected, its bonds are the minimal cuts of its components. (See also Lemma 3.1.1.)
In analogy to Proposition 1.9.2, bonds and disjoint unions suffice to generate \( {\mathcal{C}}^{ * } \) :
Lemma 1.9.4. Every cut is a disjoint union of bonds.
\( \left\lbrack {4.6.2}\right\rbrack \)
Proof. Consider first a connected graph \( H = \left( {V, E}\right) \), a connected subgraph \( C \subseteq H \), and a component \( D \) of \( H - C \) . Then \( H - D \), too, is connected (Fig. 1.9.2), so the edges between \( D \) and \( H - D \) form a minimal cut. By the choice of \( D \), this cut is precisely the set \( E\left( {C, D}\right) \) of all \( C - D \) edges in \( H \) .
Fig. 1.9.2. \( H - D \) is connected, and \( E\left( {C, D}\right) \) a minimal cut
To prove the lemma, let a cut in an arbitrary graph \( G = \left( {V, E}\right) \) be given, with partition \( \left\{ {{V}_{1},{V}_{2}}\right\} \) of \( V \) say. Consider a component \( C \) of \( G\left\lbrack {V}_{1}\right\rbrack \), and let \( H \) be the component of \( G \) containing \( C \) . Then \( E\left( {C,{V}_{2}}\right) = E\left( {C, H - C}\right) \) is the disjoint union of the edge sets \( E\left( {C, D}\right) \) over all the components \( D \) of \( H - C \) . By our earlier considerations these sets are minimal cuts in \( H \), and hence bonds in \( G \) . Now the disjoint union of all these edge sets \( E\left( {C,{V}_{2}}\right) \), taken over all the components \( C \) of \( G\left\lbrack {V}_{1}\right\rbrack \), is precisely our cut \( E\left( {{V}_{1},{V}_{2}}\right) \) .
Theorem 1.9.5. The cycle space \( \mathcal{C} \) and the cut space \( {\mathcal{C}}^{ * } \) of any graph satisfy
\[
\mathcal{C} = {\mathcal{C}}^{* \bot }\;\text{ and }\;{\mathcal{C}}^{ * } = {\mathcal{C}}^{ \bot }.
\]
Proof. (See also Exercise 30.) Let us consider a graph \( G = \left( {V, E}\right) \) . Clearly, any cycle in \( G \) has an even number of edges in each cut. This implies \( \mathcal{C} \subseteq {\mathcal{C}}^{* \bot } \) .
Conversely, recall from Proposition 1.9.2 that for every edge set \( F \notin \mathcal{C} \) there exists a vertex \( v \) incident with an odd number of edges in \( F \) . Then \( \langle E\left( v\right), F\rangle = 1 \), so \( E\left( v\right) \in {\mathcal{C}}^{ * } \) implies \( F \notin {\mathcal{C}}^{* \bot } \) . This completes the proof of \( \mathcal{C} = {\mathcal{C}}^{* \bot } \) .
To prove \( {\mathcal{C}}^{ * } = {\mathcal{C}}^{ \bot } \), it now suffices to show \( {\mathcal{C}}^{ * } = {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot } \) . Here \( {\mathcal{C}}^{ * } \subseteq {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot } \) follows directly from the definition of \( \bot \) . But \( {\mathcal{C}}^{ * } \) has the same dimension as \( {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot } \), since \( \left( \dagger \right) \) implies
\[
\dim {\mathcal{C}}^{ * } + \dim {\mathcal{C}}^{* \bot } = m = \dim {\mathcal{C}}^{* \bot } + \dim {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot }.
\]
Hence \( {\mathcal{C}}^{ * } = {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot } \) as claimed.
Consider a connected graph \( G = \left( {V, E}\right) \) with a spanning tree \( T \subseteq G \) .
---
fundamental cycles
---
Recall that for every edge \( e \in E \smallsetminus E\left( T\right) \) there is a unique cycle \( {C}_{e} \) in \( T + e \) (Fig. 1.6.3); these cycles \( {C}_{e} \) are the fundamental cycles of \( G \) with respect to \( T \) . On the other hand, given an edge \( e \in T \), the graph \( T - e \)
(1.5.1)
has exactly two components (Theorem 1.5.1 (iii)), and the set \( {D}_{e} \subseteq E \)
---
fundamental cuts
---
of edges between these two components form a bond in \( G \) (Fig.1.9.3). Th |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 10 | ension as \( {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot } \), since \( \left( \dagger \right) \) implies
\[
\dim {\mathcal{C}}^{ * } + \dim {\mathcal{C}}^{* \bot } = m = \dim {\mathcal{C}}^{* \bot } + \dim {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot }.
\]
Hence \( {\mathcal{C}}^{ * } = {\left( {\mathcal{C}}^{* \bot }\right) }^{ \bot } \) as claimed.
Consider a connected graph \( G = \left( {V, E}\right) \) with a spanning tree \( T \subseteq G \) .
---
fundamental cycles
---
Recall that for every edge \( e \in E \smallsetminus E\left( T\right) \) there is a unique cycle \( {C}_{e} \) in \( T + e \) (Fig. 1.6.3); these cycles \( {C}_{e} \) are the fundamental cycles of \( G \) with respect to \( T \) . On the other hand, given an edge \( e \in T \), the graph \( T - e \)
(1.5.1)
has exactly two components (Theorem 1.5.1 (iii)), and the set \( {D}_{e} \subseteq E \)
---
fundamental cuts
---
of edges between these two components form a bond in \( G \) (Fig.1.9.3). These bonds \( {D}_{e} \) are the fundamental cuts of \( G \) with respect to \( T \) .
It is not difficult to show directly that the fundamental cycles and cuts span the cycle and cut space of \( G \), respectively (Ex. 31-32). In the proof of the following more comprehensive theorem, this information comes for free as a consequence of Theorem 1.9.5 and the dimension formula \( \left( \dagger \right) \) for orthogonal subspaces.
\( \left\lbrack {4.5.1}\right\rbrack \)
Theorem 1.9.6. Let \( G \) be a connected graph and \( T \subseteq G \) a spanning tree. Then the corresponding fundamental cycles and cuts form a basis of \( \mathcal{C}\left( G\right) \) and of \( {\mathcal{C}}^{ * }\left( G\right) \), respectively. If \( G \) has \( n \) vertices and \( m \) edges, then
\[
\dim \mathcal{C}\left( G\right) = m - n + 1\;\text{ and }\;\dim {\mathcal{C}}^{ * }\left( G\right) = n - 1.
\]
Fig. 1.9.3. The fundamental cut \( {D}_{e} \)
Proof. Since an edge \( e \in T \) lies in \( {D}_{e} \) but not in \( {D}_{{e}^{\prime }} \) for any \( {e}^{\prime } \neq e \), the cut
\( \left( {1.5.3}\right) \)
\( {D}_{e} \) cannot be generated by other fundamental cuts. The fundamental cuts therefore form a linearly independent subset of \( {\mathcal{C}}^{ * } \), of size \( n - 1 \) (Corollary 1.5.3). Similarly, an edge \( e \in E \smallsetminus E\left( T\right) \) lies on \( {C}_{e} \) but not on any other fundamental cycle; so the fundamental cycles form a linearly independent subset of \( \mathcal{C} \), of size \( m - n + 1 \) . Thus,
\[
\dim {\mathcal{C}}^{ * } \geq n - 1\;\text{ and }\;\dim \mathcal{C} \geq m - n + 1.
\]
But
\[
\dim {\mathcal{C}}^{ * } + \dim \mathcal{C} = m = \left( {n - 1}\right) + \left( {m - n + 1}\right)
\]
by Theorem 1.9.5 and \( \left( \dagger \right) \), so the two inequalities above can hold only with equality. Hence the sets of fundamental cuts and cycles are maximal as linearly independent subsets of \( {\mathcal{C}}^{ * } \) and \( \mathcal{C} \), and hence are bases.
The incidence matrix \( B = {\left( {b}_{ij}\right) }_{n \times m} \) of a graph \( G = \left( {V, E}\right) \) with
---
incidence matrix
---
\( V = \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) and \( E = \left\{ {{e}_{1},\ldots ,{e}_{m}}\right\} \) is defined over \( {\mathbb{F}}_{2} \) by
\[
{b}_{ij} \mathrel{\text{:=}} \left\{ \begin{array}{ll} 1 & \text{ if }{v}_{i} \in {e}_{j} \\ 0 & \text{ otherwise. } \end{array}\right.
\]
As usual, let \( {B}^{t} \) denote the transpose of \( B \) . Then \( B \) and \( {B}^{t} \) define linear maps \( B : \mathcal{E}\left( G\right) \rightarrow \mathcal{V}\left( G\right) \) and \( {B}^{t} : \mathcal{V}\left( G\right) \rightarrow \mathcal{E}\left( G\right) \) with respect to the standard bases.
Proposition 1.9.7.
(i) The kernel of \( B \) is \( \mathcal{C}\left( G\right) \) .
(ii) The image of \( {B}^{t} \) is \( {\mathcal{C}}^{ * }\left( G\right) \) .
---
adjacency matrix
---
The adjacency matrix \( A = {\left( {a}_{ij}\right) }_{n \times n} \) of \( G \) is defined by
\[
{a}_{ij} \mathrel{\text{:=}} \left\{ \begin{array}{ll} 1 & \text{if }{v}_{i}{v}_{j} \in E \\ 0 & \text{otherwise.} \end{array}\right.
\]
Our last proposition establishes a simple connection between \( A \) and \( B \) (now viewed as real matrices). Let \( D \) denote the real diagonal matrix \( {\left( {d}_{ij}\right) }_{n \times n} \) with \( {d}_{ii} = d\left( {v}_{i}\right) \) and \( {d}_{ij} = 0 \) otherwise.
Proposition 1.9.8. \( B{B}^{t} = A + D \) .
## 1.10 Other notions of graphs
For completeness, we now mention a few other notions of graphs which feature less frequently or not at all in this book.
---
hypergraph
---
A hypergraph is a pair \( \left( {V, E}\right) \) of disjoint sets, where the elements of \( E \) are non-empty subsets (of any cardinality) of \( V \) . Thus, graphs are special hypergraphs.
directed A directed graph (or digraph) is a pair \( \left( {V, E}\right) \) of disjoint sets (of graph vertices and edges) together with two maps init: \( E \rightarrow V \) and ter: \( E \rightarrow V \)
\( \operatorname{init}\left( e\right) \) assigning to every edge \( e \) an initial vertex init(e) and a terminal vertex
\( \operatorname{ter}\left( e\right) \) \( \operatorname{ter}\left( e\right) \) . The edge \( e \) is said to be directed from init \( \left( e\right) \) to \( \operatorname{ter}\left( e\right) \) . Note that a directed graph may have several edges between the same two vertices \( x, y \) . Such edges are called multiple edges; if they have the same direction (say from \( x \) to \( y \) ), they are parallel. If \( \operatorname{init}\left( e\right) = \operatorname{ter}\left( e\right) \), the edge \( e \) is called
loop a loop.
orientation A directed graph \( D \) is an orientation of an (undirected) graph \( G \) if \( V\left( D\right) = V\left( G\right) \) and \( E\left( D\right) = E\left( G\right) \), and if \( \{ \operatorname{init}\left( e\right) ,\operatorname{ter}\left( e\right) \} = \{ x, y\} \) for
oriented every edge \( e = {xy} \) . Intuitively, such an oriented graph arises from an graph undirected graph simply by directing every edge from one of its ends to the other. Put differently, oriented graphs are directed graphs without loops or multiple edges.
multigraph A multigraph is a pair \( \left( {V, E}\right) \) of disjoint sets (of vertices and edges) together with a map \( E \rightarrow V \cup {\left\lbrack V\right\rbrack }^{2} \) assigning to every edge either one or two vertices, its ends. Thus, multigraphs too can have loops and multiple edges: we may think of a multigraph as a directed graph whose edge directions have been ’forgotten’. To express that \( x \) and \( y \) are the ends of an edge \( e \) we still write \( e = {xy} \), though this no longer determines \( e \) uniquely.
A graph is thus essentially the same as a multigraph without loops or multiple edges. Somewhat surprisingly, proving a graph theorem more generally for multigraphs may, on occasion, simplify the proof. Moreover, there are areas in graph theory (such as plane duality; see Chapters 4.6 and 6.5) where multigraphs arise more naturally than graphs, and where any restriction to the latter would seem artificial and be technically complicated. We shall therefore consider multigraphs in these cases, but without much technical ado: terminology introduced earlier for graphs will be used correspondingly.
A few differences, however, should be pointed out. A multigraph may have cycles of length 1 or 2: loops, and pairs of multiple edges (or double edges). A loop at a vertex makes it its own neighbour, and contributes 2 to its degree; in Figure 1.10.1, we thus have \( d\left( {v}_{e}\right) = 6 \) . And the notion of edge contraction is simpler in multigraphs than in graphs. If we contract an edge \( e = {xy} \) in a multigraph \( G = \left( {V, E}\right) \) to a new vertex \( {v}_{e} \), there is no longer a need to delete any edges other than \( e \) itself: edges parallel to \( e \) become loops at \( {v}_{e} \), while edges \( {xv} \) and \( {yv} \) become parallel edges between \( {v}_{e} \) and \( v \) (Fig. 1.10.1). Thus, formally, \( E\left( {G/e}\right) = E \smallsetminus \{ e\} \), and only the incidence map \( {e}^{\prime } \mapsto \left\{ {\operatorname{init}\left( {e}^{\prime }\right) ,\operatorname{ter}\left( {e}^{\prime }\right) }\right\} \) of \( G \) has to be adjusted to the new vertex set in \( G/e \) . The notion of a minor adapts to multigraphs accordingly.
Fig. 1.10.1. Contracting the edge \( e \) in the multigraph corresponding to Fig. 1.8.1
If \( v \) is a vertex of degree 2 in a multigraph \( G \), then by suppressing \( v \) suppressing a vertex we mean deleting \( v \) and adding an edge between its two neighbours. \( {}^{11} \) (If its two incident edges are identical, i.e. form a loop at \( v \), we add no edge and obtain just \( G - v \) . If they go to the same vertex \( w \neq v \), the added edge will be a loop at \( w \) . See Figure 1.10.2.) Since the degrees of all vertices other than \( v \) remain unchanged when \( v \) is suppressed, suppressing several vertices of \( G \) always yields a well-defined multigraph that is independent of the order in which those vertices are suppressed.
Fig. 1.10.2. Suppressing the white vertices
---
11 This is just a clumsy combinatorial paraphrase of the topological notion of amalgamating the two edges at \( v \) into one edge, of which \( v \) becomes an inner point.
---
Finally, it should be pointed out that authors who usually work with multigraphs tend to call them 'graphs'; in their terminology, ou |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 11 | ding an edge between its two neighbours. \( {}^{11} \) (If its two incident edges are identical, i.e. form a loop at \( v \), we add no edge and obtain just \( G - v \) . If they go to the same vertex \( w \neq v \), the added edge will be a loop at \( w \) . See Figure 1.10.2.) Since the degrees of all vertices other than \( v \) remain unchanged when \( v \) is suppressed, suppressing several vertices of \( G \) always yields a well-defined multigraph that is independent of the order in which those vertices are suppressed.
Fig. 1.10.2. Suppressing the white vertices
---
11 This is just a clumsy combinatorial paraphrase of the topological notion of amalgamating the two edges at \( v \) into one edge, of which \( v \) becomes an inner point.
---
Finally, it should be pointed out that authors who usually work with multigraphs tend to call them 'graphs'; in their terminology, our graphs would be called 'simple graphs'.
## Exercises
1. \( {}^{ - } \) What is the number of edges in a \( {K}^{n} \) ?
2. Let \( d \in \mathbb{N} \) and \( V \mathrel{\text{:=}} \{ 0,1{\} }^{d} \) ; thus, \( V \) is the set of all \( 0 - 1 \) sequences of length \( d \) . The graph on \( V \) in which two such sequences form an edge if and only if they differ in exactly one position is called the \( d \) -dimensional cube. Determine the average degree, number of edges, diameter, girth and circumference of this graph.
(Hint for the circumference: induction on \( d \) .)
3. Let \( G \) be a graph containing a cycle \( C \), and assume that \( G \) contains a path of length at least \( k \) between two vertices of \( C \) . Show that \( G \) contains a cycle of length at least \( \sqrt{k} \) . Is this best possible?
4. \( {}^{ - } \) Is the bound in Proposition 1.3.2 best possible?
5. Show that \( \operatorname{rad}G \leq \operatorname{diam}G \leq 2\operatorname{rad}G \) for every graph \( G \) .
6. Prove the weakening of Theorem 1.3.4 obtained by replacing average with minimum degree. Deduce that \( \left| G\right| \geq {n}_{0}\left( {d/2, g}\right) \) for every graph \( G \) as given in the theorem.
7. \( {}^{ + } \) Show that every connected graph \( G \) contains a path of length at least \( \min \{ {2\delta }\left( G\right) ,\left| G\right| - 1\} \) .
8. \( {}^{ + } \) Find a good lower bound for the order of a connected graph in terms of its diameter and minimum degree.
9. \( {}^{ - } \) Show that the components of a graph partition its vertex set. (In other words, show that every vertex belongs to exactly one component.)
10. \( {}^{ - } \) Show that every 2-connected graph contains a cycle.
11. Determine \( \kappa \left( G\right) \) and \( \lambda \left( G\right) \) for \( G = {P}^{m},{C}^{n},{K}^{n},{K}_{m, n} \) and the \( d \) - dimensional cube (Exercise 2); \( d, m, n \geq 3 \) .
12. \( {}^{ - } \) Is there a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that, for all \( k \in \mathbb{N} \), every graph of minimum degree at least \( f\left( k\right) \) is \( k \) -connected?
13. \( {}^{ + } \) Let \( \alpha ,\beta \) be two graph invariants with positive integer values. Formalize the two statements below, and show that each implies the other:
(i) \( \alpha \) is bounded above by a function of \( \beta \) ;
(ii) \( \beta \) can be forced up by making \( \alpha \) large enough.
Show that the statement
(iii) \( \beta \) is bounded below by a function of \( \alpha \)
is not equivalent to (i) and (ii). Which small change will make it so?
14. \( {}^{ + } \) What is the deeper reason behind the fact that the proof of Theorem 1.4.3 is based on an assumption of the form \( m \geq {c}_{k}n - {b}_{k} \) rather than more simply \( m \geq {c}_{k}n \) ?
15. Prove Theorem 1.5.1.
16. \( {}^{ - } \) Show that every tree \( T \) has at least \( \Delta \left( T\right) \) leaves.
17. Show that a tree without a vertex of degree 2 has more leaves than other vertices. Can you find a very short proof that does not use induction?
18. Show that the tree-order associated with a rooted tree \( T \) is indeed a partial order on \( V\left( T\right) \), and verify the claims made about this partial order in the text.
19. \( {}^{ + } \) Let \( G \) be a connected graph, and let \( r \in G \) be a vertex. Starting from \( r \), move along the edges of \( G \), going whenever possible to a vertex not visited so far. If there is no such vertex, go back along the edge by which the current vertex was first reached (unless the current vertex is \( r \) ; then stop). Show that the edges traversed form a normal spanning tree in \( G \) with root \( r \) .
(This procedure has earned those trees the name of depth-first search trees.)
20. Let \( \mathcal{T} \) be a set of subtrees of a tree \( T \) . Assume that the trees in \( \mathcal{T} \) have pairwise non-empty intersection. Show that their overall intersection \( \bigcap \mathcal{T} \) is non-empty.
21. Show that every automorphism of a tree fixes a vertex or an edge.
22. \( {}^{ - } \) Are the partition classes of a regular bipartite graph always of the same size?
23. Show that a graph is bipartite if and only if every induced cycle has even length.
24. \( {}^{ + } \) Find a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that, for all \( k \in \mathbb{N} \), every graph of average degree at least \( f\left( k\right) \) has a bipartite subgraph of minimum degree at least \( k \) .
25. Show that the minor relation \( \preccurlyeq \) defines a partial ordering on any set of (finite) graphs. Is the same true for infinite graphs?
26. Prove or disprove that every connected graph contains a walk that traverses each of its edges exactly once in each direction.
27. Given a graph \( G \), find among all cuts of the form \( E\left( v\right) \) a basis for the cut space of \( G \) .
28. Show that the bonds of a graph are precisely the minimal cuts of its components.
29. Prove that the cycles and the cuts in a graph together generate its entire edge space, or find a counterexample.
30. \( {}^{ + } \) In the proof of Theorem 1.9.5, the only implication that is not proved directly (but via dimension) is \( {\mathcal{C}}^{ \bot } \subseteq {\mathcal{C}}^{ * } \) . Prove this implication directly.
31. Give a direct proof of the fact that the fundamental cycles of a connected graph span its cycle space.
32. Give a direct proof of the fact that the fundamental cuts of a connected graph span its cut space.
33. What are the dimensions of the cycle and the cut space of a graph with \( k \) components?
34. Let \( A = {\left( {a}_{ij}\right) }_{n \times n} \) be the adjacency matrix of the graph \( G \) . Show that the matrix \( {A}^{k} = {\left( {a}_{ij}^{\prime }\right) }_{n \times n} \) displays, for all \( i, j \leq n \), the number \( {a}_{ij}^{\prime } \) of walks of length \( k \) from \( {v}_{i} \) to \( {v}_{j} \) in \( G \) .
35. \( {}^{ + } \) Prove Gallai’s cycle-cocycle partition theorem that the vertex set of any graph \( G = \left( {V, E}\right) \) can be written as a disjoint union \( V = {V}^{\prime } \cup {V}^{\prime \prime } \) of possibly empty subsets such that the edge sets of both \( G\left\lbrack {V}^{\prime }\right\rbrack \) and \( G\left\lbrack {V}^{\prime \prime }\right\rbrack \) lie in the cycle space of \( G \) .
## Notes
The terminology used in this book is mostly standard. Alternatives do exist, of course, and some of these are stated when a concept is first defined. There is one small point where our notation deviates slightly from standard usage. Whereas complete graphs, paths, cycles etc. of given order are mostly denoted by \( {K}_{n},{P}_{k},{C}_{\ell } \) and so on, we use superscripts instead of subscripts. This has the advantage of leaving the variables \( K, P, C \) etc. free for ad-hoc use: we may now enumerate components as \( {C}_{1},{C}_{2},\ldots \), speak of paths \( {P}_{1},\ldots ,{P}_{k} \), and so on without any danger of confusion.
Theorem \( {}^{12} \) 1.3.4 was proved by N. Alon, S. Hoory and N. Linial, The Moore bound for irregular graphs, Graphs Comb. 18 (2002), 53-57. The proof uses an ingenious argument counting random walks along the edges of the graph considered.
The main assertion of Theorem 1.4.3, that an average degree of at least \( {4k} \) forces a \( k \) -connected subgraph, is from W. Mader, Existenz \( n \) -fach zusammen-hängender Teilgraphen in Graphen genügend großer Kantendichte, Abh. Math. Sem. Univ. Hamburg 37 (1972) 86-97. The stronger form stated here was obtained in 2005 by Ph. Sprüssel with a different proof (unpublished); our proof is due to Mader. For the history of the Königsberg bridge problem, and Euler's actual part in its solution, see N.L. Biggs, E.K. Lloyd & R.J. Wilson, Graph Theory 1736-1936, Oxford University Press 1976.
Of the large subject of algebraic methods in graph theory, Section 1.9 does not convey an adequate impression. A good introduction is N.L. Biggs, Algebraic Graph Theory (2nd edn.), Cambridge University Press 1993. A more comprehensive account is given by C.D. Godsil & G.F. Royle, Algebraic Graph Theory, Springer GTM 207, 2001. Surveys on the use of algebraic methods can also be found in the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995.
---
12 In the interest of readability, the end-of-chapter notes in this book give references only for Theorems, and only in cases where these references cannot be found in a monograph or survey cited for that chapter.
---
Matching
Covering
and Packing
Suppose we are given a graph and are asked to find in it as many independent edges as possible. How should we go about this? Will we be able to pair up all its vertices in this way? If not, how can we be sure that this is indeed impossible? Somewhat surprisingly, this basic problem does not only lie at the heart of nu |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 12 | 1.9 does not convey an adequate impression. A good introduction is N.L. Biggs, Algebraic Graph Theory (2nd edn.), Cambridge University Press 1993. A more comprehensive account is given by C.D. Godsil & G.F. Royle, Algebraic Graph Theory, Springer GTM 207, 2001. Surveys on the use of algebraic methods can also be found in the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995.
---
12 In the interest of readability, the end-of-chapter notes in this book give references only for Theorems, and only in cases where these references cannot be found in a monograph or survey cited for that chapter.
---
Matching
Covering
and Packing
Suppose we are given a graph and are asked to find in it as many independent edges as possible. How should we go about this? Will we be able to pair up all its vertices in this way? If not, how can we be sure that this is indeed impossible? Somewhat surprisingly, this basic problem does not only lie at the heart of numerous applications, it also gives rise to some rather interesting graph theory.
A set \( M \) of independent edges in a graph \( G = \left( {V, E}\right) \) is called a matching. \( M \) is a matching of \( U \subseteq V \) if every vertex in \( U \) is incident
---
matching matched
---
with an edge in \( M \) . The vertices in \( U \) are then called matched (by \( M \) ); vertices not incident with any edge of \( M \) are unmatched.
A \( k \) -regular spanning subgraph is called a \( k \) -factor. Thus, a sub- factor graph \( H \subseteq G \) is a 1 -factor of \( G \) if and only if \( E\left( H\right) \) is a matching of \( V \) . The problem of how to characterize the graphs that have a 1-factor, i.e. a matching of their entire vertex set, will be our main theme in the first two sections of this chapter.
A generalization of the matching problem is to find in a given graph \( G \) as many disjoint subgraphs as possible that are each isomorphic to an element of a given class \( \mathcal{H} \) of graphs. This is known as the packing
---
packing covering
---
problem. It is related to the covering problem, which asks how few vertices of \( G \) suffice to meet all its subgraphs isomorphic to a graph in \( \mathcal{H} \) : clearly, we need at least as many vertices for such a cover as the maximum number \( k \) of \( \mathcal{H} \) -graphs that we can pack disjointly into \( G \) . If there is no cover by just \( k \) vertices, perhaps there is always a cover by at most \( f\left( k\right) \) vertices, where \( f\left( k\right) \) may depend on \( \mathcal{H} \) but not on \( G \) ? In Section 2.3 we shall prove that when \( \mathcal{H} \) is the class of cycles, then there is such a function \( f \) .
In Section 2.4 we consider packing and covering in terms of edges: we ask how many edge-disjoint spanning trees we can find in a given graph, and how few trees in it will cover all its edges. In Section 2.5 we prove a path cover theorem for directed graphs, which implies the well-known duality theorem of Dilworth for partial orders.
## 2.1 Matching in bipartite graphs
---
\( G = \left( {V, E}\right) \)
\( A, B \)
\( a, b \) etc.
---
For this whole section, we let \( G = \left( {V, E}\right) \) be a fixed bipartite graph with bipartition \( \{ A, B\} \) . Vertices denoted as \( a,{a}^{\prime } \) etc. will be assumed to lie in \( A \), vertices denoted as \( b \) etc. will lie in \( B \) .
How can we find a matching in \( G \) with as many edges as possible? Let us start by considering an arbitrary matching \( M \) in \( G \) . A path in \( G \) which starts in \( A \) at an unmatched vertex and then contains, alternately,
---
alternating path
---
edges from \( E \smallsetminus M \) and from \( M \), is an alternating path with respect to \( M \) . An alternating path \( P \) that ends in an unmatched vertex of \( B \) is called
augment- an augmenting path (Fig. 2.1.1), because we can use it to turn \( M \) into ing path a larger matching: the symmetric difference of \( M \) with \( E\left( P\right) \) is again a matching (consider the edges at a given vertex), and the set of matched vertices is increased by two, the ends of \( P \) .
Fig. 2.1.1. Augmenting the matching \( M \) by the alternating path \( P \)
Alternating paths play an important role in the practical search for large matchings. In fact, if we start with any matching and keep applying augmenting paths until no further such improvement is possible, the matching obtained will always be an optimal one, a matching with the largest possible number of edges (Exercise 1). The algorithmic problem of finding such matchings thus reduces to that of finding augmenting paths - which is an interesting and accessible algorithmic problem.
Our first theorem characterizes the maximal cardinality of a matching in \( G \) by a kind of duality condition. Let us call a set \( U \subseteq V \) a (vertex) cover of \( E \) if every edge of \( G \) is incident with a vertex in \( U \) .
## Theorem 2.1.1. (König 1931)
The maximum cardinality of a matching in \( G \) is equal to the minimum cardinality of a vertex cover of its edges.
Proof. Let \( M \) be a matching in \( G \) of maximum cardinality. From every edge in \( M \) let us choose one of its ends: its end in \( B \) if some alternating path ends in that vertex, and its end in \( A \) otherwise (Fig. 2.1.2). We shall prove that the set \( U \) of these \( \left| M\right| \) vertices covers \( E \) ; since any vertex cover of \( E \) must cover \( M \), there can be none with fewer than \( \left| M\right| \) vertices, and so the theorem will follow.
Fig. 2.1.2. The vertex cover \( U \)
Let \( {ab} \in E \) be an edge; we show that either \( a \) or \( b \) lies in \( U \) . If \( {ab} \in M \), this holds by definition of \( U \), so we assume that \( {ab} \notin M \) . Since \( M \) is a maximal matching, it contains an edge \( {a}^{\prime }{b}^{\prime } \) with \( a = {a}^{\prime } \) or \( b = {b}^{\prime } \) . In fact, we may assume that \( a = {a}^{\prime } \) : for if \( a \) is unmatched (and \( b = {b}^{\prime } \) ), then \( {ab} \) is an alternating path, and so the end of \( {a}^{\prime }{b}^{\prime } \in M \) chosen for \( U \) was the vertex \( {b}^{\prime } = b \) . Now if \( {a}^{\prime } = a \) is not in \( U \), then \( {b}^{\prime } \in U \), and some alternating path \( P \) ends in \( {b}^{\prime } \) . But then there is also an alternating path \( {P}^{\prime } \) ending in \( b \) : either \( {P}^{\prime } \mathrel{\text{:=}} {Pb} \) (if \( b \in P \) ) or \( {P}^{\prime } \mathrel{\text{:=}} P{b}^{\prime }{a}^{\prime }b \) . By the maximality of \( M \), however, \( {P}^{\prime } \) is not an augmenting path. So \( b \) must be matched, and was chosen for \( U \) from the edge of \( M \) containing it.
Let us return to our main problem, the search for some necessary and sufficient conditions for the existence of a 1-factor. In our present case of a bipartite graph, we may as well ask more generally when \( G \) contains a matching of \( A \) ; this will define a 1 -factor of \( G \) if \( \left| A\right| = \left| B\right| \) , a condition that has to hold anyhow if \( G \) is to have a 1 -factor.
A condition clearly necessary for the existence of a matching of \( A \) is that every subset of \( A \) has enough neighbours in \( B \), i.e. that
\[
\left| {N\left( S\right) }\right| \geq \left| S\right| \;\text{ for all }S \subseteq A.
\]
---
marriage condition
---
The following marriage theorem says that this obvious necessary condition is in fact sufficient:
\( \left\lbrack {2.2.3}\right\rbrack \) Theorem 2.1.2. (Hall 1935)
\( G \) contains a matching of \( A \) if and only if \( \left| {N\left( S\right) }\right| \geq \left| S\right| \) for all \( S \subseteq A \) .
We give three proofs, ranging from the natural and pedestrian to the slick and elegant. The theorem can also be derived easily from König's theorem (Exercise 4).
Our first proof is algorithmic and uses alternating paths.
\( M \) First proof. Consider a matching \( M \) of \( G \) that leaves a vertex of \( A \) unmatched; we shall construct an augmenting path with respect to \( M \) .
Let \( {a}_{0},{b}_{1},{a}_{1},{b}_{2},{a}_{2},\ldots \) be a maximal sequence of distinct vertices \( {a}_{i} \in A \) and \( {b}_{i} \in B \) satisfying the following conditions for all \( i \geq 1 \) (Fig. 2.1.3):
(i) \( {a}_{0} \) is unmatched;
(ii) \( {b}_{i} \) is adjacent to some vertex \( {a}_{f\left( i\right) } \in \left\{ {{a}_{0},\ldots ,{a}_{i - 1}}\right\} \) ;
(iii) \( {a}_{i}{b}_{i} \in M \) .
Fig. 2.1.3. Proving the marriage theorem by alternating paths
By the marriage condition, our sequence cannot end in a vertex of \( A \) : the \( i \) vertices \( {a}_{0},\ldots ,{a}_{i - 1} \) together have at least \( i \) neighbours in \( B \) , so we can always find a new vertex \( {b}_{i} \neq {b}_{1},\ldots ,{b}_{i - 1} \) that satisfies (ii). Let \( {b}_{k} \in B \) be the last vertex of the sequence. By (i)-(iii),
\[
P \mathrel{\text{:=}} {b}_{k}{a}_{f\left( k\right) }{b}_{f\left( k\right) }{a}_{{f}^{2}\left( k\right) }{b}_{{f}^{2}\left( k\right) }{a}_{{f}^{3}\left( k\right) }\ldots {a}_{{f}^{r}\left( k\right) }
\]
with \( {f}^{r}\left( k\right) = 0 \) is an alternating path.
What is it that prevents us from extending our sequence further? If \( {b}_{k} \) is matched, say to \( a \), we can indeed extend it by setting \( {a}_{k} \mathrel{\text{:=}} a \) , unless \( a = {a}_{i} \) with \( 0 < i < k \), in which case (iii) would imply \( {b}_{k} = {b}_{i} \) with a contradiction. So \( {b}_{k} \) is unmatc |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 13 | ge theorem by alternating paths
By the marriage condition, our sequence cannot end in a vertex of \( A \) : the \( i \) vertices \( {a}_{0},\ldots ,{a}_{i - 1} \) together have at least \( i \) neighbours in \( B \) , so we can always find a new vertex \( {b}_{i} \neq {b}_{1},\ldots ,{b}_{i - 1} \) that satisfies (ii). Let \( {b}_{k} \in B \) be the last vertex of the sequence. By (i)-(iii),
\[
P \mathrel{\text{:=}} {b}_{k}{a}_{f\left( k\right) }{b}_{f\left( k\right) }{a}_{{f}^{2}\left( k\right) }{b}_{{f}^{2}\left( k\right) }{a}_{{f}^{3}\left( k\right) }\ldots {a}_{{f}^{r}\left( k\right) }
\]
with \( {f}^{r}\left( k\right) = 0 \) is an alternating path.
What is it that prevents us from extending our sequence further? If \( {b}_{k} \) is matched, say to \( a \), we can indeed extend it by setting \( {a}_{k} \mathrel{\text{:=}} a \) , unless \( a = {a}_{i} \) with \( 0 < i < k \), in which case (iii) would imply \( {b}_{k} = {b}_{i} \) with a contradiction. So \( {b}_{k} \) is unmatched, and hence \( P \) is an augmenting path between \( {a}_{0} \) and \( {b}_{k} \) .
Second proof. We apply induction on \( \left| A\right| \) . For \( \left| A\right| = 1 \) the assertion is true. Now let \( \left| A\right| \geq 2 \), and assume that the marriage condition is sufficient for the existence of a matching of \( A \) when \( \left| A\right| \) is smaller.
If \( \left| {N\left( S\right) }\right| \geq \left| S\right| + 1 \) for every non-empty set \( S \subsetneqq A \), we pick an edge \( {ab} \in G \) and consider the graph \( {G}^{\prime } \mathrel{\text{:=}} G - \{ a, b\} \) . Then every non-empty set \( S \subseteq A \smallsetminus \{ a\} \) satisfies
\[
\left| {{N}_{{G}^{\prime }}\left( S\right) }\right| \geq \left| {{N}_{G}\left( S\right) }\right| - 1 \geq \left| S\right|
\]
so by the induction hypothesis \( {G}^{\prime } \) contains a matching of \( A \smallsetminus \{ a\} \) . Together with the edge \( {ab} \), this yields a matching of \( A \) in \( G \) .
Suppose now that \( A \) has a non-empty proper subset \( {A}^{\prime } \) with \( \left| {B}^{\prime }\right| = \) \( \left| {A}^{\prime }\right| \) for \( {B}^{\prime } \mathrel{\text{:=}} N\left( {A}^{\prime }\right) \) . By the induction hypothesis, \( {G}^{\prime } \mathrel{\text{:=}} G\left\lbrack {{A}^{\prime } \cup {B}^{\prime }}\right\rbrack \) contains a matching of \( {A}^{\prime } \) . But \( G - {G}^{\prime } \) satisfies the marriage condition too: for any set \( S \subseteq A \smallsetminus {A}^{\prime } \) with \( \left| {{N}_{G - {G}^{\prime }}\left( S\right) }\right| < \left| S\right| \) we would have \( \left| {{N}_{G}\left( {S \cup {A}^{\prime }}\right) }\right| < \left| {S \cup {A}^{\prime }}\right| \), contrary to our assumption. Again by induction, \( G - {G}^{\prime } \) contains a matching of \( A \smallsetminus {A}^{\prime } \) . Putting the two matchings together, we obtain a matching of \( A \) in \( G \) .
For our last proof, let \( H \) be a spanning subgraph of \( G \) that satisfies the marriage condition and is edge-minimal with this property. Note that \( {d}_{H}\left( a\right) \geq 1 \) for every \( a \in A \), by the marriage condition with \( S = \{ a\} \) .
Third proof. We show that \( {d}_{H}\left( a\right) = 1 \) for every \( a \in A \) . The edges of \( H \) then form a matching of \( A \), since by the marriage condition no two such edges can share a vertex in \( B \) .
Suppose \( a \) has distinct neighbours \( {b}_{1},{b}_{2} \) in \( H \) . By definition of \( H \) , the graphs \( H - a{b}_{1} \) and \( H - a{b}_{2} \) violate the marriage condition. So for \( i = 1,2 \) there is a set \( {A}_{i} \subseteq A \) containing \( a \) such that \( \left| {A}_{i}\right| > \left| {B}_{i}\right| \) for \( {B}_{i} \mathrel{\text{:=}} {N}_{H - a{b}_{i}}\left( {A}_{i}\right) \) . Since \( {b}_{1} \in {B}_{2} \) and \( {b}_{2} \in {B}_{1} \), we obtain
\[
\left| {{N}_{H}\left( {{A}_{1} \cap {A}_{2}\smallsetminus \{ a\} }\right) }\right| \leq \left| {{B}_{1} \cap {B}_{2}}\right|
\]
\[
= \left| {B}_{1}\right| + \left| {B}_{2}\right| - \left| {{B}_{1} \cup {B}_{2}}\right|
\]
\[
= \left| {B}_{1}\right| + \left| {B}_{2}\right| - \left| {{N}_{H}\left( {{A}_{1} \cup {A}_{2}}\right) }\right|
\]
\[
\leq \left| {A}_{1}\right| - 1 + \left| {A}_{2}\right| - 1 - \left| {{A}_{1} \cup {A}_{2}}\right|
\]
\[
= \left| {{A}_{1} \cap {A}_{2}\smallsetminus \{ a\} }\right| - 1\text{.}
\]
Hence \( H \) violates the marriage condition, contrary to assumption.
This last proof has a pretty 'dual', which begins by showing that \( {d}_{H}\left( b\right) \leq 1 \) for every \( b \in B \) . See Exercise 5 and its hint for details.
Corollary 2.1.3. If \( G \) is \( k \) -regular with \( k \geq 1 \), then \( G \) has a 1 -factor. Proof. If \( G \) is \( k \) -regular, then clearly \( \left| A\right| = \left| B\right| \) ; it thus suffices to show by Theorem 2.1.2 that \( G \) contains a matching of \( A \) . Now every set \( S \subseteq A \) is joined to \( N\left( S\right) \) by a total of \( k\left| S\right| \) edges, and these are among the \( k\left| {N\left( S\right) }\right| \) edges of \( G \) incident with \( N\left( S\right) \) . Therefore \( k\left| S\right| \leq k\left| {N\left( S\right) }\right| \), so \( G \) does indeed satisfy the marriage condition.
In some real-life applications, matchings are not chosen on the basis of global criteria for the entire graph but evolve as the result of independent decisions made locally by the participating vertices. A typical situation is that vertices are not indifferent to which of their incident edges are picked to match them, but prefer some to others. Then if \( M \) is a matching and \( e = {ab} \) is an edge not in \( M \) such that both \( a \) and \( b \) prefer \( e \) to their current matching edge (if they are matched), then \( a \) and \( b \) may agree to change \( M \) locally by including \( e \) and discarding their earlier matching edges. The matching \( M \), although perhaps of maximal size, would thus be unstable.
preferences More formally, call a family \( {\left( { \leq }_{v}\right) }_{v \in V} \) of linear orderings \( { \leq }_{v} \) on \( E\left( v\right) \) a set of preferences for \( G \) . Then call a matching \( M \) in \( G \) stable if for
stable every edge \( e \in E \smallsetminus M \) there exists an edge \( f \in M \) such that \( e \) and \( f \) matching have a common vertex \( v \) with \( e{ < }_{v}f \) . The following result is sometimes called the stable marriage theorem:
\( \left\lbrack {5.4.4}\right\rbrack \)
Theorem 2.1.4. (Gale & Shapley 1962)
For every set of preferences, \( G \) has a stable matching.
Proof. Call a matching \( M \) in \( G \) better than a matching \( {M}^{\prime } \neq M \) if \( M \) makes the vertices in \( B \) happier than \( {M}^{\prime } \) does, that is, if every vertex \( b \) in an edge \( {f}^{\prime } \in {M}^{\prime } \) is incident also with some \( f \in M \) such that \( {f}^{\prime }{ \leq }_{b}f \) . Given a matching \( M \), call a vertex \( a \in A \) acceptable to \( b \in B \) if \( e = {ab} \in \) \( E \smallsetminus M \) and any edge \( f \in M \) at \( b \) satisfies \( f{ < }_{b}e \) . Call \( a \in A \) happy with \( M \) if \( a \) is unmatched or its matching edge \( f \in M \) satisfies \( f{ > }_{a}e \) for all edges \( e = {ab} \) such that \( a \) is acceptable to \( b \) .
Starting with the empty matching, let us construct a sequence of matchings that each keep all the vertices in \( A \) happy. Given such a matching \( M \), consider a vertex \( a \in A \) that is unmatched but acceptable to some \( b \in B \) . (If no such \( a \) exists, terminate the sequence.) Add to \( M \) the \( { \leq }_{a} \) -maximal edge \( {ab} \) such that \( a \) is acceptable to \( b \), and discard from \( M \) any other edge at \( b \) .
Clearly, each matching in our sequence is better than the previous, and it is easy to check inductively that they all keep the vertices in \( A \) happy. So the sequence continues until it terminates with a matching \( M \) such that every unmatched vertex in \( A \) is inacceptable to all its neighbours in \( B \) . As every matched vertex in \( A \) is happy with \( M \), this matching is stable.
Despite its seemingly narrow formulation, the marriage theorem counts among the most frequently applied graph theorems, both outside graph theory and within. Often, however, recasting a problem in the setting of bipartite matching requires some clever adaptation. As a simple example, we now use the marriage theorem to derive one of the earliest results of graph theory, a result whose original proof is not all that simple, and certainly not short:
## Corollary 2.1.5. (Petersen 1891)
Every regular graph of positive even degree has a 2-factor.
Proof. Let \( G \) be any \( {2k} \) -regular graph \( \left( {k \geq 1}\right) \), without loss of generality
(1.8.1)
connected. By Theorem 1.8.1, \( G \) contains an Euler tour \( {v}_{0}{e}_{0}\ldots {e}_{\ell - 1}{v}_{\ell } \) , with \( {v}_{\ell } = {v}_{0} \) . We replace every vertex \( v \) by a pair \( \left( {{v}^{ - },{v}^{ + }}\right) \), and every edge \( {e}_{i} = {v}_{i}{v}_{i + 1} \) by the edge \( {v}_{i}^{ + }{v}_{i + 1}^{ - } \) (Fig. 2.1.4). The resulting bipartite graph \( {G}^{\prime } \) is \( k \) -regular, so by Corollary 2.1.3 it has a 1-factor. Collapsing every vertex pair \( \left( {{v}^{ - },{v}^{ + }}\right) \) back into a single vertex \( v \), we turn this 1- factor of \( {G}^{\prime } \) into a 2 -factor of \( G \) .
Fig. 2.1.4. Splitting vertices in the proof of Corollary 2.1.5
## 2.2 Matching in general graphs
Given a graph \( G \), let us denote by \( {\mathcal{C}}_{G} \) the set of its components, and by \( {\mathcal{C}}_{G} \) \( q\left( G\right) \) the number of its odd components, tho |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 14 | \( G \) contains an Euler tour \( {v}_{0}{e}_{0}\ldots {e}_{\ell - 1}{v}_{\ell } \) , with \( {v}_{\ell } = {v}_{0} \) . We replace every vertex \( v \) by a pair \( \left( {{v}^{ - },{v}^{ + }}\right) \), and every edge \( {e}_{i} = {v}_{i}{v}_{i + 1} \) by the edge \( {v}_{i}^{ + }{v}_{i + 1}^{ - } \) (Fig. 2.1.4). The resulting bipartite graph \( {G}^{\prime } \) is \( k \) -regular, so by Corollary 2.1.3 it has a 1-factor. Collapsing every vertex pair \( \left( {{v}^{ - },{v}^{ + }}\right) \) back into a single vertex \( v \), we turn this 1- factor of \( {G}^{\prime } \) into a 2 -factor of \( G \) .
Fig. 2.1.4. Splitting vertices in the proof of Corollary 2.1.5
## 2.2 Matching in general graphs
Given a graph \( G \), let us denote by \( {\mathcal{C}}_{G} \) the set of its components, and by \( {\mathcal{C}}_{G} \) \( q\left( G\right) \) the number of its odd components, those of odd order. If \( G \) has a
\( q\left( G\right) \)
1-factor, then clearly
Tutte's
\[
q\left( {G - S}\right) \leq \left| S\right| \;\text{ for all }S \subseteq V\left( G\right) ,
\]
condition
since every odd component of \( G - S \) will send a factor edge to \( S \) .
Again, this obvious necessary condition for the existence of a 1-factor is also sufficient:
Theorem 2.2.1. (Tutte 1947)
A graph \( G \) has a 1-factor if and only if \( q\left( {G - S}\right) \leq \left| S\right| \) for all \( S \subseteq V\left( G\right) \) .
Fig. 2.2.1. Tutte’s condition \( q\left( {G - S}\right) \leq \left| S\right| \) for \( q = 3 \), and the contracted graph \( {G}_{S} \) from Theorem 2.2.3.
\( V, E \) Proof. Let \( G = \left( {V, E}\right) \) be a graph without a 1-factor. Our task is to
bad set find a bad set \( S \subseteq V \), one that violates Tutte’s condition.
We may assume that \( G \) is edge-maximal without a 1-factor. Indeed, if \( {G}^{\prime } \) is obtained from \( G \) by adding edges and \( S \subseteq V \) is bad for \( {G}^{\prime } \), then \( S \) is also bad for \( G \) : any odd component of \( {G}^{\prime } - S \) is the union of components of \( G - S \), and one of these must again be odd.
What does \( G \) look like? Clearly, if \( G \) contains a bad set \( S \) then, by its edge-maximality and the trivial forward implication of the theorem,
all the components of \( G - S \) are complete and every vertex
\( \left( *\right) \)
\( s \in S \) is adjacent to all the vertices of \( G - s \) .
But also conversely, if a set \( S \subseteq V \) satisfies \( \left( *\right) \) then either \( S \) or the empty set must be bad: if \( S \) is not bad we can join the odd components of \( G - S \) disjointly to \( S \) and pair up all the remaining vertices - unless \( \left| G\right| \) is odd, in which case \( \varnothing \) is bad.
So it suffices to prove that \( G \) has a set \( S \) of vertices satisfying \( \left( *\right) \) . \( S \) Let \( S \) be the set of vertices that are adjacent to every other vertex. If this set \( S \) does not satisfy \( \left( *\right) \), then some component of \( G - S \) has non- \( a, b, c \) adjacent vertices \( a,{a}^{\prime } \) . Let \( a, b, c \) be the first three vertices on a shortest \( a - {a}^{\prime } \) path in this component; then \( {ab},{bc} \in E \) but \( {ac} \notin E \) . Since \( b \notin S \) , \( d \) there is a vertex \( d \in V \) such that \( {bd} \notin E \) . By the maximality of \( G \), there \( {M}_{1},{M}_{2} \) is a matching \( {M}_{1} \) of \( V \) in \( G + {ac} \), and a matching \( {M}_{2} \) of \( V \) in \( G + {bd} \) .
Fig. 2.2.2. Deriving a contradiction if \( S \) does not satisfy \( \left( *\right) \)
Let \( P = d\ldots v \) be a maximal path in \( G \) starting at \( d \) with an edge from \( {M}_{1} \) and containing alternately edges from \( {M}_{1} \) and \( {M}_{2} \) (Fig. 2.2.2). If the last edge of \( P \) lies in \( {M}_{1} \), then \( v = b \), since otherwise we could continue \( P \) . Let us then set \( C \mathrel{\text{:=}} P + {bd} \) . If the last edge of \( P \) lies in \( {M}_{2} \) , then by the maximality of \( P \) the \( {M}_{1} \) -edge at \( v \) must be \( {ac} \), so \( v \in \{ a, c\} \) ; then let \( C \) be the cycle \( {dPvbd} \) . In each case, \( C \) is an even cycle with every other edge in \( {M}_{2} \), and whose only edge not in \( E \) is \( {bd} \) . Replacing in \( {M}_{2} \) its edges on \( C \) with the edges of \( C - {M}_{2} \), we obtain a matching of \( V \) contained in \( E \), a contradiction.
Corollary 2.2.2. (Petersen 1891)
Every bridgeless cubic graph has a 1-factor.
Proof. We show that any bridgeless cubic graph \( G \) satisfies Tutte’s condition. Let \( S \subseteq V\left( G\right) \) be given, and consider an odd component \( C \) of \( G - S \) . Since \( G \) is cubic, the degrees (in \( G \) ) of the vertices in \( C \) sum to an odd number, but only an even part of this sum arises from edges of \( C \) . So \( G \) has an odd number of \( S - C \) edges, and therefore has at least 3 such edges (since \( G \) has no bridge). The total number of edges between \( S \) and \( G - S \) thus is at least \( {3q}\left( {G - S}\right) \) . But it is also at most \( 3\left| S\right| \), because \( G \) is cubic. Hence \( q\left( {G - S}\right) \leq \left| S\right| \), as required.
In order to shed a little more light on the techniques used in matching theory, we now give a second proof of Tutte's theorem. In fact, we shall prove a slightly stronger result, a result that places a structure interesting from the matching point of view on an arbitrary graph. If the graph happens to satisfy the condition of Tutte's theorem, this structure will at once yield a 1-factor.
A graph \( G = \left( {V, E}\right) \) is called factor-critical if \( G \neq \varnothing \) and \( G - v \) factor- critical has a 1-factor for every vertex \( v \in G \) . Then \( G \) itself has no 1-factor,
because it has odd order. We call a vertex set \( S \subseteq V \) matchable to matchable \( {\mathcal{C}}_{G - S} \) if the (bipartite \( {}^{1} \) ) graph \( {G}_{S} \), which arises from \( G \) by contracting the components \( C \in {\mathcal{C}}_{G - S} \) to single vertices and deleting all the edges inside \( S \), contains a matching of \( S \) . (Formally, \( {G}_{S} \) is the graph with \( {G}_{S} \) vertex set \( S \cup {\mathcal{C}}_{G - S} \) and edge set \( \{ {sC} \mid \exists c \in C : {sc} \in E\} \) ; see Fig. 2.2.1.)
Theorem 2.2.3. Every graph \( G = \left( {V, E}\right) \) contains a vertex set \( S \) with the following two properties:
(i) \( S \) is matchable to \( {\mathcal{C}}_{G - S} \) ;
(ii) Every component of \( G - S \) is factor-critical.
Given any such set \( S \), the graph \( G \) contains a 1 -factor if and only if \( \left| S\right| = \left| {\mathcal{C}}_{G - S}\right| \) .
1 except for the-permitted-case that \( S \) or \( {\mathcal{C}}_{G - S} \) is empty
For any given \( G \), the assertion of Tutte’s theorem follows easily from this result. Indeed, by (i) and (ii) we have \( \left| S\right| \leq \left| {\mathcal{C}}_{G - S}\right| = q\left( {G - S}\right) \) (since factor-critical graphs have odd order); thus Tutte's condition of \( q\left( {G - S}\right) \leq \left| S\right| \) implies \( \left| S\right| = \left| {\mathcal{C}}_{G - S}\right| \), and the existence of a 1-factor follows from the last statement of Theorem 2.2.3.
\( \left( {2.1.2}\right) \)
Proof of Theorem 2.2.3. Note first that the last assertion of the theorem follows at once from the assertions (i) and (ii): if \( G \) has a 1-factor, we have \( q\left( {G - S}\right) \leq \left| S\right| \) and hence \( \left| S\right| = \left| {\mathcal{C}}_{G - S}\right| \) as above; conversely if \( \left| S\right| = \left| {\mathcal{C}}_{G - S}\right| \), then the existence of a 1-factor follows straight from (i) and (ii).
We now prove the existence of a set \( S \) satisfying (i) and (ii), by induction on \( \left| G\right| \) . For \( \left| G\right| = 0 \) we may take \( S = \varnothing \) . Now let \( G \) be given with \( \left| G\right| > 0 \), and assume the assertion holds for graphs with fewer vertices.
Consider the sets \( T \subseteq V \) for which Tutte’s condition fails worst, i.e. for which
\[
d\left( T\right) \mathrel{\text{:=}} {d}_{G}\left( T\right) \mathrel{\text{:=}} q\left( {G - T}\right) - \left| T\right|
\]
is maximum, and let \( S \) be a largest such set \( T \) . Note that \( d\left( S\right) \geq d\left( \varnothing \right) \geq 0 \) .
We first show that every component \( C \in {\mathcal{C}}_{G - S} = : \mathcal{C} \) is odd. If \( \left| C\right| \) is even, pick a vertex \( c \in C \), and consider \( T \mathrel{\text{:=}} S \cup \{ c\} \) . As \( C - c \) has odd order it has at least one odd component, which is also a component of \( G - T \) . Therefore
\[
q\left( {G - T}\right) \geq q\left( {G - S}\right) + 1\;\text{ while }\;\left| T\right| = \left| S\right| + 1,
\]
so \( d\left( T\right) \geq d\left( S\right) \) contradicting the choice of \( S \) .
Next we prove the assertion (ii), that every \( C \in \mathcal{C} \) is factor-critical. Suppose there exist \( C \in \mathcal{C} \) and \( c \in C \) such that \( {C}^{\prime } \mathrel{\text{:=}} C - c \) has no 1-factor. By the induction hypothesis (and the fact that, as shown earlier, for fixed \( G \) our theorem implies Tutte’s theorem) there exists a set \( {S}^{\prime } \subseteq V\left( {C}^{\prime }\right) \) with
\[
q\left( {{C}^{\prime } - {S}^{\prime }}\right) > \left| {S}^{\prime }\right| .
\]
Since \( \left| C\right| \) is odd and hence \( \left| {C}^{\prime }\right| \) is even, the num |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 15 | \in C \), and consider \( T \mathrel{\text{:=}} S \cup \{ c\} \) . As \( C - c \) has odd order it has at least one odd component, which is also a component of \( G - T \) . Therefore
\[
q\left( {G - T}\right) \geq q\left( {G - S}\right) + 1\;\text{ while }\;\left| T\right| = \left| S\right| + 1,
\]
so \( d\left( T\right) \geq d\left( S\right) \) contradicting the choice of \( S \) .
Next we prove the assertion (ii), that every \( C \in \mathcal{C} \) is factor-critical. Suppose there exist \( C \in \mathcal{C} \) and \( c \in C \) such that \( {C}^{\prime } \mathrel{\text{:=}} C - c \) has no 1-factor. By the induction hypothesis (and the fact that, as shown earlier, for fixed \( G \) our theorem implies Tutte’s theorem) there exists a set \( {S}^{\prime } \subseteq V\left( {C}^{\prime }\right) \) with
\[
q\left( {{C}^{\prime } - {S}^{\prime }}\right) > \left| {S}^{\prime }\right| .
\]
Since \( \left| C\right| \) is odd and hence \( \left| {C}^{\prime }\right| \) is even, the numbers \( q\left( {{C}^{\prime } - {S}^{\prime }}\right) \) and \( \left| {S}^{\prime }\right| \) are either both even or both odd, so they cannot differ by exactly 1 . We may therefore sharpen the above inequality to
\[
q\left( {{C}^{\prime } - {S}^{\prime }}\right) \geq \left| {S}^{\prime }\right| + 2
\]
giving \( {d}_{{C}^{\prime }}\left( {S}^{\prime }\right) \geq 2 \) . Then for \( T \mathrel{\text{:=}} S \cup \{ c\} \cup {S}^{\prime } \) we have
\[
d\left( T\right) \geq d\left( S\right) - 1 - 1 + {d}_{{C}^{\prime }}\left( {S}^{\prime }\right) \geq d\left( S\right)
\]
where the first ’ -1 ’ comes from the loss of \( C \) as an odd component and the second comes from including \( c \) in the set \( T \) . As before, this contradicts the choice of \( S \) .
It remains to show that \( S \) is matchable to \( {\mathcal{C}}_{G - S} \) . If not, then by the marriage theorem there exists a set \( {S}^{\prime } \subseteq S \) that sends edges to fewer than \( \left| {S}^{\prime }\right| \) components in \( \mathcal{C} \) . Since the other components in \( \mathcal{C} \) are also components of \( G - \left( {S \smallsetminus {S}^{\prime }}\right) \), the set \( T = S \smallsetminus {S}^{\prime } \) satisfies \( d\left( T\right) > d\left( S\right) \) , contrary to the choice of \( S \) .
Let us consider once more the set \( S \) from Theorem 2.2.3, together with any matching \( M \) in \( G \) . As before, we write \( \mathcal{C} \mathrel{\text{:=}} {\mathcal{C}}_{G - S} \) . Let us denote by \( {k}_{S} \) the number of edges in \( M \) with at least one end in \( S \), and by \( {k}_{\mathcal{C}} \) the number of edges in \( M \) with both ends in \( G - S \) . Since each \( {k}_{S},{k}_{\mathcal{C}} \) \( C \in \mathcal{C} \) is odd, at least one of its vertices is not incident with an edge of the second type. Therefore every matching \( M \) satisfies
\[
{k}_{S} \leq \left| S\right| \text{ and }{k}_{\mathcal{C}} \leq \frac{1}{2}\left( {\left| V\right| - \left| S\right| - \left| \mathcal{C}\right| }\right) .
\]
(1)
Moreover, \( G \) contains a matching \( {M}_{0} \) with equality in both cases: first choose \( \left| S\right| \) edges between \( S \) and \( \bigcup \mathcal{C} \) according to (i), and then use (ii) to find a suitable set of \( \frac{1}{2}\left( {\left| C\right| - 1}\right) \) edges in every component \( C \in \mathcal{C} \) . This matching \( {M}_{0} \) thus has exactly
\[
\left| {M}_{0}\right| = \left| S\right| + \frac{1}{2}\left( {\left| V\right| - \left| S\right| - \left| \mathcal{C}\right| }\right)
\]
(2)
edges.
Now (1) and (2) together imply that every matching \( M \) of maximum cardinality satisfies both parts of (1) with equality: by \( \left| M\right| \geq \left| {M}_{0}\right| \) and (2), \( M \) has at least \( \left| S\right| + \frac{1}{2}\left( {\left| V\right| - \left| S\right| - \left| \mathcal{C}\right| }\right) \) edges, which implies by (1) that neither of the inequalities in (1) can be strict. But equality in (1), in turn, implies that \( M \) has the structure described above: by \( {k}_{S} = \left| S\right| \), every vertex \( s \in S \) is the end of an edge \( {st} \in M \) with \( t \in G - S, \) and by \( {k}_{\mathcal{C}} = \frac{1}{2}\left( {\left| V\right| - \left| S\right| - \left| \mathcal{C}\right| }\right) \) exactly \( \frac{1}{2}\left( {\left| C\right| - 1}\right) \) edges of \( M \) lie in \( C \) , for every \( C \in \mathcal{C} \) . Finally, since these latter edges miss only one vertex in each \( C \), the ends \( t \) of the edges \( {st} \) above lie in different components \( C \) for different \( s \) .
The seemingly technical Theorem 2.2.3 thus hides a wealth of structural information: it contains the essence of a detailed description of all maximum-cardinality matchings in all graphs. A reference to the full statement of this structural result, known as the Gallai-Edmonds matching theorem, is given in the notes at the end of this chapter.
## 2.3 Packing and covering
Much of the charm of König's and Hall's theorems in Section 2.1 lies in the fact that they guarantee the existence of the desired matching as soon as some obvious obstruction does not occur. In König's theorem, we can find \( k \) independent edges in our graph unless we can cover all its edges by fewer than \( k \) vertices (in which case it is obviously impossible).
More generally, if \( G \) is an arbitrary graph, not necessarily bipartite, and \( \mathcal{H} \) is any class of graphs, we might compare the largest number \( k \) of graphs from \( \mathcal{H} \) (not necessarily distinct) that we can pack disjointly into \( G \) with the smallest number \( s \) of vertices of \( G \) that will cover all its
---
Erdős-Pósa property
---
subgraphs in \( \mathcal{H} \) . If \( s \) can be bounded by a function of \( k \), i.e. independently of \( G \), we say that \( \mathcal{H} \) has the Erdős-Pósa property. (Thus, formally, \( \mathcal{H} \) has this property if there exists an \( \mathbb{N} \rightarrow \mathbb{R} \) function \( k \mapsto f\left( k\right) \) such that, for every \( k \) and \( G \), either \( G \) contains \( k \) disjoint subgraphs each isomorphic to a graph in \( \mathcal{H} \), or there is a set \( U \subseteq V\left( G\right) \) of at most \( f\left( k\right) \) vertices such that \( G - U \) has no subgraph in \( \mathcal{H} \) .)
Our aim in this section is to prove the theorem of Erdős and Pósa that the class of all cycles has this property: we shall find a function \( f \) (about \( {4k}\log k \) ) such that every graph contains either \( k \) disjoint cycles or a set of at most \( f\left( k\right) \) vertices covering all its cycles.
We begin by proving a stronger assertion for cubic graphs. For \( k \in \mathbb{N} \), put
\( {r}_{k},{s}_{k} \)
\[
{r}_{k} \mathrel{\text{:=}} \log k + \log \log k + 4\;\text{ and }\;{s}_{k} \mathrel{\text{:=}} \left\{ \begin{array}{ll} {4k}{r}_{k} & \text{ if }k \geq 2 \\ 1 & \text{ if }k \leq 1 \end{array}\right.
\]
Lemma 2.3.1. Let \( k \in \mathbb{N} \), and let \( H \) be a cubic multigraph. If \( \left| H\right| \geq {s}_{k} \) , then \( H \) contains \( k \) disjoint cycles.
\( \left( {1.3.5}\right) \)
Proof. We apply induction on \( k \) . For \( k \leq 1 \) the assertion is trivial, so let \( k \geq 2 \) be given for the induction step. Let \( C \) be a shortest cycle in \( H \) .
We first show that \( H - C \) contains a subdivision of a cubic multigraph \( {H}^{\prime } \) with \( \left| {H}^{\prime }\right| \geq \left| H\right| - 2\left| C\right| \) . Let \( m \) be the number of edges between \( C \) and \( H - C \) . Since \( H \) is cubic and \( d\left( C\right) = 2 \), we have \( m \leq \left| C\right| \) . We now consider bipartitions \( \left\{ {{V}_{1},{V}_{2}}\right\} \) of \( V\left( H\right) \), beginning with \( {V}_{1} \mathrel{\text{:=}} V\left( C\right) \) . If \( H\left\lbrack {V}_{2}\right\rbrack \) has a vertex of degree at most 1 we move this vertex to \( {V}_{1} \) , obtaining a new partition \( \left\{ {{V}_{1},{V}_{2}}\right\} \) crossed by fewer edges. Suppose we can perform a sequence of \( n \) such moves, but no more. Then the resulting partition \( \left\{ {{V}_{1},{V}_{2}}\right\} \) is crossed by at most \( m - n \) edges. And \( H\left\lbrack {V}_{2}\right\rbrack \) has at most \( m - n \) vertices of degree less than 3, because each of these is incident with a cut edge. These vertices have degree exactly 2 in \( H\left\lbrack {V}_{2}\right\rbrack \), since we could not move them to \( {V}_{1} \) . Let \( {H}^{\prime } \) be the cubic multigraph obtained from \( H\left\lbrack {V}_{2}\right\rbrack \) by suppressing these vertices. Then
\[
\left| {H}^{\prime }\right| \geq \left| H\right| - \left| C\right| - n - \left( {m - n}\right) \geq \left| H\right| - 2\left| C\right|
\]
as desired.
To complete the proof, it suffices to show that \( \left| {H}^{\prime }\right| \geq {s}_{k - 1} \) . Since \( \left| C\right| \leq 2\log \left| H\right| \) by Corollary 1.3.5 (or by \( \left| H\right| \geq {s}_{k} \), if \( \left| C\right| = g\left( H\right) \leq 2 \) ), and \( \left| H\right| \geq {s}_{k} \geq 6 \), we have
\[
\left| {H}^{\prime }\right| \geq \left| H\right| - 2\left| C\right| \geq \left| H\right| - 4\log \left| H\right| \geq {s}_{k} - 4\log {s}_{k}.
\]
(In the last inequality we use that the function \( x \mapsto x - 4\log x \) increases for \( x \geq 6 \) .)
It thus remains to show that \( {s}_{k} - 4\log {s}_{k} \geq {s}_{k - 1} \) . For \( k = 2 \) this is clear, so we assume that \( k \geq 3 \) . Then \( {r}_{k} \leq 4\log k \) (which is obvious for \( k \geq 4 \), while the case of \( k = 3 \) has to be calculated), and hence
\[
{s}_{k} - 4\log {s}_{k} = 4\left( {k - 1}\right) {r}_{k} + 4\log k + 4\log \log k + {16}
\]
\[
- \left( {8 + 4\log k + 4\log {r}_{k}}\right)
\]
\[
\geq {s}_{k - 1} + 4\log \log k + 8 - 4\log \left( {4\log k}\right)
\]
\[
= {s}_{k - 1}\text{.}
\]
Theore |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 16 | \leq 2\log \left| H\right| \) by Corollary 1.3.5 (or by \( \left| H\right| \geq {s}_{k} \), if \( \left| C\right| = g\left( H\right) \leq 2 \) ), and \( \left| H\right| \geq {s}_{k} \geq 6 \), we have
\[
\left| {H}^{\prime }\right| \geq \left| H\right| - 2\left| C\right| \geq \left| H\right| - 4\log \left| H\right| \geq {s}_{k} - 4\log {s}_{k}.
\]
(In the last inequality we use that the function \( x \mapsto x - 4\log x \) increases for \( x \geq 6 \) .)
It thus remains to show that \( {s}_{k} - 4\log {s}_{k} \geq {s}_{k - 1} \) . For \( k = 2 \) this is clear, so we assume that \( k \geq 3 \) . Then \( {r}_{k} \leq 4\log k \) (which is obvious for \( k \geq 4 \), while the case of \( k = 3 \) has to be calculated), and hence
\[
{s}_{k} - 4\log {s}_{k} = 4\left( {k - 1}\right) {r}_{k} + 4\log k + 4\log \log k + {16}
\]
\[
- \left( {8 + 4\log k + 4\log {r}_{k}}\right)
\]
\[
\geq {s}_{k - 1} + 4\log \log k + 8 - 4\log \left( {4\log k}\right)
\]
\[
= {s}_{k - 1}\text{.}
\]
Theorem 2.3.2. (Erdős & Pósa 1965)
There is a function \( f : \mathbb{N} \rightarrow \mathbb{R} \) such that, given any \( k \in \mathbb{N} \), every graph contains either \( k \) disjoint cycles or a set of at most \( f\left( k\right) \) vertices meeting all its cycles.
Proof. We show the result for \( f\left( k\right) \mathrel{\text{:=}} {s}_{k} + k - 1 \) . Let \( k \) be given, and let \( G \) be any graph. We may assume that \( G \) contains a cycle, and so it has a maximal subgraph \( H \) in which every vertex has degree 2 or 3 . Let \( U \) be its set of degree 3 vertices.
Let \( \mathcal{C} \) be the set of all cycles in \( G \) that avoid \( U \) and meet \( H \) in exactly one vertex. Let \( Z \subseteq V\left( H\right) \smallsetminus U \) be the set of those vertices. For each \( z \in Z \) pick a cycle \( {C}_{z} \in \mathcal{C} \) that meets \( H \) in \( z \), and put \( {\mathcal{C}}^{\prime } \mathrel{\text{:=}} \left\{ {{C}_{z} \mid z \in Z}\right\} \) . By the maximality of \( H \), the cycles in \( {\mathcal{C}}^{\prime } \) are disjoint.
Let \( \mathcal{D} \) be the set of the 2-regular components of \( H \) that avoid \( Z \) . Then \( {\mathcal{C}}^{\prime } \cup \mathcal{D} \) is another set of disjoint cycles. If \( \left| {{\mathcal{C}}^{\prime } \cup \mathcal{D}}\right| \geq k \), we are done. Otherwise we can add to \( Z \) one vertex from each cycle in \( \mathcal{D} \) to obtain a set \( X \) of at most \( k - 1 \) vertices that meets all the cycles in \( \mathcal{C} \) and all the 2-regular components of \( H \) . Now consider any cycle of \( G \) that avoids \( X \) . By the maximality of \( H \) it meets \( H \) . But it is not a component of \( H \), it does not lie in \( \mathcal{C} \), and it does not contain an \( H \) -path between distinct vertices outside \( U \) (by the maximality of \( H \) ). So this cycle meets \( U \) .
We have shown that every cycle in \( G \) meets \( X \cup U \) . As \( \left| X\right| \leq k - 1 \) , it thus suffices to show that \( \left| U\right| < {s}_{k} \) unless \( H \) contains \( k \) disjoint cycles. But this follows from Lemma 2.3.1 applied to the graph obtained from \( H \) by suppressing its vertices of degree 2 .
We shall meet the Erdős-Pósa property again in Chapter 12. There, a considerable extension of Theorem 2.3.2 will appear as an unexpected and easy corollary of the theory of graph minors.
## 2.4 Tree-packing and arboricity
In this section we consider packing and covering in terms of edges rather than vertices. How many edge-disjoint spanning trees can we find in a given graph? And how few trees in it, not necessarily edge-disjoint, suffice to cover all its edges?
To motivate the tree-packing problem, assume for a moment that our graph represents a communication network, and that for every choice of two vertices we want to be able to find \( k \) edge-disjoint paths between them. Menger’s theorem (3.3.6(ii)) in the next chapter will tell us that such paths exist as soon as our graph is \( k \) -edge-connected, which is clearly also necessary. This is a good theorem, but it does not tell us how to find those paths; in particular, having found them for one pair of endvertices we are not necessarily better placed to find them for another pair. If our graph has \( k \) edge-disjoint spanning trees, however, there will always be \( k \) canonical such paths, one in each tree. Once we have stored those trees in our computer, we shall always be able to find the \( k \) paths quickly, for any given pair of endvertices.
When does a graph \( G \) have \( k \) edge-disjoint spanning trees? If it does, it clearly must be \( k \) -edge-connected. The converse, however, is easily seen to be false (try \( k = 2 \) ); indeed it is not even clear at that any edge-connectivity will imply the existence of \( k \) edge-disjoint spanning trees. (But see Corollary 2.4.2 below.)
Here is another necessary condition. If \( G \) has \( k \) edge-disjoint spanning trees, then with respect to any partition of \( V\left( G\right) \) into \( r \) sets, every
---
cross-edges
---
spanning tree of \( G \) has at least \( r - 1 \) cross-edges, edges whose ends lie in different partition sets (why?). Thus if \( G \) has \( k \) edge-disjoint spanning trees, it has at least \( k\left( {r - 1}\right) \) cross-edges. This condition is also sufficient:
Theorem 2.4.1. (Nash-Williams 1961; Tutte 1961)
A multigraph contains \( k \) edge-disjoint spanning trees if and only if for every partition \( P \) of its vertex set it has at least \( k\left( {\left| P\right| - 1}\right) \) cross-edges.
Before we prove Theorem 2.4.1, let us note a surprising corollary: to ensure the existence of \( k \) edge-disjoint spanning trees, it suffices to raise the edge-connectivity to just \( {2k} \) :
\( \left\lbrack {6.4.4}\right\rbrack \)
Corollary 2.4.2. Every \( {2k} \) -edge-connected multigraph \( G \) has \( k \) edge-disjoint spanning trees.
Proof. Every set in a vertex partition of \( G \) is joined to other partition sets by at least \( {2k} \) edges. Hence, for any partition into \( r \) sets, \( G \) has at least \( \frac{1}{2}\mathop{\sum }\limits_{{i = 1}}^{r}{2k} = {kr} \) cross-edges. The assertion thus follows from Theorem 2.4.1.
For the proof of Theorem 2.4.1, let a multigraph \( G = \left( {V, E}\right) \) and \( G = \left( {V, E}\right) \) \( k \in \mathbb{N} \) be given. Let \( \mathcal{F} \) be the set of all \( k \) -tuples \( F = \left( {{F}_{1},\ldots ,{F}_{k}}\right) \) of
\( k,\mathcal{F} \)
edge-disjoint spanning forests in \( G \) with the maximum total number of edges, i.e. such that \( \parallel F\parallel \mathrel{\text{:=}} \left| {E\left\lbrack F\right\rbrack }\right| \) with \( E\left\lbrack F\right\rbrack \mathrel{\text{:=}} E\left( {F}_{1}\right) \cup \ldots \cup E\left( {F}_{k}\right) \)
\( E\left\lbrack F\right\rbrack ,\parallel F\parallel \)
is as large as possible.
If \( F = \left( {{F}_{1},\ldots ,{F}_{k}}\right) \in \mathcal{F} \) and \( e \in E \smallsetminus E\left\lbrack F\right\rbrack \), then every \( {F}_{i} + e \) contains a cycle \( \left( {i = 1,\ldots, k}\right) \) : otherwise we could replace \( {F}_{i} \) by \( {F}_{i} + e \) in \( F \) and obtain a contradiction to the maximality of \( \parallel F\parallel \) . Let us consider an edge \( {e}^{\prime } \neq e \) of this cycle, for some fixed \( i \) . Putting \( {F}_{i}^{\prime } \mathrel{\text{:=}} {F}_{i} + e - {e}^{\prime } \) , and \( {F}_{j}^{\prime } \mathrel{\text{:=}} {F}_{j} \) for all \( j \neq i \), we see that \( {F}^{\prime } \mathrel{\text{:=}} \left( {{F}_{1}^{\prime },\ldots ,{F}_{k}^{\prime }}\right) \) is again in \( \mathcal{F} \) ;
---
edge replacement
---
we say that \( {F}^{\prime } \) has been obtained from \( F \) by the replacement of the edge \( {e}^{\prime } \) with \( e \) . Note that the component of \( {F}_{i} \) containing \( {e}^{\prime } \) keeps its vertex set when it changes into a component of \( {F}_{i}^{\prime } \) . Hence for every path \( x\ldots y \subseteq {F}_{i}^{\prime } \) there is a unique path \( x{F}_{i}y \) in \( {F}_{i} \) ; this will be used later. \( x{F}_{i}y \)
We now consider a fixed \( k \) -tuple \( {F}^{0} = \left( {{F}_{1}^{0},\ldots ,{F}_{k}^{0}}\right) \in \mathcal{F} \) . The set \( {F}^{0} \) of all \( k \) -tuples in \( \mathcal{F} \) that can be obtained from \( {F}^{0} \) by a series of edge replacements will be denoted by \( {\mathcal{F}}^{0} \) . Finally, we let \( {\mathcal{F}}^{0} \)
\[
{E}^{0} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{F \in {\mathcal{F}}^{0}}}\left( {E \smallsetminus E\left\lbrack F\right\rbrack }\right)
\]
\( {E}^{0} \)
and \( {G}^{0} \mathrel{\text{:=}} \left( {V,{E}^{0}}\right) \) . \( {G}^{0} \)
Lemma 2.4.3. For every \( {e}^{0} \in E \smallsetminus E\left\lbrack {F}^{0}\right\rbrack \) there exists a set \( U \subseteq V \) that is connected in every \( {F}_{i}^{0}\left( {i = 1,\ldots, k}\right) \) and contains the ends of \( {e}^{0} \) .
Proof. As \( {F}^{0} \in {\mathcal{F}}^{0} \), we have \( {e}^{0} \in {E}^{0} \) ; let \( {C}^{0} \) be the component of \( {G}^{0} \) \( {C}^{0} \) containing \( {e}^{0} \) . We shall prove the assertion for \( U \mathrel{\text{:=}} V\left( {C}^{0}\right) \) .
Let \( i \in \{ 1,\ldots, k\} \) be given; we have to show that \( U \) is connected in \( {F}_{i}^{0} \) . To this end, we first prove the following:
Let \( F = \left( {{F}_{1},\ldots ,{F}_{k}}\right) \in {\mathcal{F}}^{0} \), and let \( \left( {{F}_{1}^{\prime },\ldots ,{F}_{k}^{\prime }}\right) \) have been
obtained from \( F \) by the replacement of an edge of \( {F}_{i} \) . If
(1)
\( x, y \) are the ends of a path in \( {F}_{i}^{\prime } \cap {C}^{0} \), then also \( x{F}_{i}y \subseteq {C}^{0} \) .
Let \( e = {vw} \) be the new edge in \( E\left( {F}_{i}^{\prime }\right) \smallsetminus E\left\lbrack F\right\rbrack \) ; this is the only edge of \( {F}_{i}^{\prime } \) no |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 17 | ) \) and contains the ends of \( {e}^{0} \) .
Proof. As \( {F}^{0} \in {\mathcal{F}}^{0} \), we have \( {e}^{0} \in {E}^{0} \) ; let \( {C}^{0} \) be the component of \( {G}^{0} \) \( {C}^{0} \) containing \( {e}^{0} \) . We shall prove the assertion for \( U \mathrel{\text{:=}} V\left( {C}^{0}\right) \) .
Let \( i \in \{ 1,\ldots, k\} \) be given; we have to show that \( U \) is connected in \( {F}_{i}^{0} \) . To this end, we first prove the following:
Let \( F = \left( {{F}_{1},\ldots ,{F}_{k}}\right) \in {\mathcal{F}}^{0} \), and let \( \left( {{F}_{1}^{\prime },\ldots ,{F}_{k}^{\prime }}\right) \) have been
obtained from \( F \) by the replacement of an edge of \( {F}_{i} \) . If
(1)
\( x, y \) are the ends of a path in \( {F}_{i}^{\prime } \cap {C}^{0} \), then also \( x{F}_{i}y \subseteq {C}^{0} \) .
Let \( e = {vw} \) be the new edge in \( E\left( {F}_{i}^{\prime }\right) \smallsetminus E\left\lbrack F\right\rbrack \) ; this is the only edge of \( {F}_{i}^{\prime } \) not lying in \( {F}_{i} \) . We assume that \( e \in x{F}_{i}^{\prime }y \) : otherwise we would have \( x{F}_{i}y = x{F}_{i}^{\prime }y \) and nothing to show. It suffices to show that \( v{F}_{i}w \subseteq {C}^{0} \) : then \( \left( {x{F}_{i}^{\prime }y - e}\right) \cup v{F}_{i}w \) is a connected subgraph of \( {F}_{i} \cap {C}^{0} \) that contains \( x, y \), and hence also \( x{F}_{i}y \) . Let \( {e}^{\prime } \) be any edge of \( v{F}_{i}w \) . Since we could replace \( {e}^{\prime } \) in \( F \in {\mathcal{F}}^{0} \) by \( e \) and obtain an element of \( {\mathcal{F}}^{0} \) not containing \( {e}^{\prime } \), we have \( {e}^{\prime } \in {E}^{0} \) . Thus \( v{F}_{i}w \subseteq {G}^{0} \), and hence \( v{F}_{i}w \subseteq {C}^{0} \) since \( v, w \in x{F}_{i}^{\prime }y \subseteq {C}^{0} \) . This proves (1).
In order to prove that \( U = V\left( {C}^{0}\right) \) is connected in \( {F}_{i}^{0} \) we show that, for every edge \( {xy} \in {C}^{0} \), the path \( x{F}_{i}^{0}y \) exists and lies in \( {C}^{0} \) . As \( {C}^{0} \) is connected, the union of all these paths will then be a connected spanning subgraph of \( {F}_{i}^{0}\left\lbrack U\right\rbrack \) .
So let \( e = {xy} \in {C}^{0} \) be given. As \( e \in {E}^{0} \), there exist an \( s \in \mathbb{N} \) and \( k \) -tuples \( {F}^{r} = \left( {{F}_{1}^{r},\ldots ,{F}_{k}^{r}}\right) \) for \( r = 1,\ldots, s \) such that each \( {F}^{r} \) is obtained from \( {F}^{r - 1} \) by edge replacement and \( e \in E \smallsetminus E\left\lbrack {F}^{s}\right\rbrack \) . Setting \( F \mathrel{\text{:=}} {F}^{s} \) in (1), we may think of \( e \) as a path of length 1 in \( {F}_{i}^{\prime } \cap {C}^{0} \) . Successive applications of (1) to \( F = {F}^{s},\ldots ,{F}^{0} \) then give \( x{F}_{i}^{0}y \subseteq {C}^{0} \) as desired.
\( \left( {1.5.3}\right) \)
Proof of Theorem 2.4.1. We prove the backward implication by induction on \( \left| G\right| \) . For \( \left| G\right| = 2 \) the assertion holds. For the induction step, we now suppose that for every partition \( P \) of \( V \) there are at least \( k\left( {\left| P\right| - 1}\right) \) cross-edges, and construct \( k \) edge-disjoint spanning trees in \( G \) .
\( {F}^{0} \) Pick a \( k \) -tuple \( {F}^{0} = \left( {{F}_{1}^{0},\ldots ,{F}_{k}^{0}}\right) \in \mathcal{F} \) . If every \( {F}_{i}^{0} \) is a tree, we are done. If not, we have
\[
\begin{Vmatrix}{F}^{0}\end{Vmatrix} = \mathop{\sum }\limits_{{i = 1}}^{k}\begin{Vmatrix}{F}_{i}^{0}\end{Vmatrix} < k\left( {\left| G\right| - 1}\right)
\]
by Corollary 1.5.3. On the other hand, we have \( \parallel G\parallel \geq k\left( {\left| G\right| - 1}\right) \) by assumption: consider the partition of \( V \) into single vertices. So there \( {e}^{0} \) exists an edge \( {e}^{0} \in E \smallsetminus E\left\lbrack {F}^{0}\right\rbrack \) . By Lemma 2.4.3, there exists a set \( U \) \( U \subseteq V \) that is connected in every \( {F}_{i}^{0} \) and contains the ends of \( {e}_{0} \) ; in particular, \( \left| U\right| \geq 2 \) . Since every partition of the contracted multigraph \( G/U \) induces a partition of \( G \) with the same cross-edges, \( {}^{2}G/U \) has at least \( k\left( {\left| P\right| - 1}\right) \) cross-edges with respect to any partition \( P \) . By the induction hypothesis, therefore, \( G/U \) has \( k \) edge-disjoint spanning trees \( {T}_{1},\ldots ,{T}_{k} \) . Replacing in each \( {T}_{i} \) the vertex \( {v}_{U} \) contracted from \( U \) by the spanning tree \( {F}_{i}^{0} \cap G\left\lbrack U\right\rbrack \) of \( G\left\lbrack U\right\rbrack \), we obtain \( k \) edge-disjoint spanning trees in \( G \) .
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graph partition
---
Let us say that subgraphs \( {G}_{1},\ldots ,{G}_{k} \) of a graph \( G \) partition \( G \) if their edge sets form a partition of \( E\left( G\right) \) . Our spanning tree problem may then be recast as follows: into how many connected spanning subgraphs can we partition a given graph? The excuse for rephrasing our simple tree problem in this more complicated way is that it now has an obvious dual (cf. Theorem 1.5.1): into how few acyclic (spanning) subgraphs can we partition a given graph? Or for given \( k \) : which graphs can be partitioned into at most \( k \) forests?
An obvious necessary condition now is that every set \( U \subseteq V\left( G\right) \) induces at most \( k\left( {\left| U\right| - 1}\right) \) edges, no more than \( \left| U\right| - 1 \) for each forest.
---
2 see Chapter 1.10 on contraction in multigraphs
---
Once more, this condition turns out to be sufficient too. And surprisingly, this can be shown with the help of Lemma 2.4.3, which was designed for the proof of our theorem on edge-disjoint spanning trees:
Theorem 2.4.4. (Nash-Williams 1964)
\( A \) multigraph \( G = \left( {V, E}\right) \) can be partitioned into at most \( k \) forests if and only if \( \parallel G\left\lbrack U\right\rbrack \parallel \leq k\left( {\left| U\right| - 1}\right) \) for every non-empty set \( U \subseteq V \) .
Proof. The forward implication was shown above. Conversely, we show
\( \left( {1.5.3}\right) \)
that every \( k \) -tuple \( F = \left( {{F}_{1},\ldots ,{F}_{k}}\right) \in \mathcal{F} \) partitions \( G \), i.e. that \( E\left\lbrack F\right\rbrack = \) \( E \) . If not, let \( e \in E \smallsetminus E\left\lbrack F\right\rbrack \) . By Lemma 2.4.3, there exists a set \( U \subseteq V \) that is connected in every \( {F}_{i} \) and contains the ends of \( e \) . Then \( G\left\lbrack U\right\rbrack \) contains \( \left| U\right| - 1 \) edges from each \( {F}_{i} \), and in addition the edge \( e \) . Thus \( \parallel G\left\lbrack U\right\rbrack \parallel > k\left( {\left| U\right| - 1}\right) \), contrary to our assumption.
The least number of forests forming a partition of a graph \( G \) is called the arboricity of \( G \) . By Theorem 2.4.4, the arboricity is a measure for
---
arboricity
---
the maximum local density: a graph has small arboricity if and only if it is ’nowhere dense’, i.e. if and only if it has no subgraph \( H \) with \( \varepsilon \left( H\right) \) large.
We shall meet Theorem 2.4.1 again in Chapter 8.5, where we prove its infinite version. This is based not on ordinary spanning trees (for which the result is false) but on 'topological spanning trees': the analogous structures in a topological space formed by the graph together with its ends.
## 2.5 Path covers
Let us return once more to König's duality theorem for bipartite graphs, Theorem 2.1.1. If we orient every edge of \( G \) from \( A \) to \( B \), the theorem tells us how many disjoint directed paths we need in order to cover all the vertices of \( G \) : every directed path has length 0 or 1, and clearly the number of paths in such a 'path cover' is smallest when it contains as many paths of length 1 as possible - in other words, when it contains a maximum-cardinality matching.
In this section we put the above question more generally: how many paths in a given directed graph will suffice to cover its entire vertex set? Of course, this could be asked just as well for undirected graphs. As it turns out, however, the result we shall prove is rather more trivial in the undirected case (exercise), and the directed case will also have an interesting corollary.
A directed path is a directed graph \( P \neq \varnothing \) with distinct vertices \( {x}_{0},\ldots ,{x}_{k} \) and edges \( {e}_{0},\ldots ,{e}_{k - 1} \) such that \( {e}_{i} \) is an edge directed from \( {x}_{i} \) to \( {x}_{i + 1} \), for all \( i < k \) . In this section, path will always mean ’directed path
path’. The vertex \( {x}_{k} \) above is the last vertex of the path \( P \), and when \( \mathcal{P} \)
---
\( \operatorname{ter}\left( \mathcal{P}\right) \) path cover
---
is a set of paths we write \( \operatorname{ter}\left( \mathcal{P}\right) \) for the set of their last vertices. A path cover of a directed graph \( G \) is a set of disjoint paths in \( G \) which together contain all the vertices of \( G \) .
Theorem 2.5.1. (Gallai & Milgram 1960)
Every directed graph \( G \) has a path cover \( \mathcal{P} \) and an independent set \( \left\{ {{v}_{P} \mid P \in \mathcal{P}}\right\} \) of vertices such that \( {v}_{P} \in P \) for every \( P \in \mathcal{P} \) .
\( \mathcal{P} \) Proof. We prove by induction on \( \left| G\right| \) that for every path cover \( \mathcal{P} = \) \( {P}_{i} \) \( \left\{ {{P}_{1},\ldots ,{P}_{m}}\right\} \) of \( G \) with \( \operatorname{ter}\left( \mathcal{P}\right) \) minimal there is a set \( \left\{ {{v}_{P} \mid P \in \mathcal{P}}\right\} \) as \( {v}_{i} \) claimed. For each \( i \), let \( {v}_{i} \) denote the last vertex of \( {P}_{i} \) .
If \( \operatorname{ter}\left( \mathcal{P}\right) = \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) is independent there is nothing more to show, so we assume that \( G \) has an edge from \( {v}_{2} \) |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 18 | cted graph \( G \) is a set of disjoint paths in \( G \) which together contain all the vertices of \( G \) .
Theorem 2.5.1. (Gallai & Milgram 1960)
Every directed graph \( G \) has a path cover \( \mathcal{P} \) and an independent set \( \left\{ {{v}_{P} \mid P \in \mathcal{P}}\right\} \) of vertices such that \( {v}_{P} \in P \) for every \( P \in \mathcal{P} \) .
\( \mathcal{P} \) Proof. We prove by induction on \( \left| G\right| \) that for every path cover \( \mathcal{P} = \) \( {P}_{i} \) \( \left\{ {{P}_{1},\ldots ,{P}_{m}}\right\} \) of \( G \) with \( \operatorname{ter}\left( \mathcal{P}\right) \) minimal there is a set \( \left\{ {{v}_{P} \mid P \in \mathcal{P}}\right\} \) as \( {v}_{i} \) claimed. For each \( i \), let \( {v}_{i} \) denote the last vertex of \( {P}_{i} \) .
If \( \operatorname{ter}\left( \mathcal{P}\right) = \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) is independent there is nothing more to show, so we assume that \( G \) has an edge from \( {v}_{2} \) to \( {v}_{1} \) . Since \( {P}_{2}{v}_{2}{v}_{1} \) is again a path, the minimality of \( \operatorname{ter}\left( \mathcal{P}\right) \) implies that \( {v}_{1} \) is not the only vertex of \( {P}_{1} \) ; let \( v \) be the vertex preceding \( {v}_{1} \) on \( {P}_{1} \) . Then \( {\mathcal{P}}^{\prime } \mathrel{\text{:=}} \) \( {\mathcal{P}}^{\prime },{G}^{\prime } \) \( \left\{ {{P}_{1}v,{P}_{2},\ldots ,{P}_{m}}\right\} \) is a path cover of \( {G}^{\prime } \mathrel{\text{:=}} G - {v}_{1} \) (Fig. 2.5.1). Clearly, any independent set of representatives for \( {\mathcal{P}}^{\prime } \) in \( {G}^{\prime } \) will also work for \( \mathcal{P} \) in \( G \), so all we have to check is that we may apply the induction hypothesis to \( {\mathcal{P}}^{\prime } \) . It thus remains to show that \( \operatorname{ter}\left( {\mathcal{P}}^{\prime }\right) = \left\{ {v,{v}_{2},\ldots ,{v}_{m}}\right\} \) is minimal among the sets of last vertices of path covers of \( {G}^{\prime } \) .
Fig. 2.5.1. Path covers of \( G \) and \( {G}^{\prime } \)
Suppose then that \( {G}^{\prime } \) has a path cover \( {\mathcal{P}}^{\prime \prime } \) with \( \operatorname{ter}\left( {\mathcal{P}}^{\prime \prime }\right) \subsetneqq \operatorname{ter}\left( {\mathcal{P}}^{\prime }\right) \) . If a path \( P \in {\mathcal{P}}^{\prime \prime } \) ends in \( v \), we may replace \( P \) in \( {\mathcal{P}}^{\prime \prime } \) by \( {Pv}{v}_{1} \) to obtain a path cover of \( G \) whose set of last vertices is a proper subset of \( \operatorname{ter}\left( \mathcal{P}\right) \) , contradicting the choice of \( \mathcal{P} \) . If a path \( P \in {\mathcal{P}}^{\prime \prime } \) ends in \( {v}_{2} \) (but none in \( v \) ), we similarly replace \( P \) in \( {\mathcal{P}}^{\prime \prime } \) by \( P{v}_{2}{v}_{1} \) to obtain a contradiction to the minimality of \( \operatorname{ter}\left( \mathcal{P}\right) \) . Hence \( \operatorname{ter}\left( {\mathcal{P}}^{\prime \prime }\right) \subseteq \left\{ {{v}_{3},\ldots ,{v}_{m}}\right\} \) . But now \( {\mathcal{P}}^{\prime \prime } \) and the trivial path \( \left\{ {v}_{1}\right\} \) together form a path cover of \( G \) that contradicts the minimality of \( \operatorname{ter}\left( \mathcal{P}\right) \) .
As a corollary to Theorem 2.5.1 we obtain a classical result from the theory of partial orders. Recall that a subset of a partially ordered
set \( \left( {P, \leq }\right) \) is a chain in \( P \) if its elements are pairwise comparable; it is
---
chain antichain
---
an antichain if they are pairwise incomparable.
Corollary 2.5.2. (Dilworth 1950)
In every finite partially ordered set \( \left( {P, \leq }\right) \), the minimum number of chains with union \( P \) is equal to the maximum cardinality of an antichain in \( P \) .
Proof. If \( A \) is an antichain in \( P \) of maximum cardinality, then clearly \( P \) cannot be covered by fewer than \( \left| A\right| \) chains. The fact that \( \left| A\right| \) chains will suffice follows from Theorem 2.5.1 applied to the directed graph on \( P \) with the edge set \( \{ \left( {x, y}\right) \mid x < y\} \) .
## Exercises
1. Let \( M \) be a matching in a bipartite graph \( G \) . Show that if \( M \) is suboptimal, i.e. contains fewer edges than some other matching in \( G \), then \( G \) contains an augmenting path with respect to \( M \) . Does this fact generalize to matchings in non-bipartite graphs?
2. Describe an algorithm that finds, as efficiently as possible, a matching of maximum cardinality in any bipartite graph.
3. Show that if there exist injective functions \( A \rightarrow B \) and \( B \rightarrow A \) between two infinite sets \( A \) and \( B \) then there exists a bijection \( A \rightarrow B \) .
4. Derive the marriage theorem from König's theorem.
5. Let \( G \) and \( H \) be defined as for the third proof of Hall’s theorem. Show that \( {d}_{H}\left( b\right) \leq 1 \) for every \( b \in B \), and deduce the marriage theorem.
6. \( {}^{ + } \) Find an infinite counterexample to the statement of the marriage theorem.
7. Let \( k \) be an integer. Show that any two partitions of a finite set into \( k \) -sets admit a common choice of representatives.
8. Let \( A \) be a finite set with subsets \( {A}_{1},\ldots ,{A}_{n} \), and let \( {d}_{1},\ldots ,{d}_{n} \in \mathbb{N} \) . Show that there are disjoint subsets \( {D}_{k} \subseteq {A}_{k} \), with \( \left| {D}_{k}\right| = {d}_{k} \) for all \( k \leq n \), if and only if
\[
\left| {\mathop{\bigcup }\limits_{{i \in I}}{A}_{i}}\right| \geq \mathop{\sum }\limits_{{i \in I}}{d}_{i}
\]
for all \( I \subseteq \{ 1,\ldots, n\} \) .
9. \( {}^{ + } \) Prove Sperner’s lemma: in an \( n \) -set \( X \) there are never more than \( \left( \begin{matrix} n \\ \lfloor n/2\rfloor \end{matrix}\right) \) subsets such that none of these contains another.
(Hint. Construct \( \left( \begin{matrix} n \\ \lfloor n/2\rfloor \end{matrix}\right) \) chains covering the power set lattice of \( X \) .)
10. \( {}^{ - } \) Find a bipartite graph and a set of preferences such that no matching of maximal size is stable and no stable matching has maximal size.
11. \( {}^{ - } \) Find a non-bipartite graph with a set of preferences that has no stable matching.
12. Show that all stable matchings of a given bipartite graph cover the same vertices. (In particular, they have the same size.)
13. Find a set \( S \) for Theorem 2.2.3 when \( G \) is a forest.
14. A graph \( G \) is called (vertex-) transitive if, for any two vertices \( v, w \in G \) , there is an automorphism of \( G \) mapping \( v \) to \( w \) . Using the observations following the proof of Theorem 2.2.3, show that every transitive connected graph of even order contains a 1-factor.
15. Show that a graph \( G \) contains \( k \) independent edges if and only if \( q\left( {G - S}\right) \leq \left| S\right| + \left| G\right| - {2k} \) for all sets \( S \subseteq V\left( G\right) \) .
16. \( {}^{ - } \) Find a cubic graph without a 1-factor.
17. Derive the marriage theorem from Tutte's theorem.
18. \( {}^{ - } \) Disprove the analogue of König’s theorem (2.1.1) for non-bipartite graphs, but show that \( \mathcal{H} = \left\{ {K}^{2}\right\} \) has the Erdős-Pósa property.
19. For cubic graphs, Lemma 2.3.1 is considerably stronger than the ErdősPósa theorem. Extend the lemma to arbitrary multigraphs of minimum degree \( \geq 3 \), by finding a function \( g : \mathbb{N} \rightarrow \mathbb{N} \) such that every multigraph of minimum degree \( \geq 3 \) and order at least \( g\left( k\right) \) contains \( k \) disjoint cycles, for all \( k \in \mathbb{N} \) . Alternatively, show that no such function \( g \) exists.
20. Given a graph \( G \), let \( \alpha \left( G\right) \) denote the largest size of a set of independent vertices in \( G \) . Prove that the vertices of \( G \) can be covered by at most \( \alpha \left( G\right) \) disjoint subgraphs each isomorphic to a cycle or a \( {K}^{2} \) or \( {K}^{1} \) .
21. Find the error in the following short 'proof' of Theorem 2.4.1. Call a partition non-trivial if it has at least two classes and at least one of the classes has more than one element. We show by induction on \( \left| V\right| + \left| E\right| \) that \( G = \left( {V, E}\right) \) has \( k \) edge-disjoint spanning trees if every non-trivial partition of \( V \) into \( r \) sets (say) has at least \( k\left( {r - 1}\right) \) cross-edges. The induction starts trivially with \( G = {K}^{1} \) if we allow \( k \) copies of \( {K}^{1} \) as a family of \( k \) edge-disjoint spanning trees of \( {K}^{1} \) . We now consider the induction step. If every non-trivial partition of \( V \) into \( r \) sets (say) has more than \( k\left( {r - 1}\right) \) cross-edges, we delete any edge of \( G \) and are done by induction. So \( V \) has a non-trivial partition \( \left\{ {{V}_{1},\ldots ,{V}_{r}}\right\} \) with exactly \( k\left( {r - 1}\right) \) cross-edges. Assume that \( \left| {V}_{1}\right| \geq 2 \) . If \( {G}^{\prime } \mathrel{\text{:=}} G\left\lbrack {V}_{1}\right\rbrack \) has \( k \) disjoint spanning trees, we may combine these with \( k \) disjoint spanning trees that exist in \( G/{V}_{1} \) by induction. We may thus assume that \( {G}^{\prime } \) has no \( k \) disjoint spanning trees. Then by induction it has a non-trivial vertex partition \( \left\{ {{V}_{1}^{\prime },\ldots ,{V}_{s}^{\prime }}\right\} \) with fewer than \( k\left( {s - 1}\right) \) cross-edges. Then \( \left\{ {{V}_{1}^{\prime },\ldots ,{V}_{s}^{\prime },{V}_{2},\ldots ,{V}_{r}}\right\} \) is a non-trivial vertex partition of \( G \) into \( r + s - 1 \) sets with fewer than \( k\left( {r - 1}\right) + k\left( {s - 1}\right) = k\left( {\left( {r + s - 1}\right) - 1}\right) \) cross-edges, a contradiction.
22. |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 19 | on. So \( V \) has a non-trivial partition \( \left\{ {{V}_{1},\ldots ,{V}_{r}}\right\} \) with exactly \( k\left( {r - 1}\right) \) cross-edges. Assume that \( \left| {V}_{1}\right| \geq 2 \) . If \( {G}^{\prime } \mathrel{\text{:=}} G\left\lbrack {V}_{1}\right\rbrack \) has \( k \) disjoint spanning trees, we may combine these with \( k \) disjoint spanning trees that exist in \( G/{V}_{1} \) by induction. We may thus assume that \( {G}^{\prime } \) has no \( k \) disjoint spanning trees. Then by induction it has a non-trivial vertex partition \( \left\{ {{V}_{1}^{\prime },\ldots ,{V}_{s}^{\prime }}\right\} \) with fewer than \( k\left( {s - 1}\right) \) cross-edges. Then \( \left\{ {{V}_{1}^{\prime },\ldots ,{V}_{s}^{\prime },{V}_{2},\ldots ,{V}_{r}}\right\} \) is a non-trivial vertex partition of \( G \) into \( r + s - 1 \) sets with fewer than \( k\left( {r - 1}\right) + k\left( {s - 1}\right) = k\left( {\left( {r + s - 1}\right) - 1}\right) \) cross-edges, a contradiction.
22. \( {}^{ - } \) Prove the undirected version of the theorem of Gallai &Milgram (without using the directed version).
23. Derive the marriage theorem from the theorem of Gallai & Milgram.
24. \( {}^{ - } \) Show that a partially ordered set of at least \( {rs} + 1 \) elements contains either a chain of size \( r + 1 \) or an antichain of size \( s + 1 \) .
25. Prove the following dual version of Dilworth's theorem: in every finite partially ordered set \( \left( {P, \leq }\right) \), the minimum number of antichains with union \( P \) is equal to the maximum cardinality of a chain in \( P \) .
26. Derive König's theorem from Dilworth's theorem.
27. Find a partially ordered set that has no infinite antichain but is not a union of finitely many chains.
## Notes
There is a very readable and comprehensive monograph about matching in finite graphs: L. Lovász & M.D. Plummer, Matching Theory, Annals of Discrete Math. 29, North Holland 1986. Another very comprehensive source is A. Schrijver, Combinatorial optimization, Springer 2003. All the references for the results in this chapter can be found in these two books.
As we shall see in Chapter 3, König's Theorem of 1931 is no more than the bipartite case of a more general theorem due to Menger, of 1929. At the time, neither of these results was nearly as well known as Hall's marriage theorem, which was proved even later, in 1935. To this day, Hall's theorem remains one of the most applied graph-theoretic results. The first two of our proofs are folklore; the third and its dual (Exercise 5) are due to Kriesell (2005). For background and applications of the stable marriage theorem, see D. Gusfield & R.W. Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press 1989.
Our proof of Tutte's 1-factor theorem is based on a proof by Lovász (1975). Our extension of Tutte's theorem, Theorem 2.2.3 (including the informal discussion following it) is a lean version of a comprehensive structure theorem for matchings, due to Gallai (1964) and Edmonds (1965). See Lovász & Plummer for a detailed statement and discussion of this theorem.
Theorem 2.3.2 is due to P. Erdős & L Pósa, On independent circuits contained in a graph, Canad. J. Math. 17 (1965), 347-352. Our proof is essentially due to M. Simonovits, A new proof and generalization of a theorem of Erdős and Pósa on graphs without \( k + 1 \) independent circuits, Acta Sci. Hungar 18 (1967), 191-206. Calculations such as in Lemma 2.3.1 are standard for proofs where one aims to bound one numerical invariant in terms of another. This book does not emphasize this aspect of graph theory, but it is not atypical.
There is also an analogue of the Erdős-Pósa theorem for directed graphs (with directed cycles), which had long been conjectured but was only recently proved by B. Reed, N. Robertson, P.D. Seymour and R. Thomas, Packing directed circuits, Combinatorica 16 (1996), 535-554. Its proof is much more difficult than the undirected case; see Chapter 12.4, and in particular Corollary 12.4.10, for a glimpse of the techniques used.
Theorem 2.4.1 was proved independently by Nash-Williams and by Tutte; both papers are contained in J. London Math. Soc. 36 (1961). Theorem 2.4.4 is due to C.St.J.A. Nash-Williams, Decompositions of finite graphs into forests, J. London Math. Soc. 39 (1964), 12. Both results can be elegantly expressed and proved in the setting of matroids; see \( §{18} \) in B. Bollobás, Combinatorics, Cambridge University Press 1986.
An interesting vertex analogue of Corollary 2.4.2 is to ask which connectivity forces the existence of \( k \) spanning trees \( {T}_{1},\ldots ,{T}_{k} \), all rooted at a given vertex \( r \), such that for every vertex \( v \) the \( k \) paths \( v{T}_{i}r \) are independent. For example, if \( G \) is a cycle then deleting the edge left or right of \( r \) produces two such spanning trees. A. Itai and A. Zehavi, Three tree-paths, J. Graph Theory 13 (1989), \( {175} - {187} \), conjectured that \( \kappa \geq k \) should suffice. This conjecture has been proved for \( k \leq 4 \) ; see S. Curran, O. Lee &X. Yu, Chain decompositions and independent trees in 4-connected graphs, Proc. 14th Ann. ACM SIAM symposium on Discrete algorithms (Baltimore 2003), 186-191.
Theorem 2.5.1 is due to T. Gallai & A.N. Milgram, Verallgemeinerung eines graphentheoretischen Satzes von Rédei, Acta Sci. Math. (Szeged) 21 (1960), 181-186.
## Connectivity
Our definition of \( k \) -connectedness, given in Chapter 1.4, is somewhat unintuitive. It does not tell us much about ’connections’ in a \( k \) -connected graph: all it says is that we need at least \( k \) vertices to disconnect it. The following definition which, incidentally, implies the one above might have been more descriptive: ’a graph is \( k \) -connected if any two of its vertices can be joined by \( k \) independent paths’.
It is one of the classic results of graph theory that these two definitions are in fact equivalent, are dual aspects of the same property. We shall study this theorem of Menger (1927) in some depth in Section 3.3.
In Sections 3.1 and 3.2, we investigate the structure of the 2-connected and the 3-connected graphs. For these small values of \( k \) it is still possible to give a simple general description of how these graphs can be constructed.
In Sections 3.4 and 3.5 we look at other concepts of connectedness, more recent than the standard one but no less important: the number of \( H \) -paths in \( G \) for a subgraph \( H \) of \( G \), and the existence of disjoint paths in \( G \) linking up specified pairs of vertices.
## 3.1 2-Connected graphs and subgraphs
A maximal connected subgraph without a cutvertex is called a block. Thus, every block of a graph \( G \) is either a maximal 2-connected subgraph, or a bridge (with its ends), or an isolated vertex. Conversely, every such subgraph is a block. By their maximality, different blocks of \( G \) overlap in at most one vertex, which is then a cutvertex of \( G \) . Hence, every edge of \( G \) lies in a unique block, and \( G \) is the union of its blocks.
Cycles and bonds, too, are confined to a single block: Lemma 3.1.1.
(i) The cycles of \( G \) are precisely the cycles of its blocks.
(ii) The bonds of \( G \) are precisely the minimal cuts of its blocks.
Proof. (i) Any cycle in \( G \) is a connected subgraph without a cutvertex, and hence lies in some maximal such subgraph. By definition, this is a block of \( G \) .
(ii) Consider any cut in \( G \) . Let \( {xy} \) be one of its edges, and \( B \) the block containing it. By the maximality of \( B \) in the definition of a block, \( G \) contains no \( B \) -path. Hence every \( x - y \) path of \( G \) lies in \( B \), so those edges of our cut that lie in \( B \) separate \( x \) from \( y \) even in \( G \) . Assertion (ii) follows easily by repeated application of this argument.
In a sense, blocks are the 2-connected analogues of components, the maximal connected subgraphs of a graph. While the structure of \( G \) is determined fully by that of its components, however, it is not captured completely by the structure of its blocks: since the blocks need not be disjoint, the way they intersect defines another structure, giving a coarse picture of \( G \) as if viewed from a distance.
The following proposition describes this coarse structure of \( G \) as formed by its blocks. Let \( A \) denote the set of cutvertices of \( G \), and \( \mathcal{B} \) the set of its blocks. We then have a natural bipartite graph on \( A \cup \mathcal{B} \) block formed by the edges \( {aB} \) with \( a \in B \) . This block graph of \( G \) is shown in graph Figure 3.1.1.
Fig. 3.1.1. A graph and its block graph
Proposition 3.1.2. The block graph of a connected graph is a tree.
Proposition 3.1.2 reduces the structure of a given graph to that of its blocks. So what can we say about the blocks themselves? The following proposition gives a simple method by which, in principle, a list of all 2-connected graphs could be compiled:
Proposition 3.1.3. A graph is 2-connected if and only if it can be
\( \left\lbrack {\;{4.2.6}\;}\right\rbrack \)
constructed from a cycle by successively adding \( H \) -paths to graphs \( H \) already constructed (Fig. 3.1.2).
Fig. 3.1.2. The construction of 2-connected graphs
Proof. Clearly, every graph constructed as described is 2-connected. Conversely, let a 2-connected graph \( G \) be given. Then \( G \) contains a cycle, and hence has a maximal subgraph \( H \) constructible as above. Since any edge \( {xy} \in E\left( G\right) \smallsetminus E\left( H\right) \) with \( x, y \in H \) would define an \( H \) - path, \( H \) is an induced subgraph of \( G \) . Thus if \( H \neq G \), th |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 20 | at of its blocks. So what can we say about the blocks themselves? The following proposition gives a simple method by which, in principle, a list of all 2-connected graphs could be compiled:
Proposition 3.1.3. A graph is 2-connected if and only if it can be
\( \left\lbrack {\;{4.2.6}\;}\right\rbrack \)
constructed from a cycle by successively adding \( H \) -paths to graphs \( H \) already constructed (Fig. 3.1.2).
Fig. 3.1.2. The construction of 2-connected graphs
Proof. Clearly, every graph constructed as described is 2-connected. Conversely, let a 2-connected graph \( G \) be given. Then \( G \) contains a cycle, and hence has a maximal subgraph \( H \) constructible as above. Since any edge \( {xy} \in E\left( G\right) \smallsetminus E\left( H\right) \) with \( x, y \in H \) would define an \( H \) - path, \( H \) is an induced subgraph of \( G \) . Thus if \( H \neq G \), then by the connectedness of \( G \) there is an edge \( {vw} \) with \( v \in G - H \) and \( w \in H \) . As \( G \) is 2-connected, \( G - w \) contains a \( v - H \) path \( P \) . Then \( {wvP} \) is an \( H \) -path in \( G \), and \( H \cup {wvP} \) is a constructible subgraph of \( G \) larger than \( H \) . This contradicts the maximality of \( H \) .
## 3.2 The structure of 3-connected graphs
In the last section we showed first how every connected graph decomposes canonically into 2-connected subgraphs (and bridges), and how these are arranged in a tree-like way to make up the whole graph. There is a similar canonical decomposition of 2-connected graphs into 3-connected pieces (and cycles), which are again organized in a tree-like way. This nontrivial structure theorem of Tutte is most naturally expressed in terms of tree-decompositions, to be introduced in Chapter 12. We therefore omit it here. \( {}^{1} \)
Instead, we shall describe how every 3-connected graph can be obtained from a \( {K}^{4} \) by a succession of elementary operations preserving 3-connectedness. We then prove a deep result of Tutte about the algebraic structure of the cycle space of 3-connected graphs; this will play an important role again in Chapter 4.5.
In Proposition 3.1.3 we saw how every 2-connected graph can be constructed inductively by a sequence of steps starting from a cycle. All
---
1 The curious reader may take a glance at Exercise 20 of Chapter 12.
---
the graphs in the sequence were themselves 2-connected, so the graphs obtainable by this construction method are precisely the 2-connected graphs. Note that the cycles as starting graphs cannot be replaced by a smaller class, because they do not have proper 2-connected subgraphs.
When we try to do the same for 3-connected graphs, we soon notice that both the set of starting graphs and the construction steps required become too complicated. If we base our construction sequences on the minor relation instead of subgraphs, however, it all works smoothly again:
4.4.3] Lemma 3.2.1. If \( G \) is 3-connected and \( \left| G\right| > 4 \), then \( G \) has an edge \( e \) such that \( G/e \) is again 3-connected.
Proof. Suppose there is no such edge \( e \) . Then, for every edge \( {xy} \in G \) , the graph \( G/{xy} \) contains a separator \( S \) of at most 2 vertices. Since \( \kappa \left( G\right) \geq 3 \), the contracted vertex \( {v}_{xy} \) of \( G/{xy} \) (see Chapter 1.7) lies in \( S \) and \( \left| S\right| = 2 \), i.e. \( G \) has a vertex \( z \notin \{ x, y\} \) such that \( \left\{ {{v}_{xy}, z}\right\} \) separates \( G/{xy} \) . Then any two vertices separated by \( \left\{ {{v}_{xy}, z}\right\} \) in \( G/{xy} \) are separated in \( G \) by \( T \mathrel{\text{:=}} \{ x, y, z\} \) . Since no proper subset of \( T \) separates \( G \), every vertex in \( T \) has a neighbour in every component \( C \) of \( G - T \) .
We choose the edge \( {xy} \), the vertex \( z \), and the component \( C \) so that \( \left| C\right| \) is as small as possible, and pick a neighbour \( v \) of \( z \) in \( C \) (Fig. 3.2.1). By assumption, \( G/{zv} \) is again not 3-connected, so again there is a vertex \( w \) such that \( \{ z, v, w\} \) separates \( G \), and as before every vertex in \( \{ z, v, w\} \) has a neighbour in every component of \( G - \{ z, v, w\} \) .
Fig. 3.2.1. Separating vertices in the proof of Lemma 3.2.1
As \( x \) and \( y \) are adjacent, \( G - \{ z, v, w\} \) has a component \( D \) such that \( D \cap \{ x, y\} = \varnothing \) . Then every neighbour of \( v \) in \( D \) lies in \( C \) (since \( v \in C \) ), so \( D \cap C \neq \varnothing \) and hence \( D \subsetneqq C \) by the choice of \( D \) . This contradicts the choice of \( {xy}, z \) and \( C \) .
Theorem 3.2.2. (Tutte 1961)
A graph \( G \) is 3-connected if and only if there exists a sequence \( {G}_{0},\ldots ,{G}_{n} \) of graphs with the following properties:
(i) \( {G}_{0} = {K}^{4} \) and \( {G}_{n} = G \) ;
(ii) \( {G}_{i + 1} \) has an edge \( {xy} \) with \( d\left( x\right), d\left( y\right) \geq 3 \) and \( {G}_{i} = {G}_{i + 1}/{xy} \), for every \( i < n \) .
Proof. If \( G \) is 3-connected, a sequence as in the theorem exists by Lemma 3.2.1. Note that all the graphs in this sequence are 3-connected.
Conversely, let \( {G}_{0},\ldots ,{G}_{n} \) be a sequence of graphs as stated; we show that if \( {G}_{i} = {G}_{i + 1}/{xy} \) is 3-connected then so is \( {G}_{i + 1} \), for every \( i < n \) . Suppose not, let \( S \) be a separator of at most 2 vertices in \( {G}_{i + 1} \), and let \( {C}_{1},{C}_{2} \) be two components of \( {G}_{i + 1} - S \) . As \( x \) and \( y \) are adjacent, we may \( {C}_{1},{C}_{2} \) assume that \( \{ x, y\} \cap V\left( {C}_{1}\right) = \varnothing \) (Fig. 3.2.2). Then \( {C}_{2} \) contains neither
Fig. 3.2.2. The position of \( {xy} \in {G}_{i + 1} \) in the proof of Theorem 3.2.2
both vertices \( x, y \) nor a vertex \( v \notin \{ x, y\} \) : otherwise \( {v}_{xy} \) or \( v \) would be separated from \( {C}_{1} \) in \( {G}_{i} \) by at most two vertices, a contradiction. But now \( {C}_{2} \) contains only one vertex: either \( x \) or \( y \) . This contradicts our assumption of \( d\left( x\right), d\left( y\right) \geq 3 \) .
Theorem 3.2.2 is the essential core of a result of Tutte known as his wheel theorem. \( {}^{2} \) Like Proposition 3.1.3 for 2-connected graphs, it enables us to construct all 3-connected graphs by a simple inductive process depending only on local information: starting with \( {K}^{4} \), we pick a vertex \( v \) in a graph constructed already, split it into two adjacent vertices \( {v}^{\prime },{v}^{\prime \prime } \) , and join these to the former neighbours of \( v \) as we please provided only that \( {v}^{\prime } \) and \( {v}^{\prime \prime } \) each acquire at least 3 incident edges, and that every former neighbour of \( v \) becomes adjacent to at least one of \( {v}^{\prime },{v}^{\prime \prime } \) .
## Theorem 3.2.3. (Tutte 1963)
\( \left\lbrack {4.5.2}\right\rbrack \)
The cycle space of a 3-connected graph is generated by its non-separating induced cycles.
Proof. We apply induction on the order of the graph \( G \) considered.
(1.9.1)
In \( {K}^{4} \), every cycle is a triangle or (in terms of edges) the symmetric difference of triangles. As these are induced and non-separating, the assertion holds for \( \left| G\right| = 4 \) .
For the induction step, let \( e = {xy} \) be an edge of \( G \) for which
---
\( e = {xy} \) \( {G}^{\prime } \)
---
\( {G}^{\prime } \mathrel{\text{:=}} G/e \) is again 3-connected; cf. Lemma 3.2.1. Then every edge \( {e}^{\prime } \in E\left( {G}^{\prime }\right) \smallsetminus E\left( G\right) \) is of the form \( {e}^{\prime } = u{v}_{e} \), where at least one of the two edges \( {ux} \) and \( {uy} \) lies in \( G \) . We pick one that does (either \( {ux} \) or \( {uy} \) ), and identify it notationally with the edge \( {e}^{\prime } \) ; thus \( {e}^{\prime } \) now denotes both wheel
---
2 Graphs of the form \( {C}^{n} * {K}^{1} \) are called wheels; thus, \( {K}^{4} \) is the smallest wheel.
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the edge \( u{v}_{e} \) of \( {G}^{\prime } \) and one of the two edges \( {ux},{uy} \) . In this way we may regard \( E\left( {G}^{\prime }\right) \) as a subset of \( E\left( G\right) \), and \( \mathcal{E}\left( {G}^{\prime }\right) \) as a subspace of \( \mathcal{E}\left( G\right) \) ; thus all additions of edge sets will take place unambiguously in \( \mathcal{E}\left( G\right) \) .
---
fundamental
triangles
---
A special role in this proof will be played by the triangles \( {uxy} \) in \( G \) that contain the edge \( e \) . We shall call these the fundamental triangles of \( G \) ; they are clearly (induced and) non-separating, as otherwise \( \left\{ {u,{v}_{e}}\right\} \) would separate \( {G}^{\prime } \), contradicting its 3-connectedness.
Now consider an induced cycle \( C \subseteq G \) that is not a fundamental triangle. If \( e \in C \), then \( C/e \) is a cycle in \( {G}^{\prime } \) . If \( e \notin C \), then at most one of \( x, y \) lies on \( C \), as otherwise \( e \) would be a chord. Then the vertices of \( C \) in order also form a cycle in \( {G}^{\prime } \) (replace \( x \) or \( y \) by \( {v}_{e} \) as necessary); this cycle, too, will be denoted by \( C/e \) . Thus, for every induced cycle \( C/e \) \( C \subseteq G \) that is not a fundamental triangle, \( C/e \) denotes a unique cycle in \( {G}^{\prime } \) . However, even in the case of \( e \notin C \) the edge set of \( C/e \) when viewed as a subset of \( E\left( G\right) \) need not coincide with \( E\left( C\right) \), or even form a cycle |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 21 | the fundamental triangles of \( G \) ; they are clearly (induced and) non-separating, as otherwise \( \left\{ {u,{v}_{e}}\right\} \) would separate \( {G}^{\prime } \), contradicting its 3-connectedness.
Now consider an induced cycle \( C \subseteq G \) that is not a fundamental triangle. If \( e \in C \), then \( C/e \) is a cycle in \( {G}^{\prime } \) . If \( e \notin C \), then at most one of \( x, y \) lies on \( C \), as otherwise \( e \) would be a chord. Then the vertices of \( C \) in order also form a cycle in \( {G}^{\prime } \) (replace \( x \) or \( y \) by \( {v}_{e} \) as necessary); this cycle, too, will be denoted by \( C/e \) . Thus, for every induced cycle \( C/e \) \( C \subseteq G \) that is not a fundamental triangle, \( C/e \) denotes a unique cycle in \( {G}^{\prime } \) . However, even in the case of \( e \notin C \) the edge set of \( C/e \) when viewed as a subset of \( E\left( G\right) \) need not coincide with \( E\left( C\right) \), or even form a cycle at all; an example is shown in Figure 3.2.3.
Fig. 3.2.3. One of the four possibilities for \( E\left( {C/e}\right) \) when \( e \notin C \)
basic cycles Let us refer to the non-separating induced cycles in \( G \) or \( {G}^{\prime } \) as basic cycles. We thus want to show that every element of \( \mathcal{C}\left( G\right) \) is a \( C \) sum of basic cycles in \( G \) . Let \( C \in \mathcal{C}\left( G\right) \) be given. By Proposition 1.9.1 and our observation that fundamental triangles are themselves basic, we may assume that \( C \) is an induced cycle but not a fundamental triangle. \( {C}^{\prime } \) Hence, \( {C}^{\prime } \mathrel{\text{:=}} C/e \) is a cycle in \( {G}^{\prime } \) .
Roughly, our plan is to generate \( {C}^{\prime } \) from basic cycles in \( {G}^{\prime } \) by induction, and lift the generators back to basic cycles in \( G \) that generate \( C \) . Now as we have seen, the edge set \( {C}^{\prime } \) can differ a little from \( C \), and similarly the basic cycles of \( {G}^{\prime } \) that generate \( {C}^{\prime } \) may differ a little from basic cycles in \( G \) . To make these differences precise, and to show that
similar they do not really matter, let us call two sets \( F,\widetilde{F} \in \mathcal{E}\left( G\right) \) similar if they differ only by fundamental triangles and possibly in \( e \), i.e., if there exists a sum \( D \) of fundamental triangles such that
\[
F + \widetilde{F} + D \in \{ \varnothing ,\{ e\} \} .
\]
Clearly, similarity is an equivalence relation.
Instead of generating \( C \) from basic cycles, it will be enough to generate a set \( \widetilde{C} \in \mathcal{C}\left( G\right) \) similar to \( C \) :
If \( C \) is similar to \( \widetilde{C} \in \mathcal{C}\left( G\right) \) and \( \widetilde{C} \) is a sum of basic cycles
(1)
in \( G \), then so is \( C \) .
For if \( D \) is a sum of fundamental triangles such that \( C + \widetilde{C} + D \in \) \( \{ \varnothing ,\{ e\} \} \), then \( C + \widetilde{C} + D = \varnothing \), because \( C + \widetilde{C} + D \) lies in \( \mathcal{C}\left( G\right) \) but \( \{ e\} \) does not. Hence, as \( D \) is a sum of basic cycles, so is \( C = \widetilde{C} + D \) . Let us begin our proof by noting that
\( {C}^{\prime } \) is similar to \( C \) .
(2)
Indeed, if \( e \in C \) or neither \( x \) nor \( y \) lies on \( C \), then \( {C}^{\prime } \) differs from \( C \) exactly in \( e \) or not at all. Otherwise, \( C \) contains one of the vertices \( x, y \) but not the other. Then \( {v}_{e} \in {C}^{\prime } \) ; let \( u, w \) be the two neighbours of \( {v}_{e} \) on \( {C}^{\prime } \), and \( {e}^{\prime } = u{v}_{e} \) and \( {e}^{\prime \prime } = {v}_{e}w \) its incident edges (as in Fig. 3.2.3). If \( {e}^{\prime } \notin C \), let \( {D}_{u} \) be the fundamental triangle \( {uxy} \) ; otherwise put \( {D}_{u} \mathrel{\text{:=}} \varnothing \) . If \( {e}^{\prime \prime } \notin C \), let \( {D}_{w} \) be the fundamental triangle \( {wxy} \) ; otherwise put \( {D}_{w} \mathrel{\text{:=}} \varnothing \) . Then \( C + {C}^{\prime } + {D}_{u} + {D}_{w} \in \{ \varnothing ,\{ e\} \} \), completing the proof of (2).
By the induction hypothesis, \( {C}^{\prime } \) is a sum of basic cycles \( {C}_{1}^{\prime },\ldots ,{C}_{k}^{\prime } \) in \( {G}^{\prime } \) . Let us lift these back to \( G \), as follows:
For every \( i = 1,\ldots, k \) there exists a basic cycle \( {C}_{i} \) in \( G \)
that is similar to \( {C}_{i}^{\prime } \) .
(3)
To prove (3), we shall choose the \( {C}_{i} \) so that \( {C}_{i}/e = {C}_{i}^{\prime } \) ; these will be similar to the \( {C}_{i}^{\prime } \) as in (2). If \( {v}_{e} \notin {C}_{i}^{\prime } \) then this holds with \( {C}_{i} \mathrel{\text{:=}} {C}_{i}^{\prime } \), so we assume that \( {v}_{e} \in {C}_{i}^{\prime } \) . Let \( u \) and \( w \) be the two neighbours of \( {v}_{e} \) on \( {C}_{i}^{\prime } \) , and let \( P \) be the \( u - w \) path in \( {C}_{i}^{\prime } \) avoiding \( {v}_{e} \) (Fig. 3.2.4). Then \( P \subseteq G \) .
Fig. 3.2.4. The search for a basic cycle \( {C}_{i} \) with \( {C}_{i}/e = {C}_{i}^{\prime } \)
We first assume that \( \{ {ux},{uy},{wx},{wy}\} \subseteq E\left( G\right) \), and consider (as candidates for \( \left. {C}_{i}\right) \) the cycles \( {C}_{x} \mathrel{\text{:=}} {uPwxu} \) and \( {C}_{y} \mathrel{\text{:=}} {uPwyu} \) . Both are \( {C}_{x},{C}_{y} \) induced cycles in \( G \) (because \( {C}_{i}^{\prime } \) is induced in \( {G}^{\prime } \) ), and clearly \( {C}_{x}/e = \) \( {C}_{i}^{\prime } = {C}_{y}/e \) . Moreover, neither of these cycles separates two vertices of \( G - \left( {V\left( P\right) \cup \{ x, y\} }\right) \) in \( G \), since \( {C}_{i}^{\prime } \) does not separate such vertices in \( {G}^{\prime } \) . Thus, if \( {C}_{x} \) (say) is a separating cycle in \( G \), then one of the components of \( G - {C}_{x} \) consists just of \( y \) . Likewise, if \( {C}_{y} \) separates \( G \) then one of the arising components contains only \( x \) . However, this cannot happen for both \( {C}_{x} \) and \( {C}_{y} \) at once: otherwise \( {N}_{G}\left( {\{ x, y\} }\right) \subseteq V\left( P\right) \) and hence \( {N}_{G}\left( {\{ x, y\} }\right) = \{ u, w\} \) (since \( {C}_{i}^{\prime } \) has no chord), which contradicts \( \kappa \left( G\right) \geq 3 \) . Hence, at least one of \( {C}_{x},{C}_{y} \) is a basic cycle in \( G \), and we choose this as \( {C}_{i} \) .
It remains to consider the case that \( \{ {ux},{uy},{wx},{wy}\} \nsubseteq E\left( G\right) \), say \( {ux} \notin E\left( G\right) \) . Using the 3-connectedness of \( G \) as above, we see that either \( {uPwyu} \) or \( {uPwxyu} \) is a basic cycle in \( G \) (which we choose as \( {C}_{i} \) ), according as \( {wy} \) is an edge of \( G \) or not. This completes the proof of (3).
By (3), \( \widetilde{C} \mathrel{\text{:=}} {C}_{1} + \ldots + {C}_{k} \) is similar to \( {C}^{\prime } = {C}_{1}^{\prime } + \ldots + {C}_{k}^{\prime } \), which in turn is similar to \( C \) by (2). By (1), this completes the proof.
## 3.3 Menger's theorem
The following theorem is one of the cornerstones of graph theory.
Theorem 3.3.1. (Menger 1927)
Let \( G = \left( {V, E}\right) \) be a graph and \( A, B \subseteq V \) . Then the minimum number of vertices separating \( A \) from \( B \) in \( G \) is equal to the maximum number of disjoint \( A - B \) paths in \( G \) .
We offer three proofs. Whenever \( G, A, B \) are given as in the theorem, we denote by \( k = k\left( {G, A, B}\right) \) the minimum number of vertices separating \( A \) from \( B \) in \( G \) . Clearly, \( G \) cannot contain more than \( k \) disjoint \( A - B \) paths; our task will be to show that \( k \) such paths exist.
First proof. We apply induction on \( \parallel G\parallel \) . If \( G \) has no edge, then \( \left| {A \cap B}\right| = k \) and we have \( k \) trivial \( A - B \) paths. So we assume that \( G \) has an edge \( e = {xy} \) . If \( G \) has no \( k \) disjoint \( A - B \) paths, then neither does \( G/e \) ; here, we count the contracted vertex \( {v}_{e} \) as an element of \( A \) (resp. \( B \) ) in \( G/e \) if in \( G \) at least one of \( x, y \) lies in \( A \) (resp. \( B \) ). By the induction hypothesis, \( G/e \) contains an \( A - B \) separator \( Y \) of fewer than \( k \) vertices. Among these must be the vertex \( {v}_{e} \), since otherwise \( Y \subseteq V \) would be an \( A - B \) separator in \( G \) . Then \( X \mathrel{\text{:=}} \left( {Y \smallsetminus \left\{ {v}_{e}\right\} }\right) \cup \{ x, y\} \) is an \( A - B \) separator in \( G \) of exactly \( k \) vertices.
We now consider the graph \( G - e \) . Since \( x, y \in X \), every \( A - X \) separator in \( G - e \) is also an \( A - B \) separator in \( G \) and hence contains at least \( k \) vertices. So by induction there are \( k \) disjoint \( A - X \) paths in \( G - e \) , and similarly there are \( k \) disjoint \( X - B \) paths in \( G - e \) . As \( X \) separates \( A \) from \( B \), these two path systems do not meet outside \( X \), and can thus be combined to \( k \) disjoint \( A - B \) paths.
Let \( \mathcal{P} \) be a set of disjoint \( A - B \) paths, and let \( \mathcal{Q} \) be another such set.
We say that \( \mathcal{Q} \) exceeds \( \mathcal{P} \) if the set of vertices in \( A \) that lie on a path in exceeds \( \mathcal{P} \) is a proper subset of the set of vertices in \( A \) that lie on a path in \( \mathcal{Q} \) , and likewise for \( B \) . Then, in particular, \( \left| \mathcal{Q}\right| \geq \left| \mathcal{P}\right| + 1 \) .
Second proof. We prove the following stronger statement:
If \( \mathcal{P} \) i |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 22 | y \in X \), every \( A - X \) separator in \( G - e \) is also an \( A - B \) separator in \( G \) and hence contains at least \( k \) vertices. So by induction there are \( k \) disjoint \( A - X \) paths in \( G - e \) , and similarly there are \( k \) disjoint \( X - B \) paths in \( G - e \) . As \( X \) separates \( A \) from \( B \), these two path systems do not meet outside \( X \), and can thus be combined to \( k \) disjoint \( A - B \) paths.
Let \( \mathcal{P} \) be a set of disjoint \( A - B \) paths, and let \( \mathcal{Q} \) be another such set.
We say that \( \mathcal{Q} \) exceeds \( \mathcal{P} \) if the set of vertices in \( A \) that lie on a path in exceeds \( \mathcal{P} \) is a proper subset of the set of vertices in \( A \) that lie on a path in \( \mathcal{Q} \) , and likewise for \( B \) . Then, in particular, \( \left| \mathcal{Q}\right| \geq \left| \mathcal{P}\right| + 1 \) .
Second proof. We prove the following stronger statement:
If \( \mathcal{P} \) is any set of fewer than \( k \) disjoint \( A - B \) paths in \( G \), then
\( G \) contains a set of \( \left| \mathcal{P}\right| + 1 \) disjoint \( A - B \) paths exceeding \( \mathcal{P} \) .
Keeping \( G \) and \( A \) fixed, we let \( B \) vary and apply induction on \( \left| {\bigcup \mathcal{P}}\right| \) . Let \( R \) be an \( A - B \) path that avoids the (fewer than \( k \) ) vertices of \( B \) that lie on a path in \( \mathcal{P} \) . If \( R \) avoids all the paths in \( \mathcal{P} \), then \( \mathcal{P} \cup \{ R\} \) exceeds \( \mathcal{P} \), as desired. (This will happen when \( \mathcal{P} = \varnothing \), so the induction starts.) If not, let \( x \) be the last vertex of \( R \) that lies on some \( P \in \mathcal{P} \) . Put \( {B}^{\prime } \mathrel{\text{:=}} B \cup V\left( {{xP} \cup {xR}}\right) \) and \( {\mathcal{P}}^{\prime } \mathrel{\text{:=}} \left( {\mathcal{P}\smallsetminus \{ P\} }\right) \cup \{ {Px}\} \) (Fig. 3.3.1). Then \( \left| {\mathcal{P}}^{\prime }\right| = \left| \mathcal{P}\right| \) (but \( \left| {\bigcup {\mathcal{P}}^{\prime }}\right| < \left| {\bigcup \mathcal{P}}\right| \) ) and \( k\left( {G, A,{B}^{\prime }}\right) \geq k\left( {G, A, B}\right) \), so by the induction hypothesis there is a set \( {\mathcal{Q}}^{\prime } \) of \( \left| {\mathcal{P}}^{\prime }\right| + 1 \) disjoint \( A - {B}^{\prime } \) paths exceeding \( {\mathcal{P}}^{\prime } \) . Then \( {\mathcal{Q}}^{\prime } \) contains a path \( Q \) ending in \( x \), and a unique path \( {Q}^{\prime } \) whose last vertex \( y \) is not among the last vertices of the paths in \( {\mathcal{P}}^{\prime } \) . If \( y \notin {xP} \), we let \( \mathcal{Q} \) be obtained from \( {\mathcal{Q}}^{\prime } \) by adding \( {xP} \) to \( Q \), and adding \( {yR} \) to \( {Q}^{\prime } \) if \( y \notin B \) . Otherwise \( y \in \mathring{x}P \), and we let \( \mathcal{Q} \) be obtained from \( {\mathcal{Q}}^{\prime } \) by adding \( {xR} \) to \( Q \) and adding \( {yP} \) to \( {Q}^{\prime } \) . In both cases \( \mathcal{Q} \) exceeds \( \mathcal{P} \), as desired.
Fig. 3.3.1. Paths in the second proof of Menger's theorem
Applied to a bipartite graph, Menger's theorem specializes to the assertion of König's theorem (2.1.1). For our third proof, we shall adapt the alternating path proof of König's theorem to the more general setup of Theorem 3.3.1. Let again \( G, A, B \) be given, and let \( \mathcal{P} \) be a set of disjoint \( A - B \) paths in \( G \) . Let us say that an \( A - B \) separator \( X \subseteq V \) lies on \( \mathcal{P} \) if it consists of a choice of exactly one vertex from each path in \( \mathcal{P} \) . If we can find such a separator \( X \), then clearly \( k \leq \left| X\right| = \left| \mathcal{P}\right| \), and Menger's theorem will be proved.
Put
\[
V\left\lbrack \mathcal{P}\right\rbrack \mathrel{\text{:=}} \bigcup \{ V\left( P\right) \mid P \in \mathcal{P}\}
\]
\[
E\left\lbrack \mathcal{P}\right\rbrack \mathrel{\text{:=}} \bigcup \{ E\left( P\right) \mid P \in \mathcal{P}\}
\]
\( W,{x}_{i},{e}_{i} \) Let a walk \( W = {x}_{0}{e}_{0}{x}_{1}{e}_{1}\ldots {e}_{n - 1}{x}_{n} \) in \( G \) with \( {e}_{i} \neq {e}_{j} \) for \( i \neq j \) be said to alternate with respect to \( \mathcal{P} \) (Fig. 3.3.2) if it starts in \( A \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) and
---
alternating walk
---
the following three conditions hold for all \( i < n \) (with \( {e}_{-1} \mathrel{\text{:=}} {e}_{0} \) in (iii)):
(i) if \( {e}_{i} = e \in E\left\lbrack \mathcal{P}\right\rbrack \), then \( W \) traverses the edge \( e \) backwards, i.e. \( {x}_{i + 1} \in P{\overset{ \circ }{x}}_{i} \) for some \( P \in \mathcal{P} \) ;
(ii) if \( {x}_{i} = {x}_{j} \) with \( i \neq j \), then \( {x}_{i} \in V\left\lbrack \mathcal{P}\right\rbrack \) ;
(iii) if \( {x}_{i} \in V\left\lbrack \mathcal{P}\right\rbrack \), then \( \left\{ {{e}_{i - 1},{e}_{i}}\right\} \cap E\left\lbrack \mathcal{P}\right\rbrack \neq \varnothing \) .
Fig. 3.3.2. An alternating walk from \( A \) to \( B \)
Note that, by (ii), any vertex outside \( V\left\lbrack \mathcal{P}\right\rbrack \) occurs at most once on \( W \) . And since the edges \( {e}_{i} \) of \( W \) are all distinct,(iii) implies that any vertex \( v \in V\left\lbrack \mathcal{P}\right\rbrack \) occurs at most twice on \( W \) . For \( v \neq {x}_{n} \), this can happen in exactly the following two ways. If \( {x}_{i} = {x}_{j} \) with \( 0 < i < j < n \), then
\[
\text{either}{e}_{i - 1},{e}_{j} \in E\left\lbrack \mathcal{P}\right\rbrack \text{and}{e}_{i},{e}_{j - 1} \notin E\left\lbrack \mathcal{P}\right\rbrack
\]
\[
\text{or}{e}_{i},{e}_{j - 1} \in E\left\lbrack \mathcal{P}\right\rbrack \text{and}{e}_{i - 1},{e}_{j} \notin E\left\lbrack \mathcal{P}\right\rbrack \text{.}
\]
Unless otherwise stated, any use of the word 'alternate' below will refer to our fixed path system \( \mathcal{P} \) .
\( \left\lbrack {8.4.5}\right\rbrack \)
Lemma 3.3.2. If an alternating walk \( W \) as above ends in \( B \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) , then \( G \) contains a set of disjoint \( A - B \) paths exceeding \( \mathcal{P} \) .
Proof. We may assume that \( W \) has only its first vertex in \( A \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) and only its last vertex in \( B \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) . Let \( H \) be the graph on \( V\left( G\right) \) whose edge set is the symmetric difference of \( E\left\lbrack \mathcal{P}\right\rbrack \) with \( \left\{ {{e}_{0},\ldots ,{e}_{n - 1}}\right\} \) . In \( H \) , the ends of the paths in \( \mathcal{P} \) and of \( W \) have degree 1 (or 0, if the path or \( W \) is trivial), and all other vertices have degree 0 or 2 .
For each vertex \( a \in \left( {A \cap V\left\lbrack \mathcal{P}\right\rbrack }\right) \cup \left\{ {x}_{0}\right\} \), therefore, the component of \( H \) containing \( a \) is a path, \( P = {v}_{0}\ldots {v}_{k} \) say, which starts in \( a \) and ends in \( A \) or \( B \) . Using conditions (i) and (iii), one easily shows by induction on \( i = 0,\ldots, k - 1 \) that \( P \) traverses each of its edges \( e = {v}_{i}{v}_{i + 1} \) in the forward direction with respect to \( \mathcal{P} \) or \( W \) . (Formally: if \( e \in {P}^{\prime } \) with \( \left. {{P}^{\prime } \in \mathcal{P}\text{, then }{v}_{i} \in {P}^{\prime }{\mathring{v}}_{i + 1}\text{; if }e = {e}_{j} \in W\text{, then }{v}_{i} = {x}_{j}\text{ and }{v}_{i + 1} = {x}_{j + 1}\text{. }}\right) \) Hence, \( P \) is an \( A - B \) path.
Similarly, for every \( b \in \left( {B \cap V\left\lbrack \mathcal{P}\right\rbrack }\right) \cup \left\{ {x}_{n}\right\} \) there is an \( A - B \) path in \( H \) that ends in \( b \) . The set of \( A - B \) paths in \( H \) therefore exceeds \( \mathcal{P} \) .
Lemma 3.3.3. If no alternating walk \( W \) as above ends in \( B \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) ,
\( \left\lbrack {\;{8.4.5}\;}\right\rbrack \)
then \( G \) contains an \( A - B \) separator on \( \mathcal{P} \) .
Proof. Let
\[
{A}_{1} \mathrel{\text{:=}} A \cap V\left\lbrack \mathcal{P}\right\rbrack \;\text{ and }\;{A}_{2} \mathrel{\text{:=}} A \smallsetminus {A}_{1},
\]
\( {A}_{1},{A}_{2} \)
and
\[
{B}_{1} \mathrel{\text{:=}} B \cap V\left\lbrack \mathcal{P}\right\rbrack \;\text{ and }\;{B}_{2} \mathrel{\text{:=}} B \smallsetminus {B}_{1}.
\]
\( {B}_{1},{B}_{2} \)
For every path \( P \in \mathcal{P} \), let \( {x}_{P} \) be the last vertex of \( P \) that lies on some \( {x}_{P} \) alternating walk; if no such vertex exists, let \( {x}_{P} \) be the first vertex of \( P \) . Our aim is to show that
\[
X \mathrel{\text{:=}} \left\{ {{x}_{P} \mid P \in \mathcal{P}}\right\}
\]
meets every \( A - B \) path in \( G \) ; then \( X \) is an \( A - B \) separator on \( \mathcal{P} \) .
Suppose there is an \( A - B \) path \( Q \) that avoids \( X \) . We know that \( Q \)
\( Q \)
meets \( V\left\lbrack \mathcal{P}\right\rbrack \), as otherwise it would be an alternating walk ending in \( {B}_{2} \) . Now the \( A - V\left\lbrack \mathcal{P}\right\rbrack \) path in \( Q \) is either an alternating walk or consists only of the first vertex of some path in \( \mathcal{P} \) . Therefore \( Q \) also meets the vertex set \( V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack \) of
\[
{\mathcal{P}}^{\prime } \mathrel{\text{:=}} \left\{ {P{x}_{P} \mid P \in \mathcal{P}}\right\}
\]
Let \( y \) be the last vertex of \( Q \) in \( V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack \), say \( y \in P \in \mathcal{P} \), and let \( x \mathrel{\text{:=}} {x}_{P} \) .
\( y, P, x \)
As \( Q \) avoids \( X \) and hence \( x \), |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 23 | {{x}_{P} \mid P \in \mathcal{P}}\right\}
\]
meets every \( A - B \) path in \( G \) ; then \( X \) is an \( A - B \) separator on \( \mathcal{P} \) .
Suppose there is an \( A - B \) path \( Q \) that avoids \( X \) . We know that \( Q \)
\( Q \)
meets \( V\left\lbrack \mathcal{P}\right\rbrack \), as otherwise it would be an alternating walk ending in \( {B}_{2} \) . Now the \( A - V\left\lbrack \mathcal{P}\right\rbrack \) path in \( Q \) is either an alternating walk or consists only of the first vertex of some path in \( \mathcal{P} \) . Therefore \( Q \) also meets the vertex set \( V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack \) of
\[
{\mathcal{P}}^{\prime } \mathrel{\text{:=}} \left\{ {P{x}_{P} \mid P \in \mathcal{P}}\right\}
\]
Let \( y \) be the last vertex of \( Q \) in \( V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack \), say \( y \in P \in \mathcal{P} \), and let \( x \mathrel{\text{:=}} {x}_{P} \) .
\( y, P, x \)
As \( Q \) avoids \( X \) and hence \( x \), we have \( y \in P\mathring{x} \) . In particular, \( x = {x}_{P} \) is
Fig. 3.3.3. Alternating walks in the proof of Lemma 3.3.3.
\( W \) not the first vertex of \( P \), and so there is an alternating walk \( W \) ending at \( x \) . Then \( W \cup {xPyQ} \) is a walk from \( {A}_{2} \) to \( B \) (Fig. 3.3.3). If this walk alternates and ends in \( {B}_{2} \), we have our desired contradiction.
How could \( W \cup {xPyQ} \) fail to alternate? For example, \( W \) might
\( {x}^{\prime },{W}^{\prime } \)
already use an edge of \( {xPy} \) . But if \( {x}^{\prime } \) is the first vertex of \( W \) on \( {xPy} \) , then \( {W}^{\prime } \mathrel{\text{:=}} W{x}^{\prime }{Py} \) is an alternating walk from \( {A}_{2} \) to \( y \) . (By \( W{x}^{\prime } \) we mean the initial segment of \( W \) ending at the first occurrence of \( {x}^{\prime } \) on \( W \) ; from there, \( {W}^{\prime } \) follows \( P \) back to \( y \) .) Even our new walk \( {W}^{\prime }{yQ} \) need not yet alternate: \( {W}^{\prime } \) might still meet \( \mathring{y}Q \) . By definition of \( {\mathcal{P}}^{\prime } \) and \( W \) , however, and the choice of \( y \) on \( Q \), we have
\[
V\left( {W}^{\prime }\right) \cap V\left\lbrack \mathcal{P}\right\rbrack \subseteq V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack \;\text{ and }\;V\left( {\mathring{y}Q}\right) \cap V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack = \varnothing .
\]
Thus, \( {W}^{\prime } \) and \( \mathring{y}Q \) can meet only outside \( \mathcal{P} \) .
\( z \) If \( {W}^{\prime } \) does indeed meet \( \mathring{y}Q \), we let \( z \) be the first vertex of \( {W}^{\prime } \) on \( \mathring{y}Q \) \( {W}^{\prime \prime } \) and set \( {W}^{\prime \prime } \mathrel{\text{:=}} {W}^{\prime }{zQ} \) . Otherwise we set \( {W}^{\prime \prime } \mathrel{\text{:=}} {W}^{\prime } \cup {yQ} \) . In both cases \( {W}^{\prime \prime } \) alternates with respect to \( {\mathcal{P}}^{\prime } \), because \( {W}^{\prime } \) does and \( \mathring{y}Q \) avoids \( V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack \) . ( \( {W}^{\prime \prime } \) satisfies condition (iii) at \( y \) in the second case even if \( y \) occurs twice on \( {W}^{\prime } \), because \( {W}^{\prime \prime } \) then contains the entire walk \( {W}^{\prime } \) and not just its initial segment \( {W}^{\prime }y \) .) By definition of \( {\mathcal{P}}^{\prime } \), therefore, \( {W}^{\prime \prime } \) avoids \( V\left\lbrack \mathcal{P}\right\rbrack \smallsetminus V\left\lbrack {\mathcal{P}}^{\prime }\right\rbrack \) . Thus \( {W}^{\prime \prime } \) also alternates with respect to \( \mathcal{P} \) and ends in \( {B}_{2} \), contrary to our assumptions.
Third proof of Menger’s theorem. Let \( \mathcal{P} \) contain as many disjoint \( A - B \) paths in \( G \) as possible. Then by Lemma 3.3.2, no alternating walk ends in \( B \smallsetminus V\left\lbrack \mathcal{P}\right\rbrack \) . By Lemma 3.3.3, this implies that \( G \) has an \( A - B \) separator \( X \) on \( \mathcal{P} \), giving \( k \leq \left| X\right| = \left| \mathcal{P}\right| \) as desired.
fan A set of \( a - B \) paths is called an \( a - B \) fan if any two of the paths have only \( a \) in common.
\( \left\lbrack {10.1.2}\right\rbrack \)
2] Corollary 3.3.4. For \( B \subseteq V \) and \( a \in V \smallsetminus B \), the minimum number of vertices \( \neq a \) separating \( a \) from \( B \) in \( G \) is equal to the maximum number of paths forming an \( a - B \) fan in \( G \) .
Proof. Apply Theorem 3.3.1 with \( A \mathrel{\text{:=}} N\left( a\right) \) .
Corollary 3.3.5. Let \( a \) and \( b \) be two distinct vertices of \( G \) .
(i) If \( {ab} \notin E \), then the minimum number of vertices \( \neq a, b \) separating \( a \) from \( b \) in \( G \) is equal to the maximum number of independent \( a - b \) paths in \( G \) .
(ii) The minimum number of edges separating \( a \) from \( b \) in \( G \) is equal to the maximum number of edge-disjoint \( a - b \) paths in \( G \) .
Proof. (i) Apply Theorem 3.3.1 with \( A \mathrel{\text{:=}} N\left( a\right) \) and \( B \mathrel{\text{:=}} N\left( b\right) \) .
(ii) Apply Theorem 3.3.1 to the line graph of \( G \), with \( A \mathrel{\text{:=}} E\left( a\right) \) and \( B \mathrel{\text{:=}} E\left( b\right) \) .
Theorem 3.3.6. (Global Version of Menger's Theorem) \( \left\lbrack \begin{array}{l} {4.2.7} \\ {6.6.1} \\ {9.4.2} \end{array}\right\rbrack \) (i) A graph is \( k \) -connected if and only if it contains \( k \) independent paths between any two vertices.
(ii) A graph is \( k \) -edge-connected if and only if it contains \( k \) edge-disjoint paths between any two vertices.
Proof. (i) If a graph \( G \) contains \( k \) independent paths between any two vertices, then \( \left| G\right| > k \) and \( G \) cannot be separated by fewer than \( k \) vertices; thus, \( G \) is \( k \) -connected.
Conversely, suppose that \( G \) is \( k \) -connected (and, in particular, has more than \( k \) vertices) but contains vertices \( a, b \) not linked by \( k \) independent paths. By Corollary 3.3.5 (i), \( a \) and \( b \) are adjacent; let \( {G}^{\prime } \mathrel{\text{:=}} G - {ab} \) . Then \( {G}^{\prime } \) contains at most \( k - 2 \) independent \( a - b \) paths. By Corollary 3.3.5 (i), we can separate \( a \) and \( b \) in \( {G}^{\prime } \) by a set \( X \) of at most \( k - 2 \) vertices. As \( \left| G\right| > k \), there is at least one further vertex \( v \notin X \cup \{ a, b\} \) in \( G \) . Now \( X \) separates \( v \) in \( {G}^{\prime } \) from either \( a \) or \( b \) -say, from \( a \) . But then \( X \cup \{ b\} \) is a set of at most \( k - 1 \) vertices separating \( v \) from \( a \) in \( G \) , contradicting the \( k \) -connectedness of \( G \) .
(ii) follows straight from Corollary 3.3.5 (ii).
## 3.4 Mader's theorem
In analogy to Menger's theorem we may consider the following question: given a graph \( G \) with an induced subgraph \( H \), up to how many independent \( H \) -paths can we find in \( G \) ?
In this section, we present without proof a deep theorem of Mader, which solves the above problem in a fashion similar to Menger's theorem. Again, the theorem says that an upper bound on the number of such paths that arises naturally from the size of certain separators is indeed attained by some suitable set of paths.
What could such an upper bound look like? Clearly, if \( X \subseteq V\left( {G - H}\right) \) and \( F \subseteq E\left( {G - H}\right) \) are such that every \( H \) -path in \( G \) has a vertex or an edge in \( X \cup F \), then \( G \) cannot contain more than \( \left| {X \cup F}\right| \) independent \( H \) -paths. Hence, the least cardinality of such a set \( X \cup F \) is a natural upper bound for the maximum number of independent \( H \) -paths. (Note that every \( H \) -path meets \( G - H \), because \( H \) is induced in \( G \) and edges of \( H \) do not count as \( H \) -paths.)
In contrast to Menger's theorem, this bound can still be improved. The minimality of \( \left| {X \cup F}\right| \) implies that no edge in \( F \) has an end in \( X \) : otherwise this edge would not be needed in the separator. Let \( Y \mathrel{\text{:=}} \) \( V\left( {G - H}\right) \smallsetminus X \), and denote by \( {\mathcal{C}}_{F} \) the set of components of the graph \( \left( {Y, F}\right) \) . Since every \( H \) -path avoiding \( X \) contains an edge from \( F \), it has
Fig. 3.4.1. An \( H \) -path in \( G - X \)
\( \partial C \) at least two vertices in \( \partial C \) for some \( C \in {\mathcal{C}}_{F} \), where \( \partial C \) denotes the set of vertices in \( C \) with a neighbour in \( G - X - C \) (Fig. 3.4.1). The number of independent \( H \) -paths in \( G \) is therefore bounded above by
\( {M}_{G}\left( H\right) \)
\[
{M}_{G}\left( H\right) \mathrel{\text{:=}} \min \left( {\left| X\right| + \mathop{\sum }\limits_{{C \in {\mathcal{C}}_{F}}}\left\lfloor {\frac{1}{2}\left| {\partial C}\right| }\right\rfloor }\right) ,
\]
\( X \) where the minimum is taken over all \( X \) and \( F \) as described above: \( X \subseteq \) \( V\left( {G - H}\right) \) and \( F \subseteq E\left( {G - H - X}\right) \) such that every \( H \) -path in \( G \) has a vertex or an edge in \( X \cup F \) .
Now Mader's theorem says that this upper bound is always attained by some set of independent \( H \) -paths:
Theorem 3.4.1. (Mader 1978)
Given a graph \( G \) with an induced subgraph \( H \), there are always \( {M}_{G}\left( H\right) \) independent \( H \) -paths in \( G \) .
In order to obtain direct analogues to the vertex and edge version of Menger's theorem, let us consider the two special ca |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 24 | 3.4.1). The number of independent \( H \) -paths in \( G \) is therefore bounded above by
\( {M}_{G}\left( H\right) \)
\[
{M}_{G}\left( H\right) \mathrel{\text{:=}} \min \left( {\left| X\right| + \mathop{\sum }\limits_{{C \in {\mathcal{C}}_{F}}}\left\lfloor {\frac{1}{2}\left| {\partial C}\right| }\right\rfloor }\right) ,
\]
\( X \) where the minimum is taken over all \( X \) and \( F \) as described above: \( X \subseteq \) \( V\left( {G - H}\right) \) and \( F \subseteq E\left( {G - H - X}\right) \) such that every \( H \) -path in \( G \) has a vertex or an edge in \( X \cup F \) .
Now Mader's theorem says that this upper bound is always attained by some set of independent \( H \) -paths:
Theorem 3.4.1. (Mader 1978)
Given a graph \( G \) with an induced subgraph \( H \), there are always \( {M}_{G}\left( H\right) \) independent \( H \) -paths in \( G \) .
In order to obtain direct analogues to the vertex and edge version of Menger's theorem, let us consider the two special cases of the above problem where either \( F \) or \( X \) is required to be empty. Given an induced
\( {\kappa }_{G}\left( H\right) \)
subgraph \( H \subseteq G \), we denote by \( {\kappa }_{G}\left( H\right) \) the least cardinality of a vertex set \( X \subseteq V\left( {G - H}\right) \) that meets every \( H \) -path in \( G \) . Similarly, we let
\( {\lambda }_{G}\left( H\right) \)
\( {\lambda }_{G}\left( H\right) \) denote the least cardinality of an edge set \( F \subseteq E\left( G\right) \) that meets every \( H \) -path in \( G \) .
Corollary 3.4.2. Given a graph \( G \) with an induced subgraph \( H \), there are at least \( \frac{1}{2}{\kappa }_{G}\left( H\right) \) independent \( H \) -paths and at least \( \frac{1}{2}{\lambda }_{G}\left( H\right) \) edge-disjoint \( H \) -paths in \( G \) .
Proof. To prove the first assertion, let \( k \) be the maximum number of independent \( H \) -paths in \( G \) . By Theorem 3.4.1, there are sets \( X \subseteq V\left( {G - H}\right) \) and \( F \subseteq E\left( {G - H - X}\right) \) with
\[
k = \left| X\right| + \mathop{\sum }\limits_{{C \in {\mathcal{C}}_{F}}}\left\lfloor {\frac{1}{2}\left| {\partial C}\right| }\right\rfloor
\]
such that every \( H \) -path in \( G \) has a vertex in \( X \) or an edge in \( F \) . For every \( C \in {\mathcal{C}}_{F} \) with \( \partial C \neq \varnothing \), pick a vertex \( v \in \partial C \) and let \( {Y}_{C} \mathrel{\text{:=}} \partial C \smallsetminus \{ v\} \) ; if \( \partial C = \varnothing \), let \( {Y}_{C} \mathrel{\text{:=}} \varnothing \) . Then \( \left\lfloor {\frac{1}{2}\left| {\partial C}\right| }\right\rfloor \geq \frac{1}{2}\left| {Y}_{C}\right| \) for all \( C \in {\mathcal{C}}_{F} \) . Moreover, for \( Y \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{C \in {\mathcal{C}}_{F}}}{Y}_{C} \) every \( H \) -path has a vertex in \( X \cup Y \) . Hence
\[
k \geq \left| X\right| + \mathop{\sum }\limits_{{C \in {\mathcal{C}}_{F}}}\frac{1}{2}\left| {Y}_{C}\right| \geq \frac{1}{2}\left| {X \cup Y}\right| \geq \frac{1}{2}{\kappa }_{G}\left( H\right)
\]
as claimed.
The second assertion follows from the first by considering the line graph of \( G \) (Exercise 18).
It may come as a surprise to see that the bounds in Corollary 3.4.2 are best possible (as general bounds): one can find examples for \( G \) and \( H \) where \( G \) contains no more than \( \frac{1}{2}{\kappa }_{G}\left( H\right) \) independent \( H \) -paths or no more than \( \frac{1}{2}{\lambda }_{G}\left( H\right) \) edge-disjoint \( H \) -paths (Exercises 19 and 20).
## 3.5 Linking pairs of vertices
Let \( G \) be a graph, and let \( X \subseteq V\left( G\right) \) be a set of vertices. We call \( X \)
linked in \( G \) if whenever we pick distinct vertices \( {s}_{1},\ldots ,{s}_{\ell },{t}_{1},\ldots ,{t}_{\ell } \) in linked \( X \) we can find disjoint paths \( {P}_{1},\ldots ,{P}_{\ell } \) in \( G \) such that each \( {P}_{i} \) links \( {s}_{i} \) to \( {t}_{i} \) and has no inner vertex in \( X \) . Thus, unlike in Menger’s theorem, we are not merely asking for disjoint paths between two sets of vertices: we insist that each of these paths shall link a specified pair of endvertices.
If \( \left| G\right| \geq {2k} \) and every set of at most \( {2k} \) vertices is linked in \( G \), then
\( G \) is \( k \) -linked. As is easily checked, this is equivalent to requiring that \( k \) -linked disjoint paths \( {P}_{i} = {s}_{i}\ldots {t}_{i} \) exist for every choice of exactly \( {2k} \) vertices \( {s}_{1},\ldots ,{s}_{k},{t}_{1},\ldots ,{t}_{k} \) . In practice, the latter is easier to prove, because we need not worry about inner vertices in \( X \) .
Clearly, every \( k \) -linked graph is \( k \) -connected. The converse, however, seems far from true: being \( k \) -linked is clearly a much stronger property than \( k \) -connectedness. Still, we shall prove in this section that we can force a graph to be \( k \) -linked by assuming that it is \( f\left( k\right) \) -connected, for some function \( f : \mathbb{N} \rightarrow \mathbb{N} \) . We first give a nice and simple proof that such a function \( f \) exists at all. In the remainder of the section we then prove that \( f \) can even be chosen linear.
The basic idea in the simple proof is as follows. If we can prove that \( G \) contains a subdivision \( K \) of a large complete graph, we can use Menger’s theorem to link the vertices of \( X \) disjointly to branch vertices of \( K \), and then hope to pair them up as desired through the subdivided edges of \( K \) . This requires, of course, that our paths do not hit too many of the subdivided edges before reaching the branch vertices of \( K \) .
To show that \( K \) exists is a lemma which more properly belongs in Chapter 7, and we shall derive an improved version there from the
linearity theorem (3.5.3) proved later in this section. Instead of assuming high connectivity, it suffices that \( G \) has large enough average degree:
Lemma 3.5.1. There is a function \( h : \mathbb{N} \rightarrow \mathbb{N} \) such that every graph of average degree at least \( h\left( r\right) \) contains \( {K}^{r} \) as a topological minor, for every \( r \in \mathbb{N} \) .
Proof. For \( r \leq 2 \), the assertion holds with \( h\left( r\right) = 1 \) ; we now assume that \( r \geq 3 \) . We show by induction on \( m = r,\ldots ,\left( \begin{array}{l} r \\ 2 \end{array}\right) \) that every graph \( G \) with average degree \( d\left( G\right) \geq {2}^{m} \) has a topological minor \( X \) with \( r \) vertices and \( m \) edges; for \( m = \left( \begin{array}{l} r \\ 2 \end{array}\right) \) this implies the assertion with \( h\left( r\right) = {2}^{\left( \begin{array}{l} r \\ 2 \end{array}\right) } \) .
If \( m = r \) then, by Propositions 1.2.2 and 1.3.1, \( G \) contains a cycle of length at least \( \varepsilon \left( G\right) + 1 \geq {2}^{r - 1} + 1 \geq r + 1 \), and the assertion follows with \( X = {C}^{r} \) .
Now let \( r < m \leq \left( \begin{array}{l} r \\ 2 \end{array}\right) \), and assume the assertion holds for smaller \( m \) . Let \( G \) with \( d\left( G\right) \geq {2}^{m} \) be given; thus, \( \varepsilon \left( G\right) \geq {2}^{m - 1} \) . Since \( G \) has a component \( C \) with \( \varepsilon \left( C\right) \geq \varepsilon \left( G\right) \), we may assume that \( G \) is connected. Consider a maximal set \( U \subseteq V\left( G\right) \) such that \( U \) is connected in \( G \) and \( \varepsilon \left( {G/U}\right) \geq {2}^{m - 1} \) ; such a set \( U \) exists, because \( G \) itself has the form \( G/U \) with \( \left| U\right| = 1 \) . Since \( G \) is connected, we have \( N\left( U\right) \neq \varnothing \) .
Let \( H \mathrel{\text{:=}} G\left\lbrack {N\left( U\right) }\right\rbrack \) . If \( H \) has a vertex \( v \) of degree \( {d}_{H}\left( v\right) < {2}^{m - 1} \), we may add it to \( U \) and obtain a contradiction to the maximality of \( U \) : when we contract the edge \( v{v}_{U} \) in \( G/U \), we lose one vertex and \( {d}_{H}\left( v\right) + 1 \leq \) \( {2}^{m - 1} \) edges, so \( \varepsilon \) will still be at least \( {2}^{m - 1} \) . Therefore \( d\left( H\right) \geq \delta \left( H\right) \geq \) \( {2}^{m - 1} \) . By the induction hypothesis, \( H \) contains a \( {TY} \) with \( \left| Y\right| = r \) and \( \parallel Y\parallel = m - 1 \) . Let \( x, y \) be two branch vertices of this \( {TY} \) that are non-adjacent in \( Y \) . Since \( x \) and \( y \) lie in \( N\left( U\right) \) and \( U \) is connected in \( G \) , \( G \) contains an \( x - y \) path whose inner vertices lie in \( U \) . Adding this path to the \( {TY} \), we obtain the desired \( {TX} \) .
Theorem 3.5.2. (Jung 1970; Larman & Mani 1970)
There is a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that every \( f\left( k\right) \) -connected graph is \( k \) -linked, for all \( k \in \mathbb{N} \) .
\( \left( {3.3.1}\right) \)
Proof. We prove the assertion for \( f\left( k\right) = h\left( {3k}\right) + {2k} \), where \( h \) is a \( G \) function as in Lemma 3.5.1. Let \( G \) be an \( f\left( k\right) \) -connected graph. Then \( K \) \( d\left( G\right) \geq \delta \left( G\right) \geq \kappa \left( G\right) \geq h\left( {3k}\right) \) ; choose \( K = T{K}^{3k} \subseteq G \) as in Lemma 3.5.1, \( U \) and let \( U \) denote its set of branch vertices.
\( {s}_{i},{t}_{i} \) For the proof that \( G \) is \( k \) -linked, let distinct vertices \( {s}_{1},\ldots ,{s}_{k} \) and \( {t}_{1},\ldots ,{t}_{k} \) be given. By definition of \( f\left( k\right) \), we have \( \kappa \left( G\right) \geq {2k} \) . Hence by Menger’s theorem (3.3.1), \( G \) contains disjoint paths \( {P}_{1},\ldots ,{P}_{k} \) , \( {P}_{i},{Q}_{i} \) \( {Q}_{1},\ldots ,{Q}_{k} \), such that each \( {P}_{i} \) starts in \( {s}_{i} \), each \( {Q}_{i} \) starts in \( {t}_{i} \), and all
\( \mathcal{P} \) thes |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 25 | \left( {3.3.1}\right) \)
Proof. We prove the assertion for \( f\left( k\right) = h\left( {3k}\right) + {2k} \), where \( h \) is a \( G \) function as in Lemma 3.5.1. Let \( G \) be an \( f\left( k\right) \) -connected graph. Then \( K \) \( d\left( G\right) \geq \delta \left( G\right) \geq \kappa \left( G\right) \geq h\left( {3k}\right) \) ; choose \( K = T{K}^{3k} \subseteq G \) as in Lemma 3.5.1, \( U \) and let \( U \) denote its set of branch vertices.
\( {s}_{i},{t}_{i} \) For the proof that \( G \) is \( k \) -linked, let distinct vertices \( {s}_{1},\ldots ,{s}_{k} \) and \( {t}_{1},\ldots ,{t}_{k} \) be given. By definition of \( f\left( k\right) \), we have \( \kappa \left( G\right) \geq {2k} \) . Hence by Menger’s theorem (3.3.1), \( G \) contains disjoint paths \( {P}_{1},\ldots ,{P}_{k} \) , \( {P}_{i},{Q}_{i} \) \( {Q}_{1},\ldots ,{Q}_{k} \), such that each \( {P}_{i} \) starts in \( {s}_{i} \), each \( {Q}_{i} \) starts in \( {t}_{i} \), and all
\( \mathcal{P} \) these paths end in \( U \) but have no inner vertices in \( U \) . Let the set \( \mathcal{P} \) of these paths be chosen so that their total number of edges outside \( E\left( K\right) \) is as small as possible.
Let \( {u}_{1},\ldots ,{u}_{k} \) be those \( k \) vertices in \( U \) that are not an end of a path in \( \mathcal{P} \) . For each \( i = 1,\ldots, k \), let \( {L}_{i} \) be the \( U \) -path in \( K \) (i.e., the subdivided edge of the \( {K}^{3k} \) ) from \( {u}_{i} \) to the end of \( {P}_{i} \) in \( U \), and let \( {v}_{i} \) be the first vertex of \( {L}_{i} \) on any path \( P \in \mathcal{P} \) . By definition of \( \mathcal{P}, P \) has no more edges outside \( E\left( K\right) \) than \( P{v}_{i}{L}_{i}{u}_{i} \) does, so \( {v}_{i}P = {v}_{i}{L}_{i} \) and hence \( P = {P}_{i} \) (Fig. 3.5.1). Similarly, if \( {M}_{i} \) denotes the \( U \) -path in \( K \) from \( {u}_{i} \) to the end of \( {Q}_{i} \) in \( U \), and \( {w}_{i} \) denotes the first vertex of \( {M}_{i} \) on any path in \( \mathcal{P} \), then this path is \( {Q}_{i} \) . Then the paths \( {s}_{i}{P}_{i}{v}_{i}{L}_{i}{u}_{i}{M}_{i}{w}_{i}{Q}_{i}{t}_{i} \) are disjoint for different \( i \) and show that \( G \) is \( k \) -linked.
Fig. 3.5.1. Constructing an \( {s}_{i} - {t}_{i} \) path via \( {u}_{i} \)
The proof of Theorem 3.5.2 yields only an exponential upper bound for the function \( f\left( k\right) \) . As \( {2\varepsilon }\left( G\right) \geq \delta \left( G\right) \geq \kappa \left( G\right) \), the following result implies the linear bound of \( f\left( k\right) = {16k} \) :
Theorem 3.5.3. (Thomas & Wollan 2005)
\( \left\lbrack {7.2.1}\right\rbrack \)
Let \( G \) be a graph and \( k \in \mathbb{N} \) . If \( G \) is \( {2k} \) -connected and \( \varepsilon \left( G\right) \geq {8k} \), then \( G \) is \( k \) -linked.
We begin our proof of Theorem 3.5.3 with a lemma.
Lemma 3.5.4. If \( \delta \left( G\right) \geq {8k} \) and \( \left| G\right| \leq {16k} \), then \( G \) has a \( k \) -linked subgraph.
Proof. If \( G \) itself is \( k \) -linked there is nothing to show, so suppose not. Then we can find a set \( X \) of \( {2k} \) vertices \( {s}_{1},\ldots ,{s}_{k},{t}_{1},\ldots ,{t}_{k} \) that cannot be linked in \( G \) by disjoint paths \( {P}_{i} = {s}_{i}\ldots {t}_{i} \) . Let \( \mathcal{P} \) be a set of as many such paths as possible, but all of length at most 7 . If there are several such sets \( \mathcal{P} \), we choose one with \( \left| {\bigcup \mathcal{P}}\right| \) minimum. We may assume that \( \mathcal{P} \) contains no path from \( {s}_{1} \) to \( {t}_{1} \) . Let \( J \) be the subgraph of \( G \) induced by \( X \) and all the vertices on the paths in \( \mathcal{P} \), and let \( H \mathrel{\text{:=}} G - J \) .
Note that each vertex \( v \in H \) has at most three neighbours on any given \( {P}_{i} \in \mathcal{P} \) : if it had four, then replacing the segment \( u{P}_{i}w \) between its first and its last neighbour on \( {P}_{i} \) by the path \( {uvw} \) would reduce \( \left| {\bigcup \mathcal{P}}\right| \) and thus contradict our choice of \( \mathcal{P} \) . Moreover, \( v \) is not adjacent to both \( {s}_{i} \) and \( {t}_{i} \) whenever \( {s}_{i},{t}_{i} \notin \bigcup \mathcal{P} \), by the maximality of \( \mathcal{P} \) . Thus if \( \left| \mathcal{P}\right| = : h \) , then \( v \) has at most \( {3h} + \left( {{2k} - {2h}}\right) /2 \leq {3k} \) neighbours in \( J \) . As \( \delta \left( G\right) \geq {8k} \) and \( \left| G\right| \leq {16k} \) by assumption, while \( \left| X\right| = {2k} \), we deduce that
\[
\delta \left( H\right) \geq {5k}\text{ and }\left| H\right| \leq {14k}.
\]
(1)
Our next aim is to show that \( H \) is disconnected. Since each of the paths in \( \mathcal{P} \) has at most eight vertices, we have \( \left| {J - \left\{ {{s}_{1},{t}_{1}}\right\} }\right| \leq 8\left( {k - 1}\right) \) . Therefore both \( {s}_{1} \) and \( {t}_{1} \) have neighbours in \( H \) . Let \( S \subseteq V\left( H\right) \) be the set of vertices at distance at most 2 (measured in \( H \) ) from a neighbour of \( {s}_{1} \) in \( H \), and let \( T \subseteq V\left( H\right) \) be the corresponding set for \( {t}_{1} \) . Since \( G - \bigcup \mathcal{P} \) contains no \( {s}_{1} - {t}_{1} \) path of length at most 7, we have \( S \cap T = \varnothing \) and there is no \( S - T \) edge in \( H \) . To prove that \( H \) is disconnected, it thus suffices to show that \( V\left( H\right) = S \cup T \) . Pick a neighbour \( s \in S \) of \( {s}_{1} \), and a neighbour \( t \in T \) of \( {t}_{1} \) . Then for any vertex \( v \in H - \left( {S \cup T}\right) \) the sets \( {N}_{H}\left( s\right) ,{N}_{H}\left( t\right) \) and \( {N}_{H}\left( v\right) \) are disjoint and each have size at least \( {5k} \) , contradicting (1).
So \( H \) is disconnected; let \( C \) be its smallest component. By (1),
\[
{2\delta }\left( C\right) \geq {2\delta }\left( H\right) \geq {7k} + {3k} \geq \frac{1}{2}\left| H\right| + {3k} \geq \left| C\right| + {3k}.
\]
(2)
We complete the proof by showing that \( C \) is \( k \) -linked. As \( \delta \left( C\right) \geq {5k} \) , we have \( \left| C\right| \geq {2k} \) . Let \( Y \) be a set of at most \( {2k} \) vertices in \( C \) . By (2), every two vertices in \( Y \) have at least \( {3k} \) common neighbours, at least \( k \) of which lie outside \( Y \) . We can therefore link any desired \( \ell \leq k \) pairs of vertices in \( Y \) inductively by paths of length 2 whose inner vertex lies outside \( Y \) .
Before we launch into the proof of Theorem 3.5.3, let us look at its main ideas. To prove that \( G \) is \( k \) -linked, we have to consider a given set \( X \) of up to \( {2k} \) vertices and show that \( X \) is linked in \( G \) . Ideally, we would like to use Lemma 3.5.4 to find a linked subgraph \( L \) somewhere in \( G \) , and then use our assumption of \( \kappa \left( G\right) \geq {2k} \) to obtain a set of \( \left| X\right| \) disjoint \( X - L \) paths by Menger’s theorem (3.3.1). Then \( X \) could be linked via these paths and \( L \), completing the proof.
Unfortunately, we cannot expect to find a subgraph \( H \) such that \( \delta \left( H\right) \geq {8k} \) and \( \left| H\right| \leq {16k} \) (in which \( L \) could be found by Lemma 3.5.4); cf. Corollary 11.2.3. However, it is not too difficult to find a minor \( H \preccurlyeq G \) that has such a subgraph (Ex. 22, Ch. 7), even so that the vertices of \( X \) come to lie in distinct branch sets of \( H \) . We may then regard \( X \) as a subset of \( V\left( H\right) \), and Lemma 3.5.4 provides us with a linked subgraph \( L \) of \( H \) . The only problem now is that \( H \) need no longer be \( {2k} \) -connected, that is, our assumption of \( \kappa \left( G\right) \geq {2k} \) will not ensure that we can link \( X \) to \( L \) by \( \left| X\right| \) disjoint paths in \( H \) .
And here comes the clever bit of the proof: it relaxes the assumption of \( \kappa \geq {2k} \) to a weaker assumption that does get passed on to \( H \) . This weaker assumption is that if we can separate \( X \) from some other subgraph by fewer than \( \left| X\right| \) vertices, then this other part must be ’light’: roughly, its own value of \( \varepsilon \) must not exceed \( {8k} \) . Now if we fail to link \( X \) to \( L \) by \( \left| X\right| \) disjoint paths, then \( H \) has such a separation \( \{ A, B\} \) , with \( X \subseteq A \) and \( L \subseteq B \) and such that \( \left| {A \cap B}\right| < \left| X\right| \) . If we choose this with \( \left| {A \cap B}\right| \) minimum, then by Menger’s theorem we can link \( A \cap B \) to \( L \) in \( H\left\lbrack B\right\rbrack \) by \( \left| {A \cap B}\right| \) disjoint paths. We may then continue our proof inside \( H\left\lbrack A\right\rbrack \), whose value of \( \varepsilon \) is still as big as before, because the \( B \) -part of \( H \) was ’light’. In fact, we may even turn \( A \cap B \) into a complete subgraph of \( H\left\lbrack A\right\rbrack \), because such new edges, if used by our linking paths, can be replaced by paths through \( B \) and \( L \) . This helps ensure that we do not in \( H\left\lbrack A\right\rbrack \) have new separations of order less than \( \left| X\right| \) that split a ’heavy’ part away from \( X \) . Hence, both our inductive assumptions - the value of \( \varepsilon \geq {8k} \) and the fact that small separators can only split light parts away from \( X \) -hold for \( H\left\lbrack A\right\rbrack \) because they did in \( H \) . This will complete the inductive proof.
Given \( k \in \mathbb{N} \), a graph \( G \), and \( A, B, X \subseteq V\left( G\right) \), call the ordered \( X \) - separation pair \( \left( {A, B}\right) \) an \( X \) -separation o |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 26 | H\left\lbrack A\right\rbrack \), whose value of \( \varepsilon \) is still as big as before, because the \( B \) -part of \( H \) was ’light’. In fact, we may even turn \( A \cap B \) into a complete subgraph of \( H\left\lbrack A\right\rbrack \), because such new edges, if used by our linking paths, can be replaced by paths through \( B \) and \( L \) . This helps ensure that we do not in \( H\left\lbrack A\right\rbrack \) have new separations of order less than \( \left| X\right| \) that split a ’heavy’ part away from \( X \) . Hence, both our inductive assumptions - the value of \( \varepsilon \geq {8k} \) and the fact that small separators can only split light parts away from \( X \) -hold for \( H\left\lbrack A\right\rbrack \) because they did in \( H \) . This will complete the inductive proof.
Given \( k \in \mathbb{N} \), a graph \( G \), and \( A, B, X \subseteq V\left( G\right) \), call the ordered \( X \) - separation pair \( \left( {A, B}\right) \) an \( X \) -separation of \( G \) if \( \{ A, B\} \) is a proper separation of \( G \)
of order at most \( \left| X\right| \) and \( X \subseteq A \) . An \( X \) -separation \( \left( {A, B}\right) \) is small if small/linked \( \left| {A \cap B}\right| < \left| X\right| \), and linked if \( A \cap B \) is linked in \( G\left\lbrack B\right\rbrack \) .
Call a set \( U \subseteq V\left( G\right) \) light in \( G \) if \( \parallel U{\parallel }^{ + } \leq {8k}\left| U\right| \), where \( \parallel U{\parallel }^{ + } \) denotes
\( \parallel \)
the number of edges of \( G \) with at least one end in \( U \) . A set of vertices light is heavy if it is not light. heavy
Proof of Theorem 3.5.3. We shall prove the following, for fixed \( k \in \mathbb{N} \) :
Let \( G = \left( {V, E}\right) \) be a graph and \( X \subseteq V \) a set of at most \( {2k} \) \( G = \left( {V, E}\right) \) vertices. If \( V \smallsetminus X \) is heavy and for every small \( X \) -separ-
\( \left( *\right) \)
ation \( \left( {A, B}\right) \) the set \( B \smallsetminus A \) is light, then \( X \) is linked in \( G \) .
To see that \( \left( *\right) \) implies the theorem, assume that \( \kappa \left( G\right) \geq {2k} \) and \( \varepsilon \left( G\right) \geq {8k} \), and let \( X \) be a set of exactly \( {2k} \) vertices. Then \( G \) has no small \( X \) -separation. And \( V \smallsetminus X \) is heavy, since
\[
\parallel V \smallsetminus X{\parallel }^{ + } \geq \parallel G\parallel - \left( \begin{matrix} {2k} \\ 2 \end{matrix}\right) \geq {8k}\left| V\right| - \left( \begin{matrix} {2k} \\ 2 \end{matrix}\right) > {8k}\left| {V \smallsetminus X}\right| .
\]
By \( \left( *\right), X \) is linked in \( G \), completing the proof that \( G \) is \( k \) -linked.
We prove \( \left( *\right) \) by induction on \( \left| G\right| \), and for each value of \( \left| G\right| \) by induction on \( \parallel V \smallsetminus X{\parallel }^{ + } \) . If \( \left| G\right| = 1 \) then \( X \) is linked in \( G \) . For the induction step, let \( G \) and \( X \) be given as in \( \left( *\right) \) . We first prove the following:
We may assume that \( G \) has no linked \( X \) -separation.
(1)
For our proof of (1), suppose that \( G \) has a linked \( X \) -separation
---
\( \left( {A, B}\right) \)
---
\( \left( {A, B}\right) \) . Let us choose one with \( A \) minimal, and put \( S \mathrel{\text{:=}} A \cap B \) .
We first consider the case that \( \left| S\right| = \left| X\right| \) . If \( G\left\lbrack A\right\rbrack \) contains \( \left| X\right| \) disjoint \( X - S \) paths, then \( X \) is linked in \( G \) because \( \left( {A, B}\right) \) is linked, completing the proof of \( \left( *\right) \) . If not, then by Menger’s theorem (3.3.1) \( G\left\lbrack A\right\rbrack \) has a small \( X \) - separation \( \left( {{A}^{\prime },{B}^{\prime }}\right) \) such that \( {B}^{\prime } \supseteq S \) . If we choose this with \( \left| {{A}^{\prime } \cap {B}^{\prime }}\right| \) minimum, we can link \( {A}^{\prime } \cap {B}^{\prime } \) to \( S \) in \( G\left\lbrack {B}^{\prime }\right\rbrack \) by \( \left| {{A}^{\prime } \cap {B}^{\prime }}\right| \) disjoint paths, again by Menger’s theorem. But then \( \left( {{A}^{\prime },{B}^{\prime } \cup B}\right) \) is a linked \( X \) -separation of \( G \) that contradicts the choice of \( \left( {A, B}\right) \) .
So \( \left| S\right| < \left| X\right| \) . Let \( {G}^{\prime } \) be obtained from \( G\left\lbrack A\right\rbrack \) by adding any missing edges on \( S \), so that \( {G}^{\prime }\left\lbrack S\right\rbrack \) is a complete subgraph of \( {G}^{\prime } \) . As \( \left( {A, B}\right) \) is now a small \( X \) - separation, our assumption in (*) says that \( B \smallsetminus A \) is light in \( G \) . Thus, \( {G}^{\prime } \) arises from \( G \) by deleting \( \left| {B \smallsetminus A}\right| \) vertices outside \( X \) and at most \( {8k}\left| {B \smallsetminus A}\right| \) edges, and possibly adding some edges. As \( V \smallsetminus X \) is heavy in \( G \), this implies that
\( A \smallsetminus X \) is heavy in \( {G}^{\prime } \) .
In order to be able to apply the induction hypothesis to \( {G}^{\prime } \), let us show next that for every small \( X \) -separation \( \left( {{A}^{\prime },{B}^{\prime }}\right) \) of \( {G}^{\prime } \) the set
\( \left( {{A}^{\prime },{B}^{\prime }}\right) \)
\( {B}^{\prime } \smallsetminus {A}^{\prime } \) is light in \( {G}^{\prime } \) . Suppose not, and choose a counterexample \( \left( {{A}^{\prime },{B}^{\prime }}\right) \) with \( {B}^{\prime } \) minimal. As \( {G}^{\prime }\left\lbrack S\right\rbrack \) is complete, we have \( S \subseteq {A}^{\prime } \) or \( S \subseteq {B}^{\prime } \) .
If \( S \subseteq {A}^{\prime } \) then \( {B}^{\prime } \cap B \subseteq S \subseteq {A}^{\prime } \), so \( \left( {{A}^{\prime } \cup B,{B}^{\prime }}\right) \) is a small \( X \) - separation of \( G \) . Moreover,
\[
{B}^{\prime } \smallsetminus \left( {{A}^{\prime } \cup B}\right) = {B}^{\prime } \smallsetminus {A}^{\prime }
\]
and no edge of \( {G}^{\prime } - E \) is incident with this set (Fig 3.5.2). Our assumption that this set is heavy in \( {G}^{\prime } \), by the choice of \( \left( {{A}^{\prime },{B}^{\prime }}\right) \), therefore implies that it is heavy also in \( G \) . As \( \left( {{A}^{\prime } \cup B,{B}^{\prime }}\right) \) is a small \( X \) -separation of \( G \), this contradicts our assumptions in \( \left( *\right) \) .
Fig. 3.5.2. If \( S \subseteq {A}^{\prime } \), then \( \left( {{A}^{\prime } \cup B,{B}^{\prime }}\right) \) is an \( X \) -separation of \( G \)
Hence \( S \subseteq {B}^{\prime } \) . By our choice of \( \left( {{A}^{\prime },{B}^{\prime }}\right) \), the graph \( {G}^{\prime \prime } \mathrel{\text{:=}} {G}^{\prime }\left\lbrack {B}^{\prime }\right\rbrack \) satisfies the premise of \( \left( *\right) \) for \( {X}^{\prime \prime } \mathrel{\text{:=}} {A}^{\prime } \cap {B}^{\prime } \) . By the induction hypothesis, \( {X}^{\prime \prime } \) is linked in \( {G}^{\prime \prime } \) . But then \( {X}^{\prime \prime } \) is also linked in \( G\left\lbrack {{B}^{\prime } \cup B}\right\rbrack \) : as \( S \) was linked in \( G\left\lbrack B\right\rbrack \), we simply replace any edges added on \( S \) in the definition of \( {G}^{\prime } \) by disjoint paths through \( B \) (Fig. 3.5.3). But now \( \left( {{A}^{\prime },{B}^{\prime } \cup B}\right) \) is a linked \( X \) -separation of \( G \) that violates the minimality of \( A \) in the choice of \( \left( {A, B}\right) \) .
Fig. 3.5.3. If \( S \subseteq {B}^{\prime } \), then \( \left( {{A}^{\prime },{B}^{\prime } \cup B}\right) \) is linked in \( G \)
We have thus shown that \( {G}^{\prime } \) satisfies the premise of \( \left( *\right) \) with respect to \( X \) . Since \( \{ A, B\} \) is a proper separation, \( {G}^{\prime } \) has fewer vertices than \( G \) . By the induction hypothesis, therefore, \( X \) is linked in \( {G}^{\prime } \) . Replacing edges of \( {G}^{\prime } - E \) on \( S \) by paths through \( B \) as before, we can turn any linkage of \( X \) in \( {G}^{\prime } \) into one in \( G \), completing the proof of \( \left( *\right) \) . This completes the proof of (1).
Our next goal is to show that, by the induction hypothesis, we may assume that \( G \) has not only large average degree but even large minimum degree. For our proof that \( X \) is linked in \( G \), let \( {s}_{1},\ldots ,{s}_{\ell },{t}_{1},\ldots ,{t}_{\ell } \) be the distinct vertices in \( X \) which we wish to link by disjoint paths \( {P}_{i} = {s}_{i}\ldots {t}_{i} \) . Since these paths must not have any inner vertices in \( X \), we may assume that \( G \) has all edges on \( X \) except possibly the edges \( {s}_{i}{t}_{i} \) : as no other
\( G\left\lbrack X\right\rbrack \)
edges on \( X \) may be used by the paths \( {P}_{i} \), we may add them without affecting either the premise or the conclusion in \( \left( *\right) \) .
After this modification, we can now prove the following:
We may assume that any two adjacent vertices \( u, v \) which
(2)
do not both lie in \( X \) have at least \( {8k} - 1 \) common neighbours.
To prove (2), let \( e = {uv} \) be such an edge, let \( n \) denote the number of common neighbours of \( u \) and \( v \), and let \( {G}^{\prime } \mathrel{\text{:=}} G/e \) be the graph obtained by contracting \( e \) . Since \( u, v \) are not both in \( X \) we may view \( X \) as a subset also of \( {V}^{\prime } \mathrel{\text{:=}} V\left( {G}^{\prime }\right) \), replacing \( u \) or \( v \) in \( X \) with the contracted vertex \( {v}_{e} \) if \( X \cap \{ u, v\} \neq \varnothing \) . Our aim is to s |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 27 | \) has all edges on \( X \) except possibly the edges \( {s}_{i}{t}_{i} \) : as no other
\( G\left\lbrack X\right\rbrack \)
edges on \( X \) may be used by the paths \( {P}_{i} \), we may add them without affecting either the premise or the conclusion in \( \left( *\right) \) .
After this modification, we can now prove the following:
We may assume that any two adjacent vertices \( u, v \) which
(2)
do not both lie in \( X \) have at least \( {8k} - 1 \) common neighbours.
To prove (2), let \( e = {uv} \) be such an edge, let \( n \) denote the number of common neighbours of \( u \) and \( v \), and let \( {G}^{\prime } \mathrel{\text{:=}} G/e \) be the graph obtained by contracting \( e \) . Since \( u, v \) are not both in \( X \) we may view \( X \) as a subset also of \( {V}^{\prime } \mathrel{\text{:=}} V\left( {G}^{\prime }\right) \), replacing \( u \) or \( v \) in \( X \) with the contracted vertex \( {v}_{e} \) if \( X \cap \{ u, v\} \neq \varnothing \) . Our aim is to show that unless \( n \geq {8k} - 1 \) as desired in (2), \( {G}^{\prime } \) satisfies the premise of \( \left( *\right) \) . Then \( X \) will be linked in \( {G}^{\prime } \) by the induction hypothesis, so the desired paths \( {P}_{1},\ldots ,{P}_{\ell } \) exist in \( {G}^{\prime } \) . If one of them contains \( {v}_{e} \), replacing \( {v}_{e} \) by \( u \) or \( v \) or \( {uv} \) turns it into a path in \( G \), completing the proof of \( \left( *\right) \) .
In order to show that \( {G}^{\prime } \) satisfies the premise of \( \left( *\right) \) with respect to \( X \), let us show first that \( {V}^{\prime } \smallsetminus X \) is heavy. Since \( V \smallsetminus X \) was heavy and \( \left| {{V}^{\prime } \smallsetminus X}\right| = \left| {V \smallsetminus X}\right| - 1 \), it suffices to show that the contraction of \( e \) resulted in the loss of at most \( {8k} \) edges incident with a vertex outside \( X \) . If \( u \) and \( v \) are both outside \( X \), then the number of such edges lost is only \( n + 1 \) : one edge at every common neighbour of \( u \) and \( v \), as well as \( e \) . But if \( u \in X \), then \( v \notin X \), and we lost all the \( X - v \) edges \( {xv} \) of \( G \), too: while \( {xv} \) counted towards \( \parallel V \smallsetminus X{\parallel }^{ + } \), the edge \( x{v}_{e} \) lies in \( {G}^{\prime }\left\lbrack X\right\rbrack \) and does not count towards \( {\begin{Vmatrix}{V}^{\prime } \smallsetminus X\end{Vmatrix}}^{ + } \) . If \( x \neq u \) and \( x \) is not a common neighbour of \( u \) and \( v \), then this is an additional loss. But \( u \) is adjacent to every \( x \in X \smallsetminus \{ u\} \) except at most one (by our assumption about \( G\left\lbrack X\right\rbrack \) ), so every such \( x \) except at most one is in fact a common neighbour of \( u \) and \( v \) . Thus in total, we lost at most \( n + 2 \) edges. Unless \( n \geq {8k} - 1 \) (which would prove (2) directly for \( u \) and \( v \) ), this means that we lost at most \( {8k} \) edges, as desired for our proof that \( {V}^{\prime } \smallsetminus X \) is heavy.
It remains to show that for every small \( X \) -separation \( \left( {{A}^{\prime },{B}^{\prime }}\right) \) of \( {G}^{\prime } \)
\( \left( {{A}^{\prime },{B}^{\prime }}\right) \)
the set \( {B}^{\prime } \smallsetminus {A}^{\prime } \) is light. Let \( \left( {{A}^{\prime },{B}^{\prime }}\right) \) be a counterexample, chosen with \( {B}^{\prime } \) minimal. Then \( {G}^{\prime }\left\lbrack {B}^{\prime }\right\rbrack \) satisfies the premise of \( \left( *\right) \) with respect to \( {X}^{\prime } \mathrel{\text{:=}} {A}^{\prime } \cap {B}^{\prime } \), so \( {X}^{\prime } \) is linked in \( {G}^{\prime }\left\lbrack {B}^{\prime }\right\rbrack \) by induction. Let \( A \) and \( B \) be obtained from \( {A}^{\prime } \) and \( {B}^{\prime } \) by replacing \( {v}_{e} \), where applicable, with both \( u \) and \( v \) . We may assume that \( u, v \in B \), since otherwise \( \left( {A, B}\right) \) is a small \( X \) -separation of \( G \) with \( B \smallsetminus A \) heavy, contradicting our assumptions in (*). We shall prove that \( {X}^{\prime \prime } \mathrel{\text{:=}} A \cap B \) is linked in \( G\left\lbrack B\right\rbrack \) ; then \( \left( {A, B}\right) \) is a linked \( X \) - separation of \( G \), which contradicts (1).
If \( {v}_{e} \notin {X}^{\prime } \), then \( u, v \in B \smallsetminus A \) . Now \( {X}^{\prime \prime } \) is linked in \( G\left\lbrack B\right\rbrack \) because \( {X}^{\prime } \) is linked in \( {G}^{\prime }\left\lbrack {B}^{\prime }\right\rbrack \) : if \( {v}_{e} \) occurs on one of the linking paths for \( {X}^{\prime } \) , just replace it by \( u \) or \( v \) or \( {uv} \) as earlier.
Now assume that \( {v}_{e} \in {X}^{\prime } \) . Our aim is to show that \( G\left\lbrack B\right\rbrack \) satisfies the premise of \( \left( *\right) \) with respect to \( {X}^{\prime \prime } \), so that \( {X}^{\prime \prime } \) is linked in \( G\left\lbrack B\right\rbrack \) by induction. We know that \( B \smallsetminus {X}^{\prime \prime } = {B}^{\prime } \smallsetminus {A}^{\prime } \) is heavy in \( G \), since it is heavy in \( {G}^{\prime } \) by the choice of \( \left( {{A}^{\prime },{B}^{\prime }}\right) \) . Consider a small \( {X}^{\prime \prime } \) -separation \( \left( {{A}^{\prime \prime },{B}^{\prime \prime }}\right) \) of \( G\left\lbrack B\right\rbrack \) . Then \( \left( {A \cup {A}^{\prime \prime },{B}^{\prime \prime }}\right) \) is a small \( X \) -separation of \( G \), so \( {B}^{\prime \prime } \smallsetminus {A}^{\prime \prime } = {B}^{\prime \prime } \smallsetminus \left( {A \cup {A}^{\prime \prime }}\right) \) is light by the assumption in \( \left( *\right) \) . This completes the proof that \( {X}^{\prime \prime } \) is linked in \( G\left\lbrack B\right\rbrack \), and hence the proof of (2).
Using induction by contracting an edge, we have just shown that the vertices in \( V \smallsetminus X \) may be assumed to have large degree. Using induction by deleting an edge, we now show that their degrees cannot be too large. Since \( \left( *\right) \) holds if \( V = X \), we may assume that \( V \smallsetminus X \neq \varnothing \) ; let \( {d}^{ * } \) denote the smallest degree in \( G \) of a vertex in \( V \smallsetminus X \) . Let us prove that
\[
{8k} \leq {d}^{ * } \leq {16k} - 1
\]
(3)
The lower bound in (3) follows from (2) if we assume that \( G \) has no isolated vertex outside \( X \), which we may clearly assume by induction. To prove the upper bound, let us see what happens if we delete an edge \( e \) whose ends \( u, v \) are not both in \( X \) . If \( G - e \) satisfies the premise of \( \left( *\right) \) with respect to \( X \), then \( X \) is linked in \( G - e \) by induction, and hence in \( G \) . If not, then either \( V \smallsetminus X \) is light in \( G - e \), or \( G - e \) has a small \( X \) -separation \( \left( {A, B}\right) \) such that \( B \smallsetminus A \) is heavy. If the latter happens then \( e \) must be an \( \left( {A \smallsetminus B}\right) - \left( {B \smallsetminus A}\right) \) edge: otherwise, \( \left( {A, B}\right) \) would be a small \( X \) - separation also of \( G \), and \( B \smallsetminus A \) would be heavy also in \( G \), in contradiction to our assumptions in \( \left( *\right) \) . But if \( e \) is such an edge then any common neighbours of \( u \) and \( v \) lie in \( A \cap B \), so there are fewer than \( \left| X\right| \leq {2k} \) such neighbours. This contradicts (2).
So \( V \smallsetminus X \) must be light in \( G - e \) . For \( G \), this yields
\[
\parallel V \smallsetminus X{\parallel }^{ + } \leq {8k}\left| {V \smallsetminus X}\right| + 1
\]
(4)
In order to show that this implies the desired upper bound for \( {d}^{ * } \), let us estimate the number \( f\left( x\right) \) of edges that a vertex \( x \in X \) sends to \( V \smallsetminus X \) .
\( f\left( x\right) \)
There must be at least one such edge, \( {xy} \) say, as otherwise \( \left( {X, V\smallsetminus \{ x\} }\right) \) would be a small \( X \) -separation of \( G \) that contradicts our assumptions in \( \left( *\right) \) . But then, by \( \left( 2\right), x \) and \( y \) have at least \( {8k} - 1 \) common neighbours, at most \( {2k} - 1 \) of which lie in \( X \) . Hence \( f\left( x\right) \geq {6k} \) . As
\[
2\parallel V \smallsetminus X{\parallel }^{ + } = \mathop{\sum }\limits_{{v \in V \smallsetminus X}}{d}_{G}\left( v\right) + \mathop{\sum }\limits_{{x \in X}}f\left( x\right) ,
\]
an assumption of \( {d}^{ * } \geq {16k} \) would thus imply that
\[
2\left( {{8k}\left| {V \smallsetminus X}\right| + 1}\right) \geq 2\parallel V \smallsetminus X{\parallel }^{ + } \geq {16k}\left| {V \smallsetminus X}\right| + {6k}\left| X\right| ,
\]
yielding the contradiction of \( 2 \geq {6k}\left| X\right| \) . This completes the proof of (3).
To complete our proof of \( \left( *\right) \), pick a vertex \( {v}_{0} \in V \smallsetminus X \) of degree \( {d}^{ * } \) , and consider the subgraph \( H \) induced in \( G \) by \( {v}_{0} \) and its neighbours. By (2) we have \( \delta \left( H\right) \geq {8k} \), and by (3) and the choice of \( {v}_{0} \) we have \( \left| H\right| \leq {16k} \) . By Lemma 3.5.4, then, \( H \) has a \( k \) -linked subgraph; let \( L \) be its vertex set. By definition of ’ \( k \) -linked’, we have \( \left| L\right| \geq {2k} \geq \left| X\right| \) . If \( G \) contains \( \left| X\right| \) disjoint \( X - L \) paths, then \( X \) is linked in \( G \), as desired. If not, then \( G \) has a small \( X \) - separation \( \left( {A, B}\right) \) with \( L \subseteq B \) . If we choose \( \left( {A, B}\right) \) of minimum order, then \( G\left\lbrack B\right\rbrack \) contains \( \left| {A \cap B}\right| \) disjoint \( \left( {A \cap B}\right) - L \) paths by Menger’s theorem (3.3.1). But |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 28 | .
To complete our proof of \( \left( *\right) \), pick a vertex \( {v}_{0} \in V \smallsetminus X \) of degree \( {d}^{ * } \) , and consider the subgraph \( H \) induced in \( G \) by \( {v}_{0} \) and its neighbours. By (2) we have \( \delta \left( H\right) \geq {8k} \), and by (3) and the choice of \( {v}_{0} \) we have \( \left| H\right| \leq {16k} \) . By Lemma 3.5.4, then, \( H \) has a \( k \) -linked subgraph; let \( L \) be its vertex set. By definition of ’ \( k \) -linked’, we have \( \left| L\right| \geq {2k} \geq \left| X\right| \) . If \( G \) contains \( \left| X\right| \) disjoint \( X - L \) paths, then \( X \) is linked in \( G \), as desired. If not, then \( G \) has a small \( X \) - separation \( \left( {A, B}\right) \) with \( L \subseteq B \) . If we choose \( \left( {A, B}\right) \) of minimum order, then \( G\left\lbrack B\right\rbrack \) contains \( \left| {A \cap B}\right| \) disjoint \( \left( {A \cap B}\right) - L \) paths by Menger’s theorem (3.3.1). But then \( \left( {A, B}\right) \) is a linked \( X \) -separation that contradicts (1).
## Exercises
For the first three exercises let \( G \) be a graph with vertices \( a \) and \( b \), and let \( X \subseteq V\left( G\right) \smallsetminus \{ a, b\} \) be an \( a - b \) separator in \( G \) .
1. \( {}^{ - } \) Show that \( X \) is minimal as an \( a - b \) separator if and only if every vertex in \( X \) has a neighbour in the component \( {C}_{a} \) of \( G - X \) containing \( a \), and another in the component \( {C}_{b} \) of \( G - X \) containing \( b \) .
2. Let \( {X}^{\prime } \subseteq V\left( G\right) \smallsetminus \{ a, b\} \) be another \( a - b \) separator, and define \( {C}_{a}^{\prime } \) and \( {C}_{b}^{\prime } \) correspondingly. Show that both
and
\[
{Y}_{a} \mathrel{\text{:=}} \left( {X \cap {C}_{a}^{\prime }}\right) \cup \left( {X \cap {X}^{\prime }}\right) \cup \left( {{X}^{\prime } \cap {C}_{a}}\right)
\]
\[
{Y}_{b} \mathrel{\text{:=}} \left( {X \cap {C}_{b}^{\prime }}\right) \cup \left( {X \cap {X}^{\prime }}\right) \cup \left( {{X}^{\prime } \cap {C}_{b}}\right)
\]
separate \( a \) from \( b \) (see figure).
3. Are \( {Y}_{a} \) and \( {Y}_{b} \) minimal \( a - b \) separators if \( X \) and \( {X}^{\prime } \) are? Are \( \left| {Y}_{a}\right| \) and \( \left| {Y}_{b}\right| \) minimal for \( a - b \) separators from \( V\left( G\right) \smallsetminus \{ a, b\} \) if \( \left| X\right| \) and \( \left| {X}^{\prime }\right| \) are?
4. Let \( X \) and \( {X}^{\prime } \) be minimal separators in \( G \) such that \( X \) meets at least two components of \( G - {X}^{\prime } \) . Show that \( {X}^{\prime } \) meets all the components of \( G - X \), and that \( X \) meets all the components of \( G - {X}^{\prime } \) .
5. \( {}^{ - } \) Prove the elementary properties of blocks mentioned at the beginning of Section 3.1.
6. Show that the block graph of any connected graph is a tree.
7. Show, without using Menger's theorem, that any two vertices of a 2- connected graph lie on a common cycle.
8. For edges \( e,{e}^{\prime } \in G \) write \( e \sim {e}^{\prime } \) if either \( e = {e}^{\prime } \) or \( e \) and \( {e}^{\prime } \) lie on some common cycle in \( G \) . Show that \( \sim \) is an equivalence relation on \( E\left( G\right) \) whose equivalence classes are the edge sets of the non-trivial blocks of \( G \) .
9. Let \( G \) be a 2-connected graph but not a triangle, and let \( e \) be an edge of \( G \) . Show that either \( G - e \) or \( G/e \) is again 2-connected. Deduce a constructive characterization of the 2-connected graphs analogous to Theorem 3.2.2.
10. Let \( G \) be a 3-connected graph, and let \( {xy} \) be an edge of \( G \) . Show that \( G/{xy} \) is 3-connected if and only if \( G - \{ x, y\} \) is 2-connected.
11. (i) \( {}^{ - } \) Show that every cubic 3-edge-connected graph is 3-connected.
(ii) Show that a graph is cubic and 3-connected if and only if it can be constructed from a \( {K}^{4} \) by successive applications of the following operation: subdivide two edges by inserting a new vertex on each of them, and join the two new subdividing vertices by an edge.
12. \( {}^{ + } \) Find a finite set of 3-connected graphs from which all 3-connected graphs can be constructed iteratively by the following operation, or show that no such set exists. The operation consists of adding a new vertex to the graph \( H \) constructed so far and joining it by at least three edges to some subdivision of \( H \) . (In other words, every new edge is either incident with a vertex of \( H \) or else with a new subdividing vertex of \( H \) created for this purpose, and the new edges should obviously not all go to the same subdivided edge of \( H \) including its ends.)
13. Find the error in the following 'simple proof' of Menger's theorem (3.3.1). Let \( X \) be an \( A - B \) separator of minimum size. Denote by \( {G}_{A} \) the subgraph of \( G \) induced by \( X \) and all the components of \( G - X \) that meet \( A \), and define \( {G}_{B} \) correspondingly. By the minimality of \( X \) , there can be no \( A - X \) separator in \( {G}_{A} \) with fewer than \( \left| X\right| \) vertices, so \( {G}_{A} \) contains \( k \) disjoint \( A - X \) paths by induction. Similarly, \( {G}_{B} \) contains \( k \) disjoint \( X - B \) paths. Together, all these paths form the desired \( A - B \) paths in \( G \) .
14. Prove Menger’s theorem by induction on \( \parallel G\parallel \), as follows. Given an edge \( e = {xy} \), consider a smallest \( A - B \) separator \( S \) in \( G - e \) . Show that the induction hypothesis implies a solution for \( G \) unless \( S \cup \{ x\} \) and \( S \cup \{ y\} \) are smallest \( A - B \) separators in \( G \) . Then show that if choosing neither of these separators as \( X \) in the previous exercise gives a valid proof, there is only one easy case left to do.
15. Work out the details of the proof of Corollary 3.3.5 (ii).
16. Let \( k \geq 2 \) . Show that every \( k \) -connected graph of order at least \( {2k} \) contains a cycle of length at least \( {2k} \) .
17. Let \( k \geq 2 \) . Show that in a \( k \) -connected graph any \( k \) vertices lie on a common cycle.
18. Derive the edge part of Corollary 3.4.2 from the vertex part.
(Hint. Consider the \( H \) -paths in the graph obtained from the disjoint union of \( H \) and the line graph \( L\left( G\right) \) by adding all the edges he such that \( h \) is a vertex of \( H \) and \( e \in E\left( G\right) \smallsetminus E\left( H\right) \) is an edge at \( h \) .)
19. \( {}^{ - } \) To the disjoint union of the graph \( H = \overline{{K}^{{2m} + 1}} \) with \( k \) copies of \( {K}^{{2m} + 1} \) add edges joining \( H \) bijectively to each of the \( {K}^{{2m} + 1} \) . Show that the resulting graph \( G \) contains at most \( {km} = \frac{1}{2}{\kappa }_{G}\left( H\right) \) independent \( H \) - paths.
20. Find a bipartite graph \( G \), with partition classes \( A \) and \( B \) say, such that for \( H \mathrel{\text{:=}} G\left\lbrack A\right\rbrack \) there are at most \( \frac{1}{2}{\lambda }_{G}\left( H\right) \) edge-disjoint \( H \) -paths in \( G \) .
21. \( {}^{ + } \) Derive Tutte’s 1-factor theorem (2.2.1) from Mader’s theorem.
(Hint. Extend the given graph \( G \) to a graph \( {G}^{\prime } \) by adding, for each vertex \( v \in G \), a new vertex \( {v}^{\prime } \) and joining \( {v}^{\prime } \) to \( v \) . Choose \( H \subseteq {G}^{\prime } \) so that the 1-factors in \( G \) correspond to the large enough sets of independent \( H \) -paths in \( {G}^{\prime } \) .)
22. \( {}^{ - } \) Show that \( k \) -linked graphs are \( \left( {{2k} - 1}\right) \) -connected. Are they even \( {2k} \) - connected?
23. For every \( k \in \mathbb{N} \) find an \( \ell = \ell \left( k\right) \), as large as possible, such that not every \( \ell \) -connected graph is \( k \) -linked.
24. Show that if \( G \) is \( k \) -linked and \( {s}_{1},\ldots ,{s}_{k},{t}_{1},\ldots ,{t}_{k} \) are not necessarily distinct vertices such that \( {s}_{i} \neq {t}_{i} \) for all \( i \), then \( G \) contains independent paths \( {P}_{i} = {s}_{i}\ldots {t}_{i} \) for \( i = 1,\ldots, k \) .
25. Use Theorem 3.5.3 to show that the function \( h \) in Lemma 3.5.1 can be chosen as \( h\left( r\right) = c{r}^{2} \), for some \( c \in \mathbb{N} \) .
## Notes
Although connectivity theorems are doubtless among the most natural, and also the most applicable, results in graph theory, there is still no monograph on this subject. The most comprehensive source is perhaps A. Schrijver, Combinatorial optimization, Springer 2003, together with a number of surveys on specific topics by A. Frank, to be found on his home page. Some areas are covered in B. Bollobás, Extremal Graph Theory, Academic Press 1978, in R. Halin, Graphentheorie, Wissenschaftliche Buchgesellschaft 1980, and in A. Frank's chapter of the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995. A survey specifically of techniques and results on minimally \( k \) -connected graphs (see below) is given by W. Mader, On vertices of degree \( n \) in minimally \( n \) -connected graphs and digraphs, in (D. Miklós, V.T. Sós & T. Szönyi, eds.) Paul Erdős is 80, Vol. 2, Proc. Colloq. Math. Soc. János Bolyai, Budapest 1996.
Our proof of Tutte's Theorem 3.2.3 is due to C. Thomassen, Planarity and duality of finite and infinite graphs, J. Combin. Theory B 29 (1980), 244-271. This paper also contains Lemma 3.2.1 and its short proof from first principles. (The lemma's assertion, of course, follows from Tutte's wheel theorem-its significance lies in its independent proof, which has shortened the proofs of both of Tutte's theorems considerably.)
An approach to the study of connectivity |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 29 | ess 1978, in R. Halin, Graphentheorie, Wissenschaftliche Buchgesellschaft 1980, and in A. Frank's chapter of the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995. A survey specifically of techniques and results on minimally \( k \) -connected graphs (see below) is given by W. Mader, On vertices of degree \( n \) in minimally \( n \) -connected graphs and digraphs, in (D. Miklós, V.T. Sós & T. Szönyi, eds.) Paul Erdős is 80, Vol. 2, Proc. Colloq. Math. Soc. János Bolyai, Budapest 1996.
Our proof of Tutte's Theorem 3.2.3 is due to C. Thomassen, Planarity and duality of finite and infinite graphs, J. Combin. Theory B 29 (1980), 244-271. This paper also contains Lemma 3.2.1 and its short proof from first principles. (The lemma's assertion, of course, follows from Tutte's wheel theorem-its significance lies in its independent proof, which has shortened the proofs of both of Tutte's theorems considerably.)
An approach to the study of connectivity not touched upon in this chapter is the investigation of minimal \( k \) -connected graphs, those that lose their \( k \) -connectedness as soon as we delete an edge. Like all \( k \) -connected graphs, these have minimum degree at least \( k \), and by a fundamental result of Halin (1969), their minimum degree is exactly \( k \) . The existence of a vertex of small degree can be particularly useful in induction proofs about \( k \) -connected graphs. Halin's theorem was the starting point for a series of more and more sophisticated studies of minimal \( k \) -connected graphs; see the books of Bollobás and Halin cited above, and in particular Mader's survey.
Our first proof of Menger's theorem is extracted from Halin's book. The second is due to T. Böhme, F. Göring and J. Harant, Menger's theorem, J. Graph Theory 37 (2001), 35-36, the third to T. Grünwald (later Gallai), Ein neuer Beweis eines Mengerschen Satzes, J. London Math. Soc. 13 (1938), 188-192. The global version of Menger's theorem (Theorem 3.3.6) was first stated and proved by Whitney (1932).
Mader's Theorem 3.4.1 is taken from W. Mader, Über die Maximalzahl kreuzungsfreier \( H \) -Wege, Arch. Math. 31 (1978),387-402; a short proof has been given by A. Schrijver, A short proof of Mader’s \( \mathcal{S} \) -paths theorem, \( J \) . Com-bin. Theory B 82 (2001), 319-321. The theorem may be viewed as a common generalization of Menger's theorem and Tutte's 1-factor theorem (Exercise 21).
Theorem 3.5.3 is due to R. Thomas and P. Wollan, An improved linear bound for graph linkages, Europ. J. Combinatorics 26 (2005), 309-324. Using a more involved version of Lemma 3.5.4, they prove that \( {2k} \) -connected graphs even with only \( \varepsilon \geq {5k} \) must be \( k \) -linked. And for graphs of large enough girth the condition on \( \varepsilon \) can be dropped altogether: as shown by W. Mader, Topological subgraphs in graphs of large girth, Combinatorica 18 (1998), 405- 412, such graphs are \( k \) -linked as soon as they are \( {2k} \) -connected, which is best possible. (Mader assumes a lower bound on the girth that depends on \( k \), but this is not necessary; see D. Kühn & D. Osthus, Topological minors in graphs of large girth, J. Combin. Theory B 86 (2002), 364-380.) In fact, for every \( s \in \mathbb{N} \) there exists a \( {k}_{s} \) such that if \( G \nsupseteq {K}_{s, s} \) and \( \kappa \left( G\right) \geq {2k} \geq {k}_{s} \) then \( G \) is \( k \) -linked; see D. Kühn &D. Osthus, Complete minors in \( {K}_{s, s} \) -free graphs, Combinatorica 25 (2005) 49-64. 4
## Planar Graphs
When we draw a graph on a piece of paper, we naturally try to do this as transparently as possible. One obvious way to limit the mess created by all the lines is to avoid intersections. For example, we may ask if we can draw the graph in such a way that no two edges meet in a point other than a common end.
Graphs drawn in this way are called plane graphs; abstract graphs that can be drawn in this way are called planar. In this chapter we study both plane and planar graphs - as well as the relationship between the two: the question of how an abstract graph might be drawn in fundamentally different ways. After collecting together in Section 4.1 the few basic topological facts that will enable us later to prove all results rigorously without too much technical ado, we begin in Section 4.2 by studying the structural properties of plane graphs. In Section 4.3, we investigate how two drawings of the same graph can differ. The main result of that section is that 3-connected planar graphs have essentially only one drawing, in some very strong and natural topological sense. The next two sections are devoted to the proofs of all the classical planarity criteria, conditions telling us when an abstract graph is planar. We complete the chapter with a section on plane duality, a notion with fascinating links to algebraic, colouring, and flow properties of graphs (Chapters 1.9 and 6.5).
The traditional notion of a graph drawing is that its vertices are represented by points in the Euclidean plane, its edges are represented by curves between these points, and different curves meet only in common endpoints. To avoid unnecessary topological complication, however, we shall only consider curves that are piecewise linear; it is not difficult to show that any drawing can be straightened out in this way, so the two notions come to the same thing.
## 4.1 Topological prerequisites
In this section we briefly review some basic topological definitions and facts needed later. All these facts have (by now) easy and well-known proofs; see the notes for sources. Since those proofs contain no graph theory, we do not repeat them here: indeed our aim is to collect precisely those topological facts that we need but do not want to prove. Later, all proofs will follow strictly from the definitions and facts stated here (and be guided by but not rely on geometric intuition), so the material presented now will help to keep elementary topological arguments in those proofs to a minimum.
A straight line segment in the Euclidean plane is a subset of \( {\mathbb{R}}^{2} \) that has the form \( \{ p + \lambda \left( {q - p}\right) \mid 0 \leq \lambda \leq 1\} \) for distinct points \( p, q \in {\mathbb{R}}^{2} \) .
polygon A polygon is a subset of \( {\mathbb{R}}^{2} \) which is the union of finitely many straight line segments and is homeomorphic to the unit circle \( {S}^{1} \), the set of points in \( {\mathbb{R}}^{2} \) at distance 1 from the origin. Here, as later, any subset of a topological space is assumed to carry the subspace topology. A polygonal arc is a subset of \( {\mathbb{R}}^{2} \) which is the union of finitely many straight line segments and is homeomorphic to the closed unit interval \( \left\lbrack {0,1}\right\rbrack \) . The images of 0 and of 1 under such a homeomorphism are the endpoints of this polygonal arc, which links them and runs between them. Instead of arc ’polygonal arc’ we shall simply say arc in this chapter. If \( P \) is an arc between \( x \) and \( y \), we denote the point set \( P \smallsetminus \{ x, y\} \), the interior of \( P \) , by \( \overset{ \circ }{P} \) .
Let \( O \subseteq {\mathbb{R}}^{2} \) be an open set. Being linked by an arc in \( O \) defines an equivalence relation on \( O \) . The corresponding equivalence classes are
---
region separate frontier
---
again open; they are the regions of \( O \) . A closed set \( X \subseteq {\mathbb{R}}^{2} \) is said to separate \( O \) if \( O \smallsetminus X \) has more than one region. The frontier of a set \( X \subseteq {\mathbb{R}}^{2} \) is the set \( Y \) of all points \( y \in {\mathbb{R}}^{2} \) such that every neighbourhood of \( y \) meets both \( X \) and \( {\mathbb{R}}^{2} \smallsetminus X \) . Note that if \( X \) is open then its frontier lies in \( {\mathbb{R}}^{2} \smallsetminus X \) .
The frontier of a region \( O \) of \( {\mathbb{R}}^{2} \smallsetminus X \), where \( X \) is a finite union of points and arcs, has two important properties. The first is accessibility: if \( x \in X \) lies on the frontier of \( O \), then \( x \) can be linked to some point in \( O \) by a straight line segment whose interior lies wholly inside \( O \) . As a consequence, any two points on the frontier of \( O \) can be linked by an arc whose interior lies in \( O \) (why?). The second notable property of the frontier of \( O \) is that it separates \( O \) from the rest of \( {\mathbb{R}}^{2} \) . Indeed, if \( \varphi : \left\lbrack {0,1}\right\rbrack \rightarrow P \subseteq {\mathbb{R}}^{2} \) is continuous, with \( \varphi \left( 0\right) \in O \) and \( \varphi \left( 1\right) \notin O \), then \( P \) meets the frontier of \( O \) at least in the point \( \varphi \left( y\right) \) for \( y \mathrel{\text{:=}} \inf \{ x \mid \varphi \left( x\right) \notin O\} \), the first point of \( P \) in \( {\mathbb{R}}^{2} \smallsetminus O \) .
Theorem 4.1.1. (Jordan Curve Theorem for Polygons)
For every polygon \( P \subseteq {\mathbb{R}}^{2} \), the set \( {\mathbb{R}}^{2} \smallsetminus P \) has exactly two regions. Each of these has the entire polygon \( P \) as its frontier.
With the help of Theorem 4.1.1, it is not difficult to prove the following lemma.
Lemma 4.1.2. Let \( {P}_{1},{P}_{2},{P}_{3} \) be three arcs, between the same two endpoint but otherwise disjoint.
(i) \( {\mathbb{R}}^{2} \smallsetminus \left( {{P}_{1} \cup {P}_{2} \cup {P}_{3}}\right) \) has exactly three regions, with frontiers \( {P}_{1} \cup {P}_{2},{P}_{2} \cup {P}_{3} \) and \( {P}_{1} \cup {P}_{3} \) .
(ii) If \( P \) is an arc between a point in \( {P}_{1} \) and a point in \( {P}_{3} \) whose interior lies in the region of \( {\mathbb{R}}^{2} \smallsetminus \left( {{P}_{1} \cup {P}_{3}}\right) \) that contains \( {P}_{2} \), then \( \overset{ \circ }{P} \cap {\overset{ \circ }{P}}_{2} \neq \varnothing \) .
Fig. 4.1.1. The arcs in Lemma 4.1.2 (ii)
Our next lemma complements the Jordan curve theorem by saying that an arc does not separate the plane. For easier application later, we phrase this a little more generally:
Lemma 4.1.3. Let \( {X}_{1},{X}_{2} \subseteq {\mathbb{R}}^{2} \) be disjoint sets, each the union of finitely many points and arcs, and let \( P \) be an arc between a point in \( {X}_{1} \) and one in \( {X}_{2} \) whose interior \( \overset{ \circ }{P} \) lies in a region \( O \) of \( {\mathbb{R}}^{2} \smallsetminus \left( {{X}_{1} \cup {X}_{2}}\right) \) . Then \( O \smallsetminus P \) is a region of \( {\mathbb{R}}^{2} \smallsetminus \left( {{X}_{1} \cup P \cup {X}_{2}}\right) \) .
Fig. 4.1.2. \( P \) does not separate the region \( O \) of \( {\mathbb{R}}^{2} \smallsetminus \left( {{X}_{1} \cup {X}_{2}}\right) \)
It remains to introduce a few terms and facts that will be used only once, when we consider notions of equivalence for graph drawings in Chapter 4.3.
As usual, we denote by \( {S}^{n} \) the \( n \) -dimensional sphere, the set of points in \( {\mathbb{R}}^{n + 1} \) at distance 1 from the origin. The 2-sphere minus its ’north pole’ \( \left( {0,0,1}\right) \) is homeomorphic to the plane; let us choose a fixed such homeomorphism \( \pi : {S}^{2} \smallsetminus \{ \left( {0,0,1}\right) \} \rightarrow {\mathbb{R}}^{2} \) (for example, stereographic projection). If \( P \subseteq {\mathbb{R}}^{2} \) is a polygon and \( O \) is the bounded region of
\( {\mathbb{R}}^{2} \smallsetminus P \), let us call \( C \mathrel{\text{:=}} {\pi }^{-1}\left( P\right) \) a circle on \( {S}^{2} \), and the sets \( {\pi }^{-1}\left( O\right) \) and \( {S}^{2} \smallsetminus {\pi }^{-1}\left( {P \cup O}\right) \) the regions of \( C \) .
Our last tool is the theorem of Jordan and Schoenflies, again adapted slightly for our purposes:
\( \left\lbrack {4.3.1}\right\rbrack \)
Theorem 4.1.4. Let \( \varphi : {C}_{1} \rightarrow {C}_{2} \) be a homeomorphism between two circles on \( {S}^{2} \), let \( {O}_{1} \) be a region of \( {C}_{1} \), and let \( {O}_{2} \) be a region of \( {C}_{2} \) . Then \( \varphi \) can be extended to a homeomorphism \( {C}_{1} \cup {O}_{1} \rightarrow {C}_{2} \cup {O}_{2} \) .
## 4.2 Plane graphs
---
plane
graph
---
A plane graph is a pair \( \left( {V, E}\right) \) of finite sets with the following properties (the elements of \( V \) are again called vertices, those of \( E \) edges):
(i) \( V \subseteq {\mathbb{R}}^{2} \) ;
(ii) every edge is an arc between two vertices;
(iii) different edges have different sets of endpoints;
(iv) the interior of an edge contains no vertex and no point of any other edge.
A plane graph \( \left( {V, E}\right) \) defines a graph \( G \) on \( V \) in a natural way. As long as no confusion can arise, we shall use the name \( G \) of this abstract graph also for the plane graph \( \left( {V, E}\right) \), or for the point set \( V \cup \bigcup E \) ; similar notational conventions will be used for abstract versus plane edges, for subgraphs, and so on. \( {}^{1} \)
For every plane graph \( G \), the set \( {\mathbb{R}}^{2} \smallsetminus G \) is open; its regions are the faces faces of \( G \) . Since \( G \) is bounded-i.e., lies inside some sufficiently large disc \( D \) -exactly one of its faces is unbounded: the face that contains \( {\mathbb{R}}^{2} \smallsetminus D \) . This face is the outer face of \( G \) ; the other faces are its inner faces. We denote the set of faces of \( G \) by \( F\left( G\right) \) .
The faces of plane graphs and their subgraphs are related in the obvious way:
Lemma 4.2.1. Let \( G \) be a plane graph, \( f \in F\left( G\right) \) a face, and \( H \subseteq G \) a subgraph.
(i) \( H \) has a face \( {f}^{\prime } \) containing \( f \) .
(ii) If the frontier of \( f \) lies in \( H \), then \( {f}^{\prime } = f \) .
---
\( {}^{1} \) However, we shall continue to use \( \smallsetminus \) for differences of point sets and - for graph differences-which may help a little to keep the two apart.
---
Proof. (i) Clearly, the points in \( f \) are equivalent also in \( {\mathbb{R}}^{2} \smallsetminus H \) ; let \( {f}^{\prime } \) be the equivalence class of \( {\mathbb{R}}^{2} \smallsetminus H \) containing them.
(ii) Recall from Section 4.1 that any arc between \( f \) and \( {f}^{\prime } \smallsetminus f \) meets the frontier \( X \) of \( f \) . If \( {f}^{\prime } \smallsetminus f \neq \varnothing \) then there is such an arc inside \( {f}^{\prime } \) , whose points in \( X \) do not lie in \( H \) . Hence \( X \nsubseteq H \) .
In order to lay the foundations for the (easy but) rigorous introduction to plane graphs that this section aims to provide, let us descend once now into the realm of truly elementary topology of the plane, and prove what seems entirely obvious: \( {}^{2} \) that the frontier of a face of a plane graph \( G \) is always a subgraph of \( G \) -not, say, half an edge.
The following lemma states this formally, together with two similarly 'obvious' properties of plane graphs:
Lemma 4.2.2. Let \( G \) be a plane graph and \( e \) an edge of \( G \) . \( \begin{array}{r} \left\lbrack {4.5.1}\right\rbrack \\ \left\lbrack {4.5.2}\right\rbrack \\ \left\lbrack {12.5.4}\right\rbrack \end{array} \) (i) If \( X \) is the frontier of a face of \( G \), then either \( e \subseteq X \) or \( X \cap \overset{ \circ }{e} = \varnothing \) .
(ii) If \( e \) lies on a cycle \( C \subseteq G \), then \( e \) lies on the frontier of exactly two faces of \( G \), and these are contained in distinct faces of \( C \) .
(iii) If \( e \) lies on no cycle, then \( e \) lies on the frontier of exactly one face of \( G \) .
Proof. We prove all three assertions together. Let us start by considering
(4.1.1)
(4.1.3)
one point \( {x}_{0} \in e \) . We show that \( {x}_{0} \) lies on the frontier of either exactly two faces or exactly one, according as \( e \) lies on a cycle in \( G \) or not. We then show that every other point in \( e \) lies on the frontier of exactly the same faces as \( {x}_{0} \) . Then the endpoints of \( e \) will also lie on the frontier of these faces - simply because every neighbourhood of an endpoint of \( e \) is also the neighbourhood of an inner point of \( e \) .
\( G \) is the union of finitely many straight line segments; we may assume that any two of these intersect in at most one point. Around every point \( x \in \overset{ \circ }{e} \) we can find an open disc \( {D}_{x} \), with centre \( x \), which meets \( {D}_{x} \) only those (one or two) straight line segments that contain \( x \) .
Let us pick an inner point \( {x}_{0} \) from a straight line segment \( S \subseteq e \) . \( {x}_{0}, S \) Then \( {D}_{{x}_{0}} \cap G = {D}_{{x}_{0}} \cap S \), so \( {D}_{{x}_{0}} \smallsetminus G \) is the union of two open half-discs. Since these half-discs do not meet \( G \), they each lie in a face of \( G \) . Let us denote these faces by \( {f}_{1} \) and \( {f}_{2} \) ; they are the only faces of \( G \) with \( {x}_{0} \) \( {f}_{1},{f}_{2} \) on their frontier, and they may coincide (Fig. 4.2.1).
If \( e \) lies on a cycle \( C \subseteq G \), then \( {D}_{{x}_{0}} \) meets both faces of \( C \) (Theorem 4.1.1). Since \( {f}_{1} \) and \( {f}_{2} \) are contained in faces of \( C \) by Lemma 4.2.1, this implies \( {f}_{1} \neq {f}_{2} \) . If \( e \) does not lie on any cycle, then \( e \) is a bridge
2 Note that even the best intuition can only ever be 'accurate', i.e., coincide with what the technical definitions imply, inasmuch as those definitions do indeed formalize what is intuitively intended. Given the complexity of definitions in elementary topology, this can hardly be taken for granted.
Fig. 4.2.1. Faces \( {f}_{1},{f}_{2} \) of \( G \) in the proof of Lemma 4.2.2
and thus links two disjoint point sets \( {X}_{1},{X}_{2} \) as in Lemma 4.1.3, with \( {X}_{1} \cup {X}_{2} = G \smallsetminus \overset{ \circ }{e} \) . Clearly, \( {f}_{1} \cup \overset{ \circ }{e} \cup {f}_{2} \) is the subset of a face \( f \) of \( G - e \) . By Lemma 4.1.3, \( f \smallsetminus \overset{ \circ }{e} \) is a face of \( G \) . But \( f \smallsetminus \overset{ \circ }{e} \) contains \( {f}_{1} \) and \( {f}_{2} \) by definition of \( f \), so \( {f}_{1} = f \smallsetminus \mathring{e} = {f}_{2} \) since \( {f}_{1},{f}_{2} \) and \( f \) are all faces of \( G \) .
\( {x}_{1} \) Now consider any other point \( {x}_{1} \in \overset{ \circ }{e} \) . Let \( P \) be the arc from \( {x}_{0} \) to \( P \) \( {x}_{1} \) contained in \( e \) . Since \( P \) is compact, finitely many of the dis |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 31 |
Fig. 4.2.1. Faces \( {f}_{1},{f}_{2} \) of \( G \) in the proof of Lemma 4.2.2
and thus links two disjoint point sets \( {X}_{1},{X}_{2} \) as in Lemma 4.1.3, with \( {X}_{1} \cup {X}_{2} = G \smallsetminus \overset{ \circ }{e} \) . Clearly, \( {f}_{1} \cup \overset{ \circ }{e} \cup {f}_{2} \) is the subset of a face \( f \) of \( G - e \) . By Lemma 4.1.3, \( f \smallsetminus \overset{ \circ }{e} \) is a face of \( G \) . But \( f \smallsetminus \overset{ \circ }{e} \) contains \( {f}_{1} \) and \( {f}_{2} \) by definition of \( f \), so \( {f}_{1} = f \smallsetminus \mathring{e} = {f}_{2} \) since \( {f}_{1},{f}_{2} \) and \( f \) are all faces of \( G \) .
\( {x}_{1} \) Now consider any other point \( {x}_{1} \in \overset{ \circ }{e} \) . Let \( P \) be the arc from \( {x}_{0} \) to \( P \) \( {x}_{1} \) contained in \( e \) . Since \( P \) is compact, finitely many of the discs \( {D}_{x} \) \( {D}_{0},\ldots ,{D}_{n} \) with \( x \in P \) cover \( P \) . Let us enumerate these discs as \( {D}_{0},\ldots ,{D}_{n} \) in the natural order of their centres along \( P \) ; adding \( {D}_{{x}_{0}} \) or \( {D}_{{x}_{1}} \) as necessary, we may assume that \( {D}_{0} = {D}_{{x}_{0}} \) and \( {D}_{n} = {D}_{{x}_{1}} \) . By induction on \( n \), one \( y \) easily proves that every point \( y \in {D}_{n} \smallsetminus e \) can be linked by an arc inside \( z \) \( \left( {{D}_{0} \cup \ldots \cup {D}_{n}}\right) \smallsetminus e \) to a point \( z \in {D}_{0} \smallsetminus e \) (Fig. 4.2.2); then \( y \) and \( z \) are equivalent in \( {\mathbb{R}}^{2} \smallsetminus G \) . Hence, every point of \( {D}_{n} \smallsetminus e \) lies in \( {f}_{1} \) or in \( {f}_{2} \), so \( {x}_{1} \) cannot lie on the frontier of any other face of \( G \) . Since both half-discs of \( {D}_{0} \smallsetminus e \) can be linked to \( {D}_{n} \smallsetminus e \) in this way (swap the roles of \( {D}_{0} \) and \( {D}_{n} \) ), we find that \( {x}_{1} \) lies on the frontier of both \( {f}_{1} \) and \( {f}_{2} \) .
Fig. 4.2.2. An arc from \( y \) to \( {D}_{0} \), close to \( P \)
Corollary 4.2.3. The frontier of a face is always the point set of a subgraph.
The subgraph of \( G \) whose point set is the frontier of a face \( f \) is said
---
boundary \( G\left\lbrack f\right\rbrack \)
---
to bound \( f \) and is called its boundary; we denote it by \( G\left\lbrack f\right\rbrack \) . A face is said to be incident with the vertices and edges of its boundary. By Lemma 4.2.1 (ii), every face of \( G \) is also a face of its boundary; we shall use this fact frequently in the proofs to come.
\( \left\lbrack {4.6.1}\right\rbrack \)
Proposition 4.2.4. A plane forest has exactly one face.
(4.1.3)
Proof. Use induction on the number of edges and Lemma 4.1.3.
With just one exception, different faces of a plane graph have different boundaries:
\( \left\lbrack {4.3.1}\right\rbrack \)
Lemma 4.2.5. If a plane graph has different faces with the same boundary, then the graph is a cycle.
Proof. Let \( G \) be a plane graph, and let \( H \subseteq G \) be the boundary of
(4.1.1)
distinct faces \( {f}_{1},{f}_{2} \) of \( G \) . Since \( {f}_{1} \) and \( {f}_{2} \) are also faces of \( H \), Proposition 4.2.4 implies that \( H \) contains a cycle \( C \) . By Lemma 4.2.2 (ii), \( {f}_{1} \) and \( {f}_{2} \) are contained in different faces of \( C \) . Since \( {f}_{1} \) and \( {f}_{2} \) both have all of \( H \) as boundary, this implies that \( H = C \) : any further vertex or edge of \( H \) would lie in one of the faces of \( C \) and hence not on the boundary of the other. Thus, \( {f}_{1} \) and \( {f}_{2} \) are distinct faces of \( C \) . As \( C \) has only two faces, it follows that \( {f}_{1} \cup C \cup {f}_{2} = {\mathbb{R}}^{2} \) and hence \( G = C \) .
## Proposition 4.2.6. In a 2-connected plane graph, every face is bounded by a cycle.
\( \begin{array}{l} \left\lbrack {4.3.1}\right\rbrack \\ \left\lbrack {4.4.3}\right\rbrack \\ \left\lbrack {4.5.1}\right\rbrack \\ \left\lbrack {4.5.2}\right\rbrack \end{array} \)
Proof. Let \( f \) be a face in a 2-connected plane graph \( G \) . We show by
(3.1.3)
induction on \( \left| G\right| \) that \( G\left\lbrack f\right\rbrack \) is a cycle. If \( G \) is itself a cycle, this holds (4.1.1) \( \left( {4.1.2}\right) \) by Theorem 4.1.1; we therefore assume that \( G \) is not a cycle.
By Proposition 3.1.3, there exist a 2-connected plane graph \( H \subseteq G \) \( H \) and a plane \( H \) -path \( P \) such that \( G = H \cup P \) . The interior of \( P \) lies in a face \( {f}^{\prime } \) of \( H \), which by the induction hypothesis is bounded by a cycle \( C \) . \( {f}^{\prime }, C \)
If \( f \) is also a face of \( H \), we are home by the induction hypothesis. If not, then the frontier of \( f \) meets \( P \smallsetminus H \), so \( f \subseteq {f}^{\prime } \) and \( G\left\lbrack f\right\rbrack \subseteq C \cup P \) . By Lemma 4.2.1 (ii), then, \( f \) is a face of \( C \cup P \) and hence bounded by a cycle (Lemma 4.1.2 (i)).
In a 3-connected graph, we can identify the face boundaries among the other cycles in purely combinatorial terms:
Proposition 4.2.7. The face boundaries in a 3-connected plane graph are precisely its non-separating induced cycles.
\( \left( {3.3.6}\right) \)
Proof. Let \( G \) be a 3-connected plane graph, and let \( C \subseteq G \) . If \( C \) is a (4.1.1) non-separating induced cycle, then by the Jordan curve theorem its two
(4.1.2)
faces cannot both contain points of \( G \smallsetminus C \) . Therefore it bounds a face of \( G \) .
Conversely, suppose that \( C \) bounds a face \( f \) . By Proposition 4.2.6, \( C \) is a cycle. If \( C \) has a chord \( e = {xy} \), then the components of \( C - \{ x, y\} \) are linked by a \( C \) -path in \( G \), because \( G \) is 3-connected. This path and \( e \) both run through the other face of \( C \) (not \( f \) ) but do not intersect, a contradiction to Lemma 4.1.2 (ii).
It remains to show that \( C \) does not separate any two vertices \( x, y \in \) \( G - C \) . By Menger’s theorem (3.3.6), \( x \) and \( y \) are linked in \( G \) by three independent paths. Clearly, \( f \) lies inside a face of their union, and by Lemma 4.1.2 (i) this face is bounded by only two of the paths. The third therefore avoids \( f \) and its boundary \( C \) .
maximal A plane graph \( G \) is called maximally plane, or just maximal, if we plane graph cannot add a new edge to form a plane graph \( {G}^{\prime } \supsetneqq G \) with \( V\left( {G}^{\prime }\right) = V\left( G\right) \) .
plane We call \( G \) a plane triangulation if every face of \( G \) (including the outer
triangulation face) is bounded by a triangle.
\( \left\lbrack \begin{array}{l} {4.4.1} \\ {5.4.2} \end{array}\right\rbrack \) Proposition 4.2.8. A plane graph of order at least 3 is maximally plane if and only if it is a plane triangulation.
\( \left( {4.1.2}\right) \)
Proof. Let \( G \) be a plane graph of order at least 3 . It is easy to see that if every face of \( G \) is bounded by a triangle, then \( G \) is maximally plane. Indeed, any additional edge \( e \) would have its interior inside a face of \( G \) and its ends on the boundary of that face. Hence these ends are already adjacent in \( G \), so \( G \cup e \) cannot satisfy condition (iii) in the definition of a plane graph.
\( f \) Conversely, assume that \( G \) is maximally plane and let \( f \in F\left( G\right) \) be \( H \) a face; let us write \( H \mathrel{\text{:=}} G\left\lbrack f\right\rbrack \) . Since \( G \) is maximal as a plane graph, \( G\left\lbrack H\right\rbrack \) is complete: any two vertices of \( H \) that are not already adjacent in \( G \) could be linked by an arc through \( f \), extending \( G \) to a larger plane \( n \) graph. Thus \( G\left\lbrack H\right\rbrack = {K}^{n} \) for some \( n \) -but we do not know yet which edges of \( G\left\lbrack H\right\rbrack \) lie in \( H \) .
Let us show first that \( H \) contains a cycle. If not, then \( G \smallsetminus H \neq \varnothing \) : by \( G \supseteq {K}^{n} \) if \( n \geq 3 \), or else by \( \left| G\right| \geq 3 \) . On the other hand we have \( f \cup H = {\mathbb{R}}^{2} \) by Proposition 4.2.4 and hence \( G = H \), a contradiction.
Since \( H \) contains a cycle, it suffices to show that \( n \leq 3 \) : then \( H = {K}^{3} \)
\( C,{v}_{i} \)
as claimed. Suppose \( n \geq 4 \), and let \( C = {v}_{1}{v}_{2}{v}_{3}{v}_{4}{v}_{1} \) be a cycle in \( G\left\lbrack H\right\rbrack \) \( \left( { = {K}^{n}}\right) \) . By \( C \subseteq G \), our face \( f \) is contained in a face \( {f}_{C} \) of \( C \) ; let \( {f}_{C}^{\prime } \) \( {f}_{C},{f}_{C}^{\prime } \) be the other face of \( C \) . Since the vertices \( {v}_{1} \) and \( {v}_{3} \) lie on the boundary of \( f \), they can be linked by an arc whose interior lies in \( {f}_{C} \) and avoids \( G \) . Hence by Lemma 4.1.2 (ii), the plane edge \( {v}_{2}{v}_{4} \) of \( G\left\lbrack H\right\rbrack \) runs through \( {f}_{C}^{\prime } \) rather than \( {f}_{C} \) (Fig. 4.2.3). Analogously, since \( {v}_{2},{v}_{4} \in G\left\lbrack f\right\rbrack \), the edge \( {v}_{1}{v}_{3} \) runs through \( {f}_{C}^{\prime } \) . But the edges \( {v}_{1}{v}_{3} \) and \( {v}_{2}{v}_{4} \) are disjoint, so this contradicts Lemma 4.1.2 (ii).
Fig. 4.2.3. The edge \( {v}_{2}{v}_{4} \) of \( G \) runs through the face \( {f}_{C}^{\prime } \)
The following classic result of Euler (1752)-here stated in its simplest form, for the plane-marks one of the common origins of graph theory and topology. T |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 32 | ) be the other face of \( C \) . Since the vertices \( {v}_{1} \) and \( {v}_{3} \) lie on the boundary of \( f \), they can be linked by an arc whose interior lies in \( {f}_{C} \) and avoids \( G \) . Hence by Lemma 4.1.2 (ii), the plane edge \( {v}_{2}{v}_{4} \) of \( G\left\lbrack H\right\rbrack \) runs through \( {f}_{C}^{\prime } \) rather than \( {f}_{C} \) (Fig. 4.2.3). Analogously, since \( {v}_{2},{v}_{4} \in G\left\lbrack f\right\rbrack \), the edge \( {v}_{1}{v}_{3} \) runs through \( {f}_{C}^{\prime } \) . But the edges \( {v}_{1}{v}_{3} \) and \( {v}_{2}{v}_{4} \) are disjoint, so this contradicts Lemma 4.1.2 (ii).
Fig. 4.2.3. The edge \( {v}_{2}{v}_{4} \) of \( G \) runs through the face \( {f}_{C}^{\prime } \)
The following classic result of Euler (1752)-here stated in its simplest form, for the plane-marks one of the common origins of graph theory and topology. The theorem relates the number of vertices, edges and faces in a plane graph: taken with the correct signs, these numbers always add up to 2 . The general form of Euler's theorem asserts the same for graphs suitably embedded in other surfaces, too: the sum obtained is always a fixed number depending only on the surface, not on the graph, and this number differs for distinct (orientable closed) surfaces. Hence, any two such surfaces can be distinguished by a simple arithmetic invariant of the graphs embedded in them! \( {}^{3} \)
Let us then prove Euler's theorem in its simplest form:
Theorem 4.2.9. (Euler's Formula)
Let \( G \) be a connected plane graph with \( n \) vertices, \( m \) edges, and \( \ell \) faces. Then
\[
n - m + \ell = 2\text{.}
\]
Proof. We fix \( n \) and apply induction on \( m \) . For \( m \leq n - 1, G \) is a tree
(1.5.1)
\( \left( {1.5.3}\right) \) and \( m = n - 1 \) (why?), so the assertion follows from Proposition 4.2.4.
Now let \( m \geq n \) . Then \( G \) has an edge \( e \) that lies on a cycle; let \( {G}^{\prime } \mathrel{\text{:=}} G - e \) . By Lemma 4.2.2 (ii), \( e \) lies on the boundary of exactly two faces \( {f}_{1},{f}_{2} \) of \( G \), and as the points in \( \overset{ \circ }{e} \) are all equivalent in \( {\mathbb{R}}^{2} \smallsetminus {G}^{\prime } \) , \( {f}_{1},{f}_{2} \) there is a face \( {f}_{e} \) of \( {G}^{\prime } \) containing \( \mathring{e} \) . We show that \( {f}_{e} \)
\[
F\left( G\right) \smallsetminus \left\{ {{f}_{1},{f}_{2}}\right\} = F\left( {G}^{\prime }\right) \smallsetminus \left\{ {f}_{e}\right\}
\]
\( \left( *\right) \)
then \( {G}^{\prime } \) has exactly one face and one edge less than \( G \), and so the assertion follows from the induction hypothesis for \( {G}^{\prime } \) .
For a proof of \( \left( *\right) \) let first \( f \in F\left( G\right) \smallsetminus \left\{ {{f}_{1},{f}_{2}}\right\} \) be given. By Lemma 4.2.2 (i) we have \( G\left\lbrack f\right\rbrack \subseteq G \smallsetminus \overset{ \circ }{e} = {G}^{\prime } \), and hence \( f \in F\left( {G}^{\prime }\right) \) by Lemma 4.2.1 (ii). As clearly \( f \neq {f}_{e} \), this establishes the forwad inclusion in (*).
Conversely, consider any face \( {f}^{\prime } \in F\left( {G}^{\prime }\right) \smallsetminus \left\{ {f}_{e}\right\} \) . Clearly \( {f}^{\prime } \neq {f}_{1},{f}_{2} \) , and \( {f}^{\prime } \cap \mathring{e} = \varnothing \) . Hence every two points of \( {f}^{\prime } \) lie in \( {\mathbb{R}}^{2} \smallsetminus G \) and are equivalent there, so \( G \) has a face \( f \) containing \( {f}^{\prime } \) . By Lemma 4.2.1 (i), however, \( f \) lies inside a face \( {f}^{\prime \prime } \) of \( {G}^{\prime } \) . Thus \( {f}^{\prime } \subseteq f \subseteq {f}^{\prime \prime } \) and hence \( {f}^{\prime } = f = {f}^{\prime \prime } \), since both \( {f}^{\prime } \) and \( {f}^{\prime \prime } \) are faces of \( {G}^{\prime } \) .
Corollary 4.2.10. A plane graph with \( n \geq 3 \) vertices has at most \( {3n} - 6 \) edges. Every plane triangulation with \( n \) vertices has \( {3n} - 6 \) edges.
Proof. By Proposition 4.2.8 it suffices to prove the second assertion. In a plane triangulation \( G \), every face boundary contains exactly three edges, and every edge lies on the boundary of exactly two faces (Lemma 4.2.2). The bipartite graph on \( E\left( G\right) \cup F\left( G\right) \) with edge set \( \{ {ef} \mid e \subseteq G\left\lbrack f\right\rbrack \} \) thus has exactly \( 2\left| {E\left( G\right) }\right| = 3\left| {F\left( G\right) }\right| \) edges. According to this identity we may replace \( \ell \) with \( {2m}/3 \) in Euler’s formula, and obtain \( m = {3n} - 6 \) .
3 This fundamental connection between graphs and surfaces lies at the heart of the proof of the famous Robertson-Seymour graph minor theorem; see Chapter 12.5.
Euler's formula can be useful for showing that certain graphs cannot occur as plane graphs. The graph \( {K}^{5} \), for example, has \( {10} > 3 \cdot 5 - 6 \) edges, more than allowed by Corollary 4.2.10. Similarly, \( {K}_{3,3} \) cannot be a plane graph. For since \( {K}_{3,3} \) is 2-connected but contains no triangle, every face of a plane \( {K}_{3,3} \) would be bounded by a cycle of length \( \geq 4 \) (Proposition 4.2.6). As in the proof of Corollary 4.2.10 this implies \( {2m} \geq 4\ell \), which yields \( m \leq {2n} - 4 \) when substituted in Euler’s formula. But \( {K}_{3,3} \) has \( 9 > 2 \cdot 6 - 4 \) edges.
Clearly, along with \( {K}^{5} \) and \( {K}_{3,3} \) themselves, their subdivisions cannot occur as plane graphs either:
\( \left\lbrack \begin{array}{l} {4.4.5} \\ {4.4.6} \end{array}\right\rbrack \) Corollary 4.2.11. A plane graph contains neither \( {K}^{5} \) nor \( {K}_{3,3} \) as a topological minor.
Surprisingly, it turns out that this simple property of plane graphs identifies them among all other graphs: as Section 4.4 will show, an arbitrary graph can be drawn in the plane if and only if it has no (topological) \( {K}^{5} \) or \( {K}_{3,3} \) minor.
## 4.3 Drawings
---
planar embedding
---
An embedding in the plane, or planar embedding, of an (abstract) graph \( G \) is an isomorphism between \( G \) and a plane graph \( H \) . The latter will be
drawing called a drawing of \( G \) . We shall not always distinguish notationally between the vertices and edges of \( G \) and of \( H \) . In this section we investigate how two planar embeddings of a graph can differ.
How should we measure the likeness of two embeddings \( \rho : G \rightarrow H \) and \( {\rho }^{\prime } : G \rightarrow {H}^{\prime } \) of a planar graph \( G \) ? An obvious way to do this is to consider the canonical isomorphism \( \sigma \mathrel{\text{:=}} {\rho }^{\prime } \circ {\rho }^{-1} \) between \( H \) and \( {H}^{\prime } \) as abstract graphs, and ask how much of their position in the plane this isomorphism respects or preserves. For example, if \( \sigma \) is induced by a simple rotation of the plane, we would hardly consider \( \rho \) and \( {\rho }^{\prime } \) as genuinely different ways of drawing \( G \) .
So let us begin by considering any abstract isomorphism \( \sigma : V \rightarrow {V}^{\prime } \)
\( H;V, E, F \) between two plane graphs \( H = \left( {V, E}\right) \) and \( {H}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \), with face
\( {H}^{\prime };{V}^{\prime },{E}^{\prime },{F}^{\prime } \) sets \( F\left( H\right) = : F \) and \( F\left( {H}^{\prime }\right) = : {F}^{\prime } \) say, and try to measure to what degree \( \sigma \) respects or preserves the features of \( H \) and \( {H}^{\prime } \) as plane graphs. In what follows we shall propose three criteria for this in decreasing order of strictness (and increasing order of ease of handling), and then prove that for most graphs these three criteria turn out to agree. In particular, applied to the isomorphism \( \sigma = {\rho }^{\prime } \circ {\rho }^{-1} \) considered earlier, all three criteria will say that there is essentially only one way to draw a 3-connected graph.
Our first criterion for measuring how well our abstract isomorphism \( \sigma \) preserves the plane features of \( H \) and \( {H}^{\prime } \) is perhaps the most natural one. Intuitively, we would like to call \( \sigma \) ’topological’ if it is induced by a homeomorphism from the plane \( {\mathbb{R}}^{2} \) to itself. To avoid having to grant the outer faces of \( H \) and \( {H}^{\prime } \) a special status, however, we take a detour via the homeomorphism \( \pi : {S}^{2} \smallsetminus \{ \left( {0,0,1}\right) \} \rightarrow {\mathbb{R}}^{2} \) chosen in Section 4.1: we call \( \sigma \) a topological isomorphism between the plane graphs \( H \) and \( {H}^{\prime } \)
---
topological isomorphism
---
if there exists a homeomorphism \( \varphi : {S}^{2} \rightarrow {S}^{2} \) such that \( \psi \mathrel{\text{:=}} \pi \circ \varphi \circ {\pi }^{-1} \) induces \( \sigma \) on \( V \cup E \) . (More formally: we ask that \( \psi \) agree with \( \sigma \) on \( V \) , and that it map every plane edge \( {xy} \in H \) onto the plane edge \( \sigma \left( x\right) \sigma \left( y\right) \in \) \( {H}^{\prime } \) . Unless \( \varphi \) fixes the point \( \left( {0,0,1}\right) \), the map \( \psi \) will be undefined at \( \pi \left( {{\varphi }^{-1}\left( {0,0,1}\right) }\right) \) .)
Fig. 4.3.1. Two drawings of a graph that are not topologically isomorphic-why not?
It can be shown that, up to topological isomorphism, inner and outer faces are indeed no longer different: if we choose as \( \varphi \) a rotation of \( {S}^{2} \) mapping the \( {\pi }^{-1} \) -image of a point of some inner face of \( H \) to the north pole \( \left( {0,0,1}\right) \) of \( {S}^{2} \), then \( \psi \) maps the rest of this face to the outer f |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 33 | -1} \) induces \( \sigma \) on \( V \cup E \) . (More formally: we ask that \( \psi \) agree with \( \sigma \) on \( V \) , and that it map every plane edge \( {xy} \in H \) onto the plane edge \( \sigma \left( x\right) \sigma \left( y\right) \in \) \( {H}^{\prime } \) . Unless \( \varphi \) fixes the point \( \left( {0,0,1}\right) \), the map \( \psi \) will be undefined at \( \pi \left( {{\varphi }^{-1}\left( {0,0,1}\right) }\right) \) .)
Fig. 4.3.1. Two drawings of a graph that are not topologically isomorphic-why not?
It can be shown that, up to topological isomorphism, inner and outer faces are indeed no longer different: if we choose as \( \varphi \) a rotation of \( {S}^{2} \) mapping the \( {\pi }^{-1} \) -image of a point of some inner face of \( H \) to the north pole \( \left( {0,0,1}\right) \) of \( {S}^{2} \), then \( \psi \) maps the rest of this face to the outer face of \( \psi \left( H\right) \) . (To ensure that the edges of \( \psi \left( H\right) \) are again piecewise linear, however, one may have to adjust \( \varphi \) a little.)
If \( \sigma \) is a topological isomorphism as above, then-except possibly for a pair of missing points where \( \psi \) or \( {\psi }^{-1} \) is undefined \( - \psi \) maps the faces of \( H \) onto those of \( {H}^{\prime } \) (proof?). In this way, \( \sigma \) extends naturally to a bijection \( \sigma : V \cup E \cup F \rightarrow {V}^{\prime } \cup {E}^{\prime } \cup {F}^{\prime } \) which preserves incidence of vertices, edges and faces.
Let us single out this last property of a topological isomorphism as the second criterion for how well an abstract isomorphism between plane graphs respects their position in the plane: let us call \( \sigma \) a combinatorial
---
combinatorial isomorphism
---
isomorphism of the plane graphs \( H \) and \( {H}^{\prime } \) if it can be extended to a bijection \( \sigma : V \cup E \cup F \rightarrow {V}^{\prime } \cup {E}^{\prime } \cup {F}^{\prime } \) that preserves incidence not only of vertices with edges but also of vertices and edges with faces. (Formally: we require that a vertex or edge \( x \in H \) shall lie on the boundary of a face \( f \in F \) if and only if \( \sigma \left( x\right) \) lies on the boundary of the face \( \sigma \left( f\right) \) .)
If \( \sigma \) is a combinatorial isomorphism of the plane graphs \( H \) and \( {H}^{\prime } \), it maps the face boundaries of \( H \) to those of \( {H}^{\prime } \) . Let us pick out this prop-
---
graph-theoretical isomorphism
---
erty as our third criterion, and call \( \sigma \) a graph-theoretical isomorphism of
Fig. 4.3.2. Two drawings of a graph that are combinatorially isomorphic but not topologically-why not?
the plane graphs \( H \) and \( {H}^{\prime } \) if
\[
\{ \sigma \left( {H\left\lbrack f\right\rbrack }\right) : f \in F\} = \left\{ {{H}^{\prime }\left\lbrack {f}^{\prime }\right\rbrack : {f}^{\prime } \in {F}^{\prime }}\right\} .
\]
Thus, we no longer keep track of which face is bounded by a given subgraph: the only information we keep is whether a subgraph bounds some face or not, and we require that \( \sigma \) map the subgraphs that do onto each other. At first glance, this third criterion may appear a little less natural than the previous two. However, it has the practical advantage of being formally weaker and hence easier to verify, and moreover, it will turn out to be equivalent to the other two in most cases.
As we have seen, every topological isomorphism between two plane graphs is also combinatorial, and every combinatorial isomorphism is also graph-theoretical. The following theorem shows that, for most graphs, the converse is true as well:
## Theorem 4.3.1.
(i) Every graph-theoretical isomorphism between two plane graphs is combinatorial. Its extension to a face bijection is unique if and only if the graph is not a cycle.
(ii) Every combinatorial isomorphism between two 2-connected plane graphs is topological.
Proof. Let \( H = \left( {V, E}\right) \) and \( {H}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \) be two plane graphs, put \( F\left( H\right) = : F \) and \( F\left( {H}^{\prime }\right) = : {F}^{\prime } \), and let \( \sigma : V \rightarrow {V}^{\prime } \) be an isomorphism between the underlying abstract graphs. Extend \( \sigma \) to a map \( V \cup E \rightarrow {V}^{\prime } \cup {E}^{\prime } \) by letting \( \sigma \left( {xy}\right) \mathrel{\text{:=}} \sigma \left( x\right) \sigma \left( y\right) \) .
(i) If \( H \) is a cycle, the assertion follows from the Jordan curve theorem. We now assume that \( H \) is not a cycle. Let \( \mathcal{B} \) and \( {\mathcal{B}}^{\prime } \) be the sets of all face boundaries in \( H \) and \( {H}^{\prime } \), respectively. If \( \sigma \) is a graph-theoretical isomorphism, then the map \( B \mapsto \sigma \left( B\right) \) is a bijection between \( \mathcal{B} \) and \( {\mathcal{B}}^{\prime } \) . By Lemma 4.2.5, the map \( f \mapsto H\left\lbrack f\right\rbrack \) is a bijection between \( F \) and \( \mathcal{B} \) , and likewise for \( {F}^{\prime } \) and \( {\mathcal{B}}^{\prime } \) . The composition of these three bijections is a bijection between \( F \) and \( {F}^{\prime } \), which we choose as \( \sigma : F \rightarrow {F}^{\prime } \) . By construction, this extension of \( \sigma \) to \( V \cup E \cup F \) preserves incidences (and is unique with this property), so \( \sigma \) is indeed a combinatorial isomorphism.
(ii) Let us assume that \( H \) is 2-connected, and that \( \sigma \) is a combinatorial isomorphism. We have to construct a homeomorphism \( \varphi : {S}^{2} \rightarrow {S}^{2} \) which, for every vertex or plane edge \( x \in H \), maps \( {\pi }^{-1}\left( x\right) \) to \( {\pi }^{-1}\left( {\sigma \left( x\right) }\right) \) . Since \( \sigma \) is a combinatorial isomorphism, \( \widetilde{\sigma } : {\pi }^{-1} \circ \sigma \circ \pi \) is an incidence preserving bijection from the vertices, edges and faces \( {}^{4} \) of \( \widetilde{H} \mathrel{\text{:=}} {\pi }^{-1}\left( H\right) \) to the vertices, edges and faces of \( {\widetilde{H}}^{\prime } \mathrel{\text{:=}} {\pi }^{-1}\left( {H}^{\prime }\right) \) . \( \widetilde{H},{\widetilde{H}}^{\prime } \) 
Fig. 4.3.3. Defining \( \widetilde{\sigma } \) via \( \sigma \)
We construct \( \varphi \) in three steps. Let us first define \( \varphi \) on the vertex set of \( \widetilde{H} \), setting \( \varphi \left( x\right) \mathrel{\text{:=}} \widetilde{\sigma }\left( x\right) \) for all \( x \in V\left( \widetilde{H}\right) \) . This is trivially a homeomorphism between \( V\left( \widetilde{H}\right) \) and \( V\left( {\widetilde{H}}^{\prime }\right) \) .
As the second step, we now extend \( \varphi \) to a homeomorphism between \( \widetilde{H} \) and \( {\widetilde{H}}^{\prime } \) that induces \( \widetilde{\sigma } \) on \( V\left( \widetilde{H}\right) \cup E\left( \widetilde{H}\right) \) . We may do this edge by edge, as follows. Every edge \( {xy} \) of \( \widetilde{H} \) is homeomorphic to the edge \( \widetilde{\sigma }\left( {xy}\right) = \varphi \left( x\right) \varphi \left( y\right) \) of \( {\widetilde{H}}^{\prime } \), by a homeomorphism mapping \( x \) to \( \varphi \left( x\right) \) and \( y \) to \( \varphi \left( y\right) \) . Then the union of all these homeomorphisms, one for every edge of \( \widetilde{H} \), is indeed a homeomorphism between \( \widetilde{H} \) and \( {\widetilde{H}}^{\prime } \) -our desired extension of \( \varphi \) to \( \widetilde{H} \) : all we have to check is continuity at the vertices (where the edge homeomorphisms overlap), and this follows at once from our assumption that the two graphs and their individual edges all carry the subspace topology in \( {\mathbb{R}}^{3} \) .
In the third step we now extend our homeomorphism \( \varphi : \widetilde{H} \rightarrow {\widetilde{H}}^{\prime } \) to all of \( {S}^{2} \) . This can be done analogously to the second step, face by face. By Proposition 4.2.6, all face boundaries in \( \widetilde{H} \) and \( {\widetilde{H}}^{\prime } \) are cycles. Now if \( f \) is a face of \( \widetilde{H} \) and \( C \) its boundary, then \( \widetilde{\sigma }\left( C\right) \mathrel{\text{:=}} \bigcup \{ \widetilde{\sigma }\left( e\right) \mid e \in E\left( C\right) \} \) bounds the face \( \widetilde{\sigma }\left( f\right) \) of \( {\widetilde{H}}^{\prime } \) . By Theorem 4.1.4, we may therefore extend the homeomorphism \( \varphi : C \rightarrow \widetilde{\sigma }\left( C\right) \) defined so far to a homeomorphism from \( C \cup f \) to \( \widetilde{\sigma }\left( C\right) \cup \widetilde{\sigma }\left( f\right) \) . We finally take the union of all these homeomorphisms, one for every face \( f \) of \( \widetilde{H} \), as our desired homeomorphism \( \varphi : {S}^{2} \rightarrow {S}^{2} \) ; as before, continuity is easily checked.
Let us return now to our original goal, the definition of equivalence for planar embeddings. Let us call two planar embeddings \( \rho ,{\rho }^{\prime } \) of a graph
4 By the ’vertices, edges and faces’ of \( \widetilde{H} \) and \( {\widetilde{H}}^{\prime } \) we mean the images under \( {\pi }^{-1} \) of the vertices, edges and faces of \( H \) and \( {H}^{\prime } \) (plus \( \left( {0,0,1}\right) \) in the case of the outer face). Their sets will be denoted by \( V\left( \widetilde{H}\right), E\left( \widetilde{H}\right), F\left( \widetilde{H}\right) \) and \( V\left( {\widetilde{H}}^{\prime }\right), E\left( {\widetilde{H}}^{\prime }\right), F\left( {\widetilde{H}}^{\prime }\right) \) , and incidence is de |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 34 | C\right) \cup \widetilde{\sigma }\left( f\right) \) . We finally take the union of all these homeomorphisms, one for every face \( f \) of \( \widetilde{H} \), as our desired homeomorphism \( \varphi : {S}^{2} \rightarrow {S}^{2} \) ; as before, continuity is easily checked.
Let us return now to our original goal, the definition of equivalence for planar embeddings. Let us call two planar embeddings \( \rho ,{\rho }^{\prime } \) of a graph
4 By the ’vertices, edges and faces’ of \( \widetilde{H} \) and \( {\widetilde{H}}^{\prime } \) we mean the images under \( {\pi }^{-1} \) of the vertices, edges and faces of \( H \) and \( {H}^{\prime } \) (plus \( \left( {0,0,1}\right) \) in the case of the outer face). Their sets will be denoted by \( V\left( \widetilde{H}\right), E\left( \widetilde{H}\right), F\left( \widetilde{H}\right) \) and \( V\left( {\widetilde{H}}^{\prime }\right), E\left( {\widetilde{H}}^{\prime }\right), F\left( {\widetilde{H}}^{\prime }\right) \) , and incidence is defined as inherited from \( H \) and \( {H}^{\prime } \) .
---
equivalent
embeddings
---
\( G \) topologically (respectively, combinatorially) equivalent if \( {\rho }^{\prime } \circ {\rho }^{-1} \) is a topological (respectively, combinatorial) isomorphism between \( \rho \left( G\right) \) and \( {\rho }^{\prime }\left( G\right) \) . If \( G \) is 2-connected, the two definitions coincide by Theorem 4.3.1, and we simply speak of equivalent embeddings. Clearly, this is indeed an equivalence relation on the set of planar embeddings of any given graph.
Note that two drawings of \( G \) resulting from inequivalent embeddings may well be topologically isomorphic (exercise): for the equivalence of two embeddings we ask not only that some (topological or combinatorial) isomorphism exist between the their images, but that the canonical isomorphism \( {\rho }^{\prime } \circ {\rho }^{-1} \) be a topological or combinatorial one.
Even in this strong sense, 3-connected graphs have only one embedding up to equivalence:
\( \left\lbrack {12.5.4}\right\rbrack \)
Theorem 4.3.2. (Whitney 1932)
Any two planar embeddings of a 3-connected graph are equivalent.
\( \left( {4.2.7}\right) \)
Proof. Let \( G \) be a 3-connected graph with planar embeddings \( \rho : G \rightarrow H \) and \( {\rho }^{\prime } : G \rightarrow {H}^{\prime } \) . By Theorem 4.3.1 it suffices to show that \( {\rho }^{\prime } \circ {\rho }^{-1} \) is a graph-theoretical isomorphism, i.e. that \( \rho \left( C\right) \) bounds a face of \( H \) if and only if \( {\rho }^{\prime }\left( C\right) \) bounds a face of \( {H}^{\prime } \), for every subgraph \( C \subseteq G \) . This follows at once from Proposition 4.2.7.
## 4.4 Planar graphs: Kuratowski's theorem
planar A graph is called planar if it can be embedded in the plane: if it is isomorphic to a plane graph. A planar graph is maximal, or maximally planar, if it is planar but cannot be extended to a larger planar graph by adding an edge (but no vertex).
Drawings of maximal planar graphs are clearly maximally plane. The converse, however, is not obvious: when we start to draw a planar graph, could it happen that we get stuck half-way with a proper subgraph that is already maximally plane? Our first proposition says that this can never happen, that is, a plane graph is never maximally plane just because it is badly drawn:
## Proposition 4.4.1.
(i) Every maximal plane graph is maximally planar.
(ii) A planar graph with \( n \geq 3 \) vertices is maximally planar if and only if it has \( {3n} - 6 \) edges.
(4.2.8)
Proof. Apply Proposition 4.2.8 and Corollary 4.2.10. (4.2.10)
Which graphs are planar? As we saw in Corollary 4.2.11, no planar graph contains \( {K}^{5} \) or \( {K}_{3,3} \) as a topological minor. Our aim in this section is to prove the surprising converse, a classic theorem of Kuratowski: any graph without a topological \( {K}^{5} \) or \( {K}_{3,3} \) minor is planar.
Before we prove Kuratowski's theorem, let us note that it suffices to consider ordinary minors rather than topological ones:
Lemma 4.4.2. A graph contains \( {K}^{5} \) or \( {K}_{3,3} \) as a minor if and only if it contains \( {K}^{5} \) or \( {K}_{3,3} \) as a topological minor.
Proof. By Proposition 1.7.2 it suffices to show that every graph \( G \)
\( \left( {1.7.2}\right) \)
with a \( {K}^{5} \) minor contains either \( {K}^{5} \) as a topological minor or \( {K}_{3,3} \) as a minor. So suppose that \( G \succcurlyeq {K}^{5} \), and let \( K \subseteq G \) be minimal such that \( K = M{K}^{5} \) . Then every branch set of \( K \) induces a tree in \( K \), and between any two branch sets \( K \) has exactly one edge. If we take the tree induced by a branch set \( {V}_{x} \) and add to it the four edges joining it to other branch sets, we obtain another tree, \( {T}_{x} \) say. By the minimality of \( K,{T}_{x} \) has exactly 4 leaves, the 4 neighbours of \( {V}_{x} \) in other branch sets (Fig. 4.4.1).
Fig. 4.4.1. Every \( M{K}^{5} \) contains a \( T{K}^{5} \) or \( M{K}_{3,3} \)
If each of the five trees \( {T}_{x} \) is a \( T{K}_{1,4} \) then \( K \) is a \( T{K}^{5} \), and we are done. If one of the \( {T}_{x} \) is not a \( T{K}_{1,4} \) then it has exactly two vertices of degree 3. Contracting \( {V}_{x} \) onto these two vertices, and every other branch set to a single vertex, we obtain a graph on 6 vertices containing a \( {K}_{3,3} \) . Thus, \( G \succcurlyeq {K}_{3,3} \) as desired.
We first prove Kuratowski's theorem for 3-connected graphs. This is the heart of the proof: the general case will then follow easily.
Lemma 4.4.3. Every 3-connected graph \( G \) without a \( {K}^{5} \) or \( {K}_{3,3} \) minor is planar.
\( \left( {3.2.1}\right) \) Proof. We apply induction on \( \left| G\right| \) . For \( \left| G\right| = 4 \) we have \( G = {K}^{4} \), and
(4.2.6)
the assertion holds. Now let \( \left| G\right| > 4 \), and assume the assertion is true \( {xy} \) for smaller graphs. By Lemma 3.2.1, \( G \) has an edge \( {xy} \) such that \( G/{xy} \) is again 3-connected. Since the minor relation is transitive, \( G/{xy} \) has no \( {K}^{5} \) or \( {K}_{3,3} \) minor either. Thus, by the induction hypothesis, \( G/{xy} \) has \( \widetilde{G} \) a drawing \( \widetilde{G} \) in the plane. Let \( f \) be the face of \( \widetilde{G} - {v}_{xy} \) containing the \( f, C \) point \( {v}_{xy} \), and let \( C \) be the boundary of \( f \) . Let \( X \mathrel{\text{:=}} {N}_{G}\left( x\right) \smallsetminus \{ y\} \) and \( X, Y \) \( Y \mathrel{\text{:=}} {N}_{G}\left( y\right) \smallsetminus \{ x\} \) ; then \( X \cup Y \subseteq V\left( C\right) \), because \( {v}_{xy} \in f \) . Clearly,
\( {\widetilde{G}}^{\prime } \)
\[
{\widetilde{G}}^{\prime } \mathrel{\text{:=}} \widetilde{G} - \left\{ {{v}_{xy}v \mid v \in Y \smallsetminus X}\right\}
\]
may be viewed as a drawing of \( G - y \), in which the vertex \( x \) is represented by the point \( {v}_{xy} \) (Fig. 4.4.2). Our aim is to add \( y \) to this drawing to obtain a drawing of \( G \) .
Fig. 4.4.2. \( {\widetilde{G}}^{\prime } \) as a drawing of \( G - y \) : the vertex \( x \) is represented by the point \( {v}_{xy} \)
Since \( \widetilde{G} \) is 3-connected, \( \widetilde{G} - {v}_{xy} \) is 2-connected, so \( C \) is a cycle
---
\( {x}_{1},\ldots ,{x}_{k} \)
\( {P}_{i} \)
---
(Proposition 4.2.6). Let \( {x}_{1},\ldots ,{x}_{k} \) be an enumeration along this cycle of the vertices in \( X \), and let \( {P}_{i} = {x}_{i}\ldots {x}_{i + 1} \) be the \( X \) -paths on \( C \) between them \( \left( {i = 1,\ldots, k\text{; with}{x}_{k + 1} \mathrel{\text{:=}} {x}_{1}}\right) \) . Let us show that \( Y \subseteq V\left( {P}_{i}\right) \) for some \( i \) . If not, then either \( x \) and \( y \) have three common neighbours on \( C \) and form a \( T{K}^{5} \) with these, or \( y \) has two neighbours on \( C \) that are separated in \( C \) by two neighbours of \( x \), and these four vertices of \( C \) form with \( x \) and \( y \) the branch vertices of a \( T{K}_{3,3} \) . In either case have a contradiction, since \( G \) contains neither a \( T{K}^{5} \) nor a \( T{K}_{3,3} \) .
Now fix \( i \) so that \( Y \subseteq {P}_{i} \) . The set \( C \smallsetminus {P}_{i} \) is contained in one of the \( {C}_{i} \) two faces of the cycle \( {C}_{i} \mathrel{\text{:=}} x{x}_{i}{P}_{i}{x}_{i + 1}x \) ; we denote the other face of \( {C}_{i} \) \( {f}_{i} \) by \( {f}_{i} \) . Since \( {f}_{i} \) contains points of \( f \) (close to \( x \) ) but no points of \( C \), we have \( {f}_{i} \subseteq f \) . Moreover, the plane edges \( x{x}_{j} \) with \( j \notin \{ i, i + 1\} \) meet \( {C}_{i} \) only in \( x \) and end outside \( {f}_{i} \) in \( C \smallsetminus {P}_{i} \), so \( {f}_{i} \) meets none of those edges. Hence \( {f}_{i} \subseteq {\mathbb{R}}^{2} \smallsetminus {\widetilde{G}}^{\prime } \), that is, \( {f}_{i} \) is contained in (and hence equal to) a face of \( {\widetilde{G}}^{\prime } \) . We may therefore extend \( {\widetilde{G}}^{\prime } \) to a drawing of \( G \) by placing \( y \) and its incident edges in \( {f}_{i} \) .
Compared with other proofs of Kuratowski's theorem, the above proof has the attractive feature that it can easily be adapted to produce a drawing in which every inner face is convex (exercise); in particular, every edge can be drawn straight. Note that 3-connectedness is essential here: a 2-connected planar graph need not have a drawing with all inner faces convex (example?), although it always has a straight-line drawing (Exercise 12).
It is not difficult, in principle, to reduce the general Kuratowski theorem to the 3-connected case by manipulating and combining partial drawing |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 35 | n \( C \smallsetminus {P}_{i} \), so \( {f}_{i} \) meets none of those edges. Hence \( {f}_{i} \subseteq {\mathbb{R}}^{2} \smallsetminus {\widetilde{G}}^{\prime } \), that is, \( {f}_{i} \) is contained in (and hence equal to) a face of \( {\widetilde{G}}^{\prime } \) . We may therefore extend \( {\widetilde{G}}^{\prime } \) to a drawing of \( G \) by placing \( y \) and its incident edges in \( {f}_{i} \) .
Compared with other proofs of Kuratowski's theorem, the above proof has the attractive feature that it can easily be adapted to produce a drawing in which every inner face is convex (exercise); in particular, every edge can be drawn straight. Note that 3-connectedness is essential here: a 2-connected planar graph need not have a drawing with all inner faces convex (example?), although it always has a straight-line drawing (Exercise 12).
It is not difficult, in principle, to reduce the general Kuratowski theorem to the 3-connected case by manipulating and combining partial drawings assumed to exist by induction. For example, if \( \kappa \left( G\right) = 2 \) and \( G = {G}_{1} \cup {G}_{2} \) with \( V\left( {{G}_{1} \cap {G}_{2}}\right) = \{ x, y\} \), and if \( G \) has no \( T{K}^{5} \) or \( T{K}_{3,3} \) subgraph, then neither \( {G}_{1} + {xy} \) nor \( {G}_{2} + {xy} \) has such a subgraph, and we may try to combine drawings of these graphs to one of \( G + {xy} \) . (If \( {xy} \) is already an edge of \( G \), the same can be done with \( {G}_{1} \) and \( {G}_{2} \) .) For \( \kappa \left( G\right) \leq 1 \), things become even simpler. However, the geometric operations involved require some cumbersome shifting and scaling, even if all the plane edges occurring are assumed to be straight.
The following more combinatorial route is just as easy, and may be a welcome alternative.
Lemma 4.4.4. Let \( \mathcal{X} \) be a set of 3-connected graphs. Let \( G \) be a graph
\( \left\lbrack {7.3.1}\right\rbrack \)
with \( \kappa \left( G\right) \leq 2 \), and let \( {G}_{1},{G}_{2} \) be proper induced subgraphs of \( G \) such that \( G = {G}_{1} \cup {G}_{2} \) and \( \left| {{G}_{1} \cap {G}_{2}}\right| = \kappa \left( G\right) \) . If \( G \) is edge-maximal without a topological minor in \( \mathcal{X} \), then so are \( {G}_{1} \) and \( {G}_{2} \), and \( {G}_{1} \cap {G}_{2} = {K}^{2} \) .
Proof. Note first that every vertex \( v \in S \mathrel{\text{:=}} V\left( {{G}_{1} \cap {G}_{2}}\right) \) has a neighbour in every component of \( {G}_{i} - S, i = 1,2 \) : otherwise \( S \smallsetminus \{ v\} \) would separate \( G \), contradicting \( \left| S\right| = \kappa \left( G\right) \) . By the maximality of \( G \), every edge \( e \) added to \( G \) lies in a \( {TX} \subseteq G + e \) with \( X \in \mathcal{X} \) . For all the choices of \( e \) considered below, the 3-connectedness of \( X \) will imply that the branch vertices of this \( {TX} \) all lie in the same \( {G}_{i} \), say in \( {G}_{1} \) . (The position of \( e \) will always be symmetrical with respect to \( {G}_{1} \) and \( {G}_{2} \), so this assumption entails no loss of generality.) Then the \( {TX} \) meets \( {G}_{2} \) at most in a path \( P \) corresponding to an edge of \( X \) .
If \( S = \varnothing \), we obtain an immediate contradiction by choosing \( e \) with one end in \( {G}_{1} \) and the other in \( {G}_{2} \) . If \( S = \{ v\} \) is a singleton, let \( e \) join a neighbour \( {v}_{1} \) of \( v \) in \( {G}_{1} - S \) to a neighbour \( {v}_{2} \) of \( v \) in \( {G}_{2} - S \) (Fig. 4.4.3). Then \( P \) contains both \( v \) and the edge \( {v}_{1}{v}_{2} \) ; replacing \( {vP}{v}_{1} \) with the edge \( v{v}_{1} \), we obtain a \( {TX} \) in \( {G}_{1} \subseteq G \), a contradiction.
So \( \left| S\right| = 2 \), say \( S = \{ x, y\} \) . If \( {xy} \notin G \), we let \( e \mathrel{\text{:=}} {xy} \), and in the arising \( {TX} \) replace \( e \) by an \( x - y \) path through \( {G}_{2} \) ; this gives a \( {TX} \) in \( G \) , a contradiction. Hence \( {xy} \in G \), and \( G\left\lbrack S\right\rbrack = {K}^{2} \) as claimed.
It remains to show that \( {G}_{1} \) and \( {G}_{2} \) are edge-maximal without a topological minor in \( \mathcal{X} \) . So let \( {e}^{\prime } \) be an additional edge for \( {G}_{1} \), say. Replacing \( {xPy} \) with the edge \( {xy} \) if necessary, we obtain a \( {TX} \) either
Fig. 4.4.3. If \( G + e \) contains a \( {TX} \), then so does \( {G}_{1} \) or \( {G}_{2} \)
in \( {G}_{1} + {e}^{\prime } \) (which shows the edge-maximality of \( {G}_{1} \), as desired) or in \( {G}_{2} \) (which contradicts \( {G}_{2} \subseteq G \) ).
Lemma 4.4.5. If \( \left| G\right| \geq 4 \) and \( G \) is edge-maximal with \( T{K}^{5}, T{K}_{3,3} \nsubseteq G \) , then \( G \) is 3-connected.
(4.2.11)
Proof. We apply induction on \( \left| G\right| \) . For \( \left| G\right| = 4 \), we have \( G = {K}^{4} \) and the assertion holds. Now let \( \left| G\right| > 4 \), and let \( G \) be edge-maximal
\( {G}_{1},{G}_{2} \)
without a \( T{K}^{5} \) or \( T{K}_{3,3} \) . Suppose \( \kappa \left( G\right) \leq 2 \), and choose \( {G}_{1} \) and \( {G}_{2} \) as in Lemma 4.4.4. For \( \mathcal{X} \mathrel{\text{:=}} \left\{ {{K}^{5},{K}_{3,3}}\right\} \), the lemma says that \( {G}_{1} \cap {G}_{2} \) is \( x, y \) a \( {K}^{2} \), with vertices \( x, y \) say. By Lemmas 4.4.4,4.4.3 and the induction hypothesis, \( {G}_{1} \) and \( {G}_{2} \) are planar. For each \( i = 1,2 \) separately, choose a \( {f}_{i} \) drawing of \( {G}_{i} \), a face \( {f}_{i} \) with the edge \( {xy} \) on its boundary, and a vertex \( {z}_{i} \) \( {z}_{i} \neq x, y \) on the boundary of \( {f}_{i} \) . Let \( K \) be a \( T{K}^{5} \) or \( T{K}_{3,3} \) in the \( K \) abstract graph \( G + {z}_{1}{z}_{2} \) (Fig. 4.4.4).
Fig. 4.4.4. A \( T{K}^{5} \) or \( T{K}_{3,3} \) in \( G + {z}_{1}{z}_{2} \)
If all the branch vertices of \( K \) lie in the same \( {G}_{i} \), then either \( {G}_{i} + x{z}_{i} \) or \( {G}_{i} + y{z}_{i} \) (or \( {G}_{i} \) itself, if \( {z}_{i} \) is already adjacent to \( x \) or \( y \), respectively) contains a \( T{K}^{5} \) or \( T{K}_{3,3} \) ; this contradicts Corollary 4.2.11, since these graphs are planar by the choice of \( {z}_{i} \) . Since \( G + {z}_{1}{z}_{2} \) does not contain four independent paths between \( \left( {{G}_{1} - {G}_{2}}\right) \) and \( \left( {{G}_{2} - {G}_{1}}\right) \), these subgraphs cannot both contain a branch vertex of a \( T{K}^{5} \), and cannot both contain two branch vertices of a \( T{K}_{3,3} \) . Hence \( K \) is a \( T{K}_{3,3} \) with only one branch vertex \( v \) in, say, \( {G}_{2} - {G}_{1} \) . But then also the graph \( {G}_{1} + v + \left\{ {{vx},{vy}, v{z}_{1}}\right\} \) , which is planar by the choice of \( {z}_{1} \), contains a \( T{K}_{3,3} \) . This contradicts Corollary 4.2.11.
Theorem 4.4.6. (Kuratowski 1930; Wagner 1937) \( \left\lbrack \begin{array}{l} {4.5.1} \\ {12.4.3} \end{array}\right\rbrack \)
The following assertions are equivalent for graphs \( G \) :
(i) \( G \) is planar;
(ii) \( G \) contains neither \( {K}^{5} \) nor \( {K}_{3,3} \) as a minor;
(iii) \( G \) contains neither \( {K}^{5} \) nor \( {K}_{3,3} \) as a topological minor.
Proof. Combine Corollary 4.2.11 with Lemmas 4.4.2, 4.4.3 and 4.4.5.
(4.2.11)
Corollary 4.4.7. Every maximal planar graph with at least four vertices is 3-connected.
Proof. Apply Lemma 4.4.5 and Theorem 4.4.6.
## 4.5 Algebraic planarity criteria
One of the most conspicuous features of a plane graph \( G \) are its facial
---
facial cycles
---
cycles, the cycles that bound a face. If \( G \) is 2-connected it is covered by its facial cycles, so in a sense these form a 'large' set. In fact, the set of facial cycles is large even in the sense that they generate the entire cycle space: every cycle in \( G \) is easily seen to be the sum of the facial cycles (see below). On the other hand, the facial cycles only cover \( G \) ’thinly’, as every edge lies on at most two of them. Our first aim in this section is to show that the existence of such a large yet thinly spread family of cycles is not only a conspicuous feature of planarity but lies at its very heart: it characterizes it.
Let \( G = \left( {V, E}\right) \) be any graph. We call a subset \( \mathcal{F} \) of its edge space \( \mathcal{E}\left( G\right) \) simple if every edge of \( G \) lies in at most two sets of \( \mathcal{F} \) . For example, simple the cut space \( {\mathcal{C}}^{ * }\left( G\right) \) has a simple basis: according to Proposition 1.9.3 it is generated by the cuts \( E\left( v\right) \) formed by all the edges at a given vertex \( v \) , and an edge \( {xy} \in G \) lies in \( E\left( v\right) \) only for \( v = x \) and for \( v = y \) .
Theorem 4.5.1. (MacLane 1937)
\( \left\lbrack {4.6.3}\right\rbrack \)
A graph is planar if and only if its cycle space has a simple basis.
Proof. The assertion being trivial for graphs of order at most 2 , we consider a graph \( G \) of order at least 3 . If \( \kappa \left( G\right) \leq 1 \), then \( G \) is the union
\( \left( {1.9.2}\right) \)
of two proper induced subgraphs \( {G}_{1},{G}_{2} \) with \( \left| {{G}_{1} \cap {G}_{2}}\right| \leq 1 \) . Then \( \mathcal{C}\left( G\right) \) (4.1.1) is the direct sum of \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \), and hence has a simple basis if (4.2.2) and only if both \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \) do (proof?). Moreover, \( G \) is planar if (4.2.6) (4.4.6) and only if both \( {G}_{1} \) and \( {G}_{2} \) are: this follows at once from Kuratowski’s theorem, but also from easy geome |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 36 | t( v\right) \) only for \( v = x \) and for \( v = y \) .
Theorem 4.5.1. (MacLane 1937)
\( \left\lbrack {4.6.3}\right\rbrack \)
A graph is planar if and only if its cycle space has a simple basis.
Proof. The assertion being trivial for graphs of order at most 2 , we consider a graph \( G \) of order at least 3 . If \( \kappa \left( G\right) \leq 1 \), then \( G \) is the union
\( \left( {1.9.2}\right) \)
of two proper induced subgraphs \( {G}_{1},{G}_{2} \) with \( \left| {{G}_{1} \cap {G}_{2}}\right| \leq 1 \) . Then \( \mathcal{C}\left( G\right) \) (4.1.1) is the direct sum of \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \), and hence has a simple basis if (4.2.2) and only if both \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \) do (proof?). Moreover, \( G \) is planar if (4.2.6) (4.4.6) and only if both \( {G}_{1} \) and \( {G}_{2} \) are: this follows at once from Kuratowski’s theorem, but also from easy geometrical considerations. The assertion for \( G \) thus follows inductively from those for \( {G}_{1} \) and \( {G}_{2} \) . For the rest of the proof, we now assume that \( G \) is 2-connected.
We first assume that \( G \) is planar and choose a drawing. By Proposition 4.2.6, the face boundaries of \( G \) are cycles, so they are elements of \( \mathcal{C}\left( G\right) \) . We shall show that the face boundaries generate all the cycles in \( G \) ; then \( \mathcal{C}\left( G\right) \) has a simple basis by Lemma 4.2.2. Let \( C \subseteq G \) be any cycle, and let \( f \) be its inner face. By Lemma 4.2.2, every edge \( e \) with \( e \subseteq f \) lies on exactly two face boundaries \( G\left\lbrack {f}^{\prime }\right\rbrack \) with \( {f}^{\prime } \subseteq f \), and every edge of \( C \) lies on exactly one such face boundary. Hence the sum in \( \mathcal{C}\left( G\right) \) of all those face boundaries is exactly \( C \) .
Conversely, let \( \left\{ {{C}_{1},\ldots ,{C}_{k}}\right\} \) be a simple basis of \( \mathcal{C}\left( G\right) \) . Then, for every edge \( e \in G \), also \( \mathcal{C}\left( {G - e}\right) \) has a simple basis. Indeed, if \( e \) lies in just one of the sets \( {C}_{i} \), say in \( {C}_{1} \), then \( \left\{ {{C}_{2},\ldots ,{C}_{k}}\right\} \) is a simple basis of \( \mathcal{C}\left( {G - e}\right) \) ; if \( e \) lies in two of the \( {C}_{i} \), say in \( {C}_{1} \) and \( {C}_{2} \), then \( \left\{ {{C}_{1} + {C}_{2},{C}_{3},\ldots ,{C}_{k}}\right\} \) is such a basis. (Note that the two bases are indeed subsets of \( \mathcal{C}\left( {G - e}\right) \) by Proposition 1.9.2.) Thus every subgraph of \( G \) has a cycle space with a simple basis. For our proof that \( G \) is planar, it thus suffices to show that the cycle spaces of \( {K}^{5} \) and \( {K}_{3,3} \) (and hence those of their subdivisions) do not have a simple basis: then \( G \) cannot contain a \( T{K}^{5} \) or \( T{K}_{3,3} \), and so is planar by Kuratowski’s theorem.
Let us consider \( {K}^{5} \) first. By Theorem 1.9.6, \( \dim \mathcal{C}\left( {K}^{5}\right) = 6 \) ; let \( \mathcal{B} = \left\{ {{C}_{1},\ldots ,{C}_{6}}\right\} \) be a simple basis, and put \( {C}_{0} \mathrel{\text{:=}} {C}_{1} + \ldots + {C}_{6} \) . As \( \mathcal{B} \) is linearly independent, none of the sets \( {C}_{0},\ldots ,{C}_{6} \) is empty, so each of them contains at least three edges (cf. Proposition 1.9.2). Moreover, as every edge from \( {C}_{0} \) lies in just one of \( {C}_{1},\ldots ,{C}_{6} \), the set \( \left\{ {{C}_{0},\ldots ,{C}_{6}}\right\} \) is still simple. But this implies that \( {K}^{5} \) should have more edges than it does, i.e. we obtain the contradiction of
\[
{21} = 7 \cdot 3 \leq \left| {C}_{0}\right| + \ldots + \left| {C}_{6}\right| \leq 2\begin{Vmatrix}{K}^{5}\end{Vmatrix} = {20}.
\]
For \( {K}_{3,3} \), Theorem 1.9.6 gives \( \dim \mathcal{C}\left( {K}_{3,3}\right) = 4 \) ; let \( \mathcal{B} = \left\{ {{C}_{1},\ldots ,{C}_{4}}\right\} \) be a simple basis, and put \( {C}_{0} \mathrel{\text{:=}} {C}_{1} + \ldots + {C}_{4} \) . As \( {K}_{3,3} \) has girth 4, each \( {C}_{i} \) contains at least four edges. We then obtain the contradiction of
\[
{20} = 5 \cdot 4 \leq \left| {C}_{0}\right| + \ldots + \left| {C}_{4}\right| \leq 2\begin{Vmatrix}{K}_{3,3}\end{Vmatrix} = {18}.
\]
It is one of the hidden beauties of planarity theory that two such abstract and seemingly unintuitive results about generating sets in cycle spaces as MacLane's theorem and Tutte's theorem 3.2.3 conspire to produce a very tangible planarity criterion for 3-connected graphs:
Theorem 4.5.2. (Kelmans 1978)
A 3-connected graph is planar if and only if every edge lies on at most (equivalently: exactly) two non-separating induced cycles.
\( \left( {4.2.2}\right) \) Proof. The forward implication follows from Propositions 4.2.7 and \( \left( {4.2.6}\right) \) \( \left( {4.2.7}\right) \) 4.2.2 (and Proposition 4.2.6 for the 'exactly two' version); the backward implication follows from Theorems 3.2.3 and 4.5.1.
## 4.6 Plane duality
In this section we shall use MacLane's theorem to uncover another connection between planarity and algebraic structure: a connection between the duality of plane graphs, defined below, and the duality of the cycle and cut space hinted at in Chapters 1.9 and 2.4.
---
plane multigraph
---
A plane multigraph is a pair \( G = \left( {V, E}\right) \) of finite sets (of vertices and edges, respectively) satisfying the following conditions:
(i) \( V \subseteq {\mathbb{R}}^{2} \) ;
(ii) every edge is either an arc between two vertices or a polygon containing exactly one vertex (its endpoint);
(iii) apart from its own endpoint(s), an edge contains no vertex and no point of any other edge.
We shall use terms defined for plane graphs freely for plane multigraphs. Note that, as in abstract multigraphs, both loops and double edges count as cycles.
Let us consider the plane multigraph \( G \) shown in Figure 4.6.1. Let us place a new vertex inside each face of \( G \) and link these new vertices up to form another plane multigraph \( {G}^{ * } \), as follows: for every edge \( e \) of \( G \) we link the two new vertices in the faces incident with \( e \) by an edge \( {e}^{ * } \) crossing \( e \) ; if \( e \) is incident with only one face, we attach a loop \( {e}^{ * } \) to the new vertex in that face, again crossing the edge \( e \) . The plane multigraph \( {G}^{ * } \) formed in this way is then dual to \( G \) in the following sense: if we apply the same procedure as above to \( {G}^{ * } \), we obtain a plane multigraph very similar to \( G \) ; in fact, \( G \) itself may be reobtained from \( {G}^{ * } \) in this way.
Fig. 4.6.1. A plane graph and its dual
To make this idea more precise, let \( G = \left( {V, E}\right) \) and \( \left( {{V}^{ * },{E}^{ * }}\right) \) be any two plane multigraphs, and put \( F\left( G\right) = : F \) and \( F\left( \left( {{V}^{ * },{E}^{ * }}\right) \right) = : {F}^{ * } \) . We
---
plane
dual \( {G}^{ * } \)
---
call \( \left( {{V}^{ * },{E}^{ * }}\right) \) a plane dual of \( G \), and write \( \left( {{V}^{ * },{E}^{ * }}\right) = : {G}^{ * } \), if there are bijections
\[
F \rightarrow {V}^{ * }\;E \rightarrow {E}^{ * }\;V \rightarrow {F}^{ * }
\]
\[
f \mapsto {v}^{ * }\left( f\right) \;e \mapsto {e}^{ * }\;v \mapsto {f}^{ * }\left( v\right)
\]
satisfying the following conditions:
(i) \( {v}^{ * }\left( f\right) \in f \) for all \( f \in F \) ;
(ii) \( \left| {{e}^{ * } \cap G}\right| = \left| {{\mathring{e}}^{ * } \cap \mathring{e}}\right| = \left| {e \cap {G}^{ * }}\right| = 1 \) for all \( e \in E \), and in each of \( e \) and \( {e}^{ * } \) this point is an inner point of a straight line segment;
(iii) \( v \in {f}^{ * }\left( v\right) \) for all \( v \in V \) .
Every connected plane multigraph has a plane dual. Indeed, to satisfy condition (i) we start by picking from each face \( f \) of \( G \) a point \( {v}^{ * }\left( f\right) \) as a vertex for \( {G}^{ * } \) . We can then link these vertices up by independent arcs as required by (ii), and using the connectedness of \( G \) show that there is indeed a bijection \( V \rightarrow {F}^{ * } \) satisfying (iii) (Exercise 27).
If \( {G}_{1}^{ * } \) and \( {G}_{2}^{ * } \) are two plane duals of \( G \), then clearly \( {G}_{1}^{ * } \simeq {G}_{2}^{ * } \) ; in fact, one can show that the natural bijection \( {v}_{1}^{ * }\left( f\right) \mapsto {v}_{2}^{ * }\left( f\right) \) is a topological isomorphism between \( {G}_{1}^{ * } \) and \( {G}_{2}^{ * } \) . In this sense, we may speak of the plane dual \( {G}^{ * } \) of \( G \) .
Finally, \( G \) is in turn a plane dual of \( {G}^{ * } \) . Indeed, this is witnessed by the inverse maps of the bijections from the definition of \( {G}^{ * } \) : setting \( {v}^{ * }\left( {{f}^{ * }\left( v\right) }\right) \mathrel{\text{:=}} v \) and \( {f}^{ * }\left( {{v}^{ * }\left( f\right) }\right) \mathrel{\text{:=}} f \) for \( {f}^{ * }\left( v\right) \in {F}^{ * } \) and \( {v}^{ * }\left( f\right) \in {V}^{ * } \), we see that conditions (i) and (iii) for \( {G}^{ * } \) transform into (iii) and (i) for \( G \) , while condition (ii) is symmetrical in \( G \) and \( {G}^{ * } \) . As duals are easily seen to be connected (Exercise 26), this symmetry implies that connectedness is also a necessary condition for \( G \) to have a dual.
Perhaps the most interesting aspect of plane duality is that it relates geometrically two types of edges sets - cycles and bonds - that we have previously seen to be algebraically related (Theorem 1.9.5):
\( \left\lbrack {6.5.2}\right\rbrack \)
Proposition 4.6.1. For any connected plane multigraph \( G \), an edge set \( E \subseteq E\left( G\right) \) is the edge set of a cycle in \( G \) if and only if |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 37 | ting \( {v}^{ * }\left( {{f}^{ * }\left( v\right) }\right) \mathrel{\text{:=}} v \) and \( {f}^{ * }\left( {{v}^{ * }\left( f\right) }\right) \mathrel{\text{:=}} f \) for \( {f}^{ * }\left( v\right) \in {F}^{ * } \) and \( {v}^{ * }\left( f\right) \in {V}^{ * } \), we see that conditions (i) and (iii) for \( {G}^{ * } \) transform into (iii) and (i) for \( G \) , while condition (ii) is symmetrical in \( G \) and \( {G}^{ * } \) . As duals are easily seen to be connected (Exercise 26), this symmetry implies that connectedness is also a necessary condition for \( G \) to have a dual.
Perhaps the most interesting aspect of plane duality is that it relates geometrically two types of edges sets - cycles and bonds - that we have previously seen to be algebraically related (Theorem 1.9.5):
\( \left\lbrack {6.5.2}\right\rbrack \)
Proposition 4.6.1. For any connected plane multigraph \( G \), an edge set \( E \subseteq E\left( G\right) \) is the edge set of a cycle in \( G \) if and only if \( {E}^{ * } \mathrel{\text{:=}} \left\{ {{e}^{ * } \mid e \in E}\right\} \) is a minimal cut in \( {G}^{ * } \) .
Proof. By conditions (i) and (ii) in the definition of \( {G}^{ * } \), two vertices \( {v}^{ * }\left( {f}_{1}\right) \) and \( {v}^{ * }\left( {f}_{2}\right) \) of \( {G}^{ * } \) lie in the same component of \( {G}^{ * } - {E}^{ * } \) if and only if \( {f}_{1} \) and \( {f}_{2} \) lie in the same region of \( {\mathbb{R}}^{2} \smallsetminus \bigcup E \) : every \( {v}^{ * }\left( {f}_{1}\right) - {v}^{ * }\left( {f}_{2}\right) \) path in \( {G}^{ * } - {E}^{ * } \) is an arc between \( {f}_{1} \) and \( {f}_{2} \) in \( {\mathbb{R}}^{2} \smallsetminus \bigcup E \), and conversely every such arc \( P \) (with \( P \cap V\left( G\right) = \varnothing \) ) defines a walk in \( {G}^{ * } - {E}^{ * } \) between \( {v}^{ * }\left( {f}_{1}\right) \) and \( {v}^{ * }\left( {f}_{2}\right) \) .
Now if \( C \subseteq G \) is a cycle and \( E = E\left( C\right) \) then, by the Jordan curve theorem and the above correspondence, \( {G}^{ * } - {E}^{ * } \) has exactly two components, so \( {E}^{ * } \) is a minimal cut in \( {G}^{ * } \) .
Conversely, if \( E \subseteq E\left( G\right) \) is such that \( {E}^{ * } \) is a cut in \( {G}^{ * } \), then, by Proposition 4.2.4 and the above correspondence, \( E \) contains the edges of a cycle \( C \subseteq G \) . If \( {E}^{ * } \) is minimal as a cut, then \( E \) cannot contain any further edges (by the implication shown before), so \( E = E\left( C\right) \) .
Proposition 4.6.1 suggests the following generalization of plane du-
---
abstract dual
---
ality to abstract multigraphs. Call a multigraph \( {G}^{ * } \) an abstract dual of a multigraph \( G \) if \( E\left( {G}^{ * }\right) = E\left( G\right) \) and the bonds in \( {G}^{ * } \) are precisely the edge sets of cycles in \( G \) . (Neither \( G \) nor \( {G}^{ * } \) need be connected now.)
This correspondence between cycles and bonds extends to the spaces they generate:
Proposition 4.6.2. If \( {G}^{ * } \) is an abstract dual of \( G \), then the cut space of \( {G}^{ * } \) is the cycle space of \( G \), i.e.,
\[
{\mathcal{C}}^{ * }\left( {G}^{ * }\right) = \mathcal{C}\left( G\right)
\]
Proof. Since the cycles of \( G \) are precisely the bonds of \( {G}^{ * } \), the subspace
(1.9.4)
\( \mathcal{C}\left( G\right) \) they generate in \( \mathcal{E}\left( G\right) = \mathcal{E}\left( {G}^{ * }\right) \) is the same as the subspace generated by the bonds in \( {G}^{ * } \) . By Lemma 1.9.4, \( {}^{5} \) this is the space \( {\mathcal{C}}^{ * }\left( {G}^{ * }\right) \) .
By Theorem 1.9.5, Proposition 4.6.2 implies at once that if \( {G}^{ * } \) is an abstract dual of \( G \) then \( G \) is an abstract dual of \( {G}^{ * } \) . One can show that if \( G \) is 3-connected, then \( {G}^{ * } \) is unique up to isomorphism.
Although the notion of abstract duality arose as a generalization of plane duality, it could have been otherwise. We knew already from Theorem 1.9.5 that the cycles and the bonds of a graph form natural and related sets of edges. It would not have been unthinkable to ask whether, for some graphs, the orthogonality between these collections of edge sets might give them sufficiently similar intersection patterns that a collection forming the cycles in one graph could form the bonds in another, and vice versa. In other words, for which graphs can we move their entire edge set to a new set of vertices, redefining incidences, so that precisely those sets of edges that used to form cycles now become bonds (and vice versa)? Put in this way, it seems surprising that this could ever be achieved, let alone for such a large and natural class of graphs as all planar graphs.
As the one of the highlights of classical planarity theory we now show that the planar graphs are precisely those for which this can be done. Admitting an abstract dual thus appears as a new planarity criterion. Conversely, the theorem can be read as a surprising topological characterization of the equally fundamental property of admitting an abstract dual:
Theorem 4.6.3. (Whitney 1933)
A graph is planar if and only if it has an abstract dual.
---
5 Although the lemma was stated for graphs only, its proof remains the same for multigraphs.
---
Proof. Let \( G \) be a planar graph, and consider any drawing. Every \( {\text{component}}^{6}C \) of this drawing has a plane dual \( {C}^{ * } \) . Consider these \( {C}^{ * } \) as abstract multigraphs, and let \( {G}^{ * } \) be their disjoint union. Then the bonds of \( {G}^{ * } \) are precisely the minimal cuts in the \( {C}^{ * } \), which by Proposition 4.6.1 correspond to the cycles in \( G \) .
Conversely, suppose that \( G \) has an abstract dual \( {G}^{ * } \) . For a proof that \( G \) is planar, it suffices by Theorem 4.5.1 and Proposition 4.6.2 to show that \( {\mathcal{C}}^{ * }\left( {G}^{ * }\right) \) has a simple basis. By Proposition 1.9.3, it does.
The duality theory for both abstract and plane graphs can be extended to infinite graphs. As these can have infinite bonds, their duals must then have 'infinite cycles'. Such things do indeed exist, and are quite fascinating: they arise as topological circles in a space formed by the graph and its ends; see Chapter 8.5.
## Exercises
1. Show that every graph can be embedded in \( {\mathbb{R}}^{3} \) with all edges straight.
2. \( {}^{ - } \) Show directly by Lemma 4.1.2 that \( {K}_{3,3} \) is not planar.
3. \( {}^{ - } \) Find an Euler formula for disconnected graphs.
4. Show that every connected planar graph with \( n \) vertices, \( m \) edges and finite girth \( g \) satisfies \( m \leq \frac{g}{g - 2}\left( {n - 2}\right) \) .
5. Show that every planar graph is a union of three forests.
6. Let \( {G}_{1},{G}_{2},\ldots \) be an infinite sequence of pairwise non-isomorphic graphs. Show that if \( \lim \sup \varepsilon \left( {G}_{i}\right) > 3 \) then the graphs \( {G}_{i} \) have unbounded genus - that is to say, there is no (closed) surface \( S \) in which all the \( {G}_{i} \) can be embedded.
(Hint. You may use the fact that for every surface \( S \) there is a constant \( \chi \left( S\right) \leq 2 \) such that every graph embedded in \( S \) satisfies the generalized Euler formula of \( n - m + \ell \geq \chi \left( S\right) \) .)
7. Find a direct proof for planar graphs of Tutte's theorem on the cycle space of 3-connected graphs (Theorem 3.2.3).
8. \( {}^{ - } \) Show that the two plane graphs in Figure 4.3.1 are not combinatorially (and hence not topologically) isomorphic.
9. Show that the two graphs in Figure 4.3.2 are combinatorially but not topologically isomorphic.
10. \( {}^{ - } \) Show that our definition of equivalence for planar embeddings does indeed define an equivalence relation.
---
6 More elegantly but less straightforwardly, use blocks instead of components and apply Lemma 3.1.1.
---
11. Find a 2-connected planar graph whose drawings are all topologically isomorphic but whose planar embeddings are not all equivalent.
12. \( {}^{ + } \) Show that every plane graph is combinatorially isomorphic to a plane graph whose edges are all straight.
(Hint. Given a plane triangulation, construct inductively a graph-theoretically isomorphic plane graph whose edges are straight. Which additional property of the inner faces could help with the induction?)
Do not use Kuratowski's theorem in the following two exercises.
13. Show that any minor of a planar graph is planar. Deduce that a graph is planar if and only if it is the minor of a grid. (Grids are defined in Chapter 12.3.)
14. (i) Show that the planar graphs can in principle be characterized as in Kuratowski’s theorem, i.e., that there exists a set \( \mathcal{X} \) of graphs such that a graph \( G \) is planar if and only if \( G \) has no topological minor in \( \mathcal{X} \) .
(ii) More generally, which graph properties can be characterized in this way?
15. \( {}^{ - } \) Does every planar graph have a drawing with all inner faces convex?
16. Modify the proof of Lemma 4.4.3 so that all inner faces become convex.
17. Does every minimal non-planar graph \( G \) (i.e., every non-planar graph \( G \) whose proper subgraphs are all planar) contain an edge \( e \) such that \( G - e \) is maximally planar? Does the answer change if we define ’minimal' with respect to minors rather than subgraphs?
18. Show that adding a new edge to a maximal planar graph of order at least 6 always produces both a \( T{K}^{5} \) and a \( T{K}_{3,3} \) subgraph.
19. Prove the general Kuratowski theorem from its 3-connected case by manipulating plane graphs, i.e. avoiding Lemma 4.4.5.
(This is not intended as an exercise in elementary topology; for the topological parts of the proof, a rough sketch will do.)
20. A graph is called outerplanar if it has a drawing in which every vertex lies on the boundary of the outer face. Show that a |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 38 | terized in this way?
15. \( {}^{ - } \) Does every planar graph have a drawing with all inner faces convex?
16. Modify the proof of Lemma 4.4.3 so that all inner faces become convex.
17. Does every minimal non-planar graph \( G \) (i.e., every non-planar graph \( G \) whose proper subgraphs are all planar) contain an edge \( e \) such that \( G - e \) is maximally planar? Does the answer change if we define ’minimal' with respect to minors rather than subgraphs?
18. Show that adding a new edge to a maximal planar graph of order at least 6 always produces both a \( T{K}^{5} \) and a \( T{K}_{3,3} \) subgraph.
19. Prove the general Kuratowski theorem from its 3-connected case by manipulating plane graphs, i.e. avoiding Lemma 4.4.5.
(This is not intended as an exercise in elementary topology; for the topological parts of the proof, a rough sketch will do.)
20. A graph is called outerplanar if it has a drawing in which every vertex lies on the boundary of the outer face. Show that a graph is outerplanar if and only if it contains neither \( {K}^{4} \) nor \( {K}_{2,3} \) as a minor.
21. Let \( G = {G}_{1} \cup {G}_{2} \), where \( \left| {{G}_{1} \cap {G}_{2}}\right| \leq 1 \) . Show that \( \mathcal{C}\left( G\right) \) has a simple basis if both \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \) have one.
22. \( {}^{ + } \) Find a cycle space basis among the face boundaries of a 2-connected plane graph.
23. Show that a 2-connected plane graph is bipartite if and only if every face is bounded by an even cycle.
24. \( {}^{ + } \) Let \( C \) be a closed curve in the plane that intersects itself at most once in any given point of the plane, and where every such self-intersection is a proper crossing. Call \( C \) alternating if we can turn these crossings into over- and underpasses in such a way that when we run along the curve the overpasses alternate with the underpasses.
(i) Prove that every such curve is alternating, or find a counterexample.
(ii) Does the solution to (i) change if the curves considered are not closed?
25. \( {}^{ - } \) What does the plane dual of a plane tree look like?
26. \( {}^{ - } \) Show that the plane dual of a plane multigraph is connected.
27. \( {}^{ + } \) Show that a connected plane multigraph has a plane dual.
28. Let \( G,{G}^{ * } \) be dual plane multigraphs, and let \( e \in E\left( G\right) \) . Prove the following statements (with a suitable definition of \( G/e \) ):
(i) If \( e \) is not a bridge, then \( {G}^{ * }/{e}^{ * } \) is a plane dual of \( G - e \) .
(ii) If \( e \) is not a loop, then \( {G}^{ * } - {e}^{ * } \) is a plane dual of \( G/e \) .
29. Show that any two plane duals of a plane multigraph are combinatorially isomorphic.
30. Let \( G,{G}^{ * } \) be dual plane graphs. Prove the following statements:
(i) If \( G \) is 2-connected, then \( {G}^{ * } \) is 2-connected.
(ii) If \( G \) is 3-connected, then \( {G}^{ * } \) is 3-connected.
(iii) If \( G \) is 4-connected, then \( {G}^{ * } \) need not be 4-connected.
31. Let \( G,{G}^{ * } \) be dual plane graphs. Let \( {B}_{1},\ldots ,{B}_{n} \) be the blocks of \( G \) . Show that \( {B}_{1}^{ * },\ldots ,{B}_{n}^{ * } \) are the blocks of \( {G}^{ * } \) .
32. Show that if \( {G}^{ * } \) is an abstract dual of a multigraph \( G \), then \( G \) is an abstract dual of \( {G}^{ * } \) .
33. Show that the following statements are equivalent for connected multi-graphs \( G = \left( {V, E}\right) \) and \( {G}^{\prime } = \left( {{V}^{\prime }, E}\right) \) with the same edge set:
(i) \( G \) and \( {G}^{\prime } \) are abstract duals of each other;
(ii) given any set \( F \subseteq E \), the multigraph \( \left( {V, F}\right) \) is a tree if and only if \( \left( {{V}^{\prime }, E \smallsetminus F}\right) \) is a tree.
## Notes
There is a very thorough monograph on the embedding of graphs in surfaces, including the plane: B. Mohar & C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press 2001. Proofs of the results cited in Section 4.1, as well as all references for this chapter, can be found there. A good account of the Jordan curve theorem, both polygonal and general, is given also in J. Stillwell, Classical topology and combinatorial group theory, Springer 1980.
The short proof of Corollary 4.2.10 uses a trick that deserves special mention: the so-called double counting of pairs, illustrated in the text by a bipartite graph whose edges can be counted alternatively by summing its degrees on the left or on the right. Double counting is a technique widely used in combinatorics, and there will be more examples later in the book.
The material of Section 4.3 is not normally standard for an introductory graph theory course, and the rest of the chapter can be read independently of this section. However, the results of Section 4.3 are by no means unimportant. In a way, they have fallen victim to their own success: the shift from a topological to a combinatorial setting for planarity problems which they achieve has made the topological techniques developed there dispensable for most of planarity theory.
In its original version, Kuratowski's theorem was stated only for topological minors; the version for general minors was added by Wagner in 1937. Our proof of the 3-connected case (Lemma 4.4.3) is a weakening of a proof due to C. Thomassen, Planarity and duality of finite and infinite graphs, J. Combin. Theory B 29 (1980), 244-271, which yields a drawing in which all the inner faces are convex (Exercise 16). The existence of such 'convex' drawings for 3-connected planar graphs follows already from the theorem of Steinitz (1922) that these graphs are precisely the 1-skeletons of 3-dimensional convex polyhedra. Compare also W.T. Tutte, How to draw a graph, Proc. London Math. Soc. 13 (1963), 743-767.
As one readily observes, adding an edge to a maximal planar graph (of order at least 6) produces not only a topological \( {K}^{5} \) or \( {K}_{3,3} \), but both. In Chapter 7.3 we shall see that, more generally, every graph with \( n \) vertices and more than \( {3n} - 6 \) edges contains a \( T{K}^{5} \) and, with one easily described class of exceptions, also a \( T{K}_{3,3} \) (Ex. 26, Ch. 7).
The simple cycle space basis constructed in the proof of MacLane's theorem, which consists of the inner face boundaries, is canonical in the following sense: for every simple basis \( \mathcal{B} \) of the cycle space of a 2-connected planar graph there exists a drawing of that graph in which \( \mathcal{B} \) is precisely the set of inner face boundaries. (This is proved in Mohar & Thomassen, who also mention some further planarity criteria.) Our proof of the backward direction of MacLane's theorem is based on Kuratowski's theorem. A more direct approach, in which a planar embedding is actually constructed from a simple basis, is adopted in K. Wagner, Graphentheorie, BI Hochschultaschenbücher 1972.
Theorem 4.5.2 is widely known as 'Tutte's planarity criterion', because it follows at once from Tutte's 1963 Theorem 3.2.3 and the even earlier planarity criterion of MacLane, Theorem 4.5.1. However, Tutte appears to have been unaware of this. Theorem 4.5.2 was first noticed in the late 1970s, and proved independently of both Theorems 3.2.3 and 4.5.1, by A.K. Kelmans, The concept of a vertex in a matroid, the non-separating cycles in a graph and a new criterion for graph planarity, in Algebraic Methods in Graph Theory, Vol. 1, Conf. Szeged 1978, Colloq. Math. Soc. János Bolyai 25 (1981) 345-388. Kelmans also reproved Theorem 3.2.3 (being unaware of Tutte's proof), and noted that it can be combined with MacLane's criterion to a proof of Theorem 4.5.2.
The proper setting for cycle-bond duality in abstract finite graphs (and beyond) is the theory of matroids; see J.G. Oxley, Matroid Theory, Oxford University Press 1992. Duality in infinite graphs is treated in H. Bruhn & R. Diestel, Duality in infinite graphs, Combinatorics, Probability and Computing (to appear).
## Colouring
How many colours do we need to colour the countries of a map in such a way that adjacent countries are coloured differently? How many days have to be scheduled for committee meetings of a parliament if every committee intends to meet for one day and some members of parliament serve on several committees? How can we find a school timetable of minimum total length, based on the information of how often each teacher has to teach each class?
---
vertex colouring
---
A vertex colouring of a graph \( G = \left( {V, E}\right) \) is a map \( c : V \rightarrow S \) such that \( c\left( v\right) \neq c\left( w\right) \) whenever \( v \) and \( w \) are adjacent. The elements of the set \( S \) are called the available colours. All that interests us about \( S \) is its size: typically, we shall be asking for the smallest integer \( k \) such that
---
chromatic number
\( \chi \left( G\right) \)
---
\( G \) has a \( k \) -colouring, a vertex colouring \( c : V \rightarrow \{ 1,\ldots, k\} \) . This \( k \) is the (vertex-) chromatic number of \( G \) ; it is denoted by \( \chi \left( G\right) \) . A graph \( G \) with \( \chi \left( G\right) = k \) is called \( k \) -chromatic; if \( \chi \left( G\right) \leq k \), we call \( {Gk} \) -colourable.
Fig. 5.0.1. A vertex colouring \( V \rightarrow \{ 1,\ldots ,4\} \)
Note that a \( k \) -colouring is nothing but a vertex partition into \( k \) independent sets, now called colour classes; the non-trivial 2-colourable
---
colour classes
---
graphs, for example, are precisely the bipartite graphs. Historically, the colouring terminology comes from the map colouring problem stated above, which leads to the problem of determining the maximum chromatic number of planar graphs. The committee scheduling problem, too,
---
edge colouring
- |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 39 | has a \( k \) -colouring, a vertex colouring \( c : V \rightarrow \{ 1,\ldots, k\} \) . This \( k \) is the (vertex-) chromatic number of \( G \) ; it is denoted by \( \chi \left( G\right) \) . A graph \( G \) with \( \chi \left( G\right) = k \) is called \( k \) -chromatic; if \( \chi \left( G\right) \leq k \), we call \( {Gk} \) -colourable.
Fig. 5.0.1. A vertex colouring \( V \rightarrow \{ 1,\ldots ,4\} \)
Note that a \( k \) -colouring is nothing but a vertex partition into \( k \) independent sets, now called colour classes; the non-trivial 2-colourable
---
colour classes
---
graphs, for example, are precisely the bipartite graphs. Historically, the colouring terminology comes from the map colouring problem stated above, which leads to the problem of determining the maximum chromatic number of planar graphs. The committee scheduling problem, too,
---
edge colouring
---
can be phrased as a vertex colouring problem-how?
An edge colouring of \( G = \left( {V, E}\right) \) is a map \( c : E \rightarrow S \) with \( c\left( e\right) \neq c\left( f\right) \) for any adjacent edges \( e, f \) . The smallest integer \( k \) for which a \( k \) -edge-
---
chromatic index \( {\chi }^{\prime }\left( G\right) \)
---
colouring exists, i.e. an edge colouring \( c : E \rightarrow \{ 1,\ldots, k\} \), is the edge-chromatic number, or chromatic index, of \( G \) ; it is denoted by \( {\chi }^{\prime }\left( G\right) \) . The third of our introductory questions can be modelled as an edge colouring problem in a bipartite multigraph (how?).
Clearly, every edge colouring of \( G \) is a vertex colouring of its line graph \( L\left( G\right) \), and vice versa; in particular, \( {\chi }^{\prime }\left( G\right) = \chi \left( {L\left( G\right) }\right) \) . The problem of finding good edge colourings may thus be viewed as a restriction of the more general vertex colouring problem to this special class of graphs. As we shall see, this relationship between the two types of colouring problem is reflected by a marked difference in our knowledge about their solutions: while there are only very rough estimates for \( \chi \) , its sister \( {\chi }^{\prime } \) always takes one of two values, either \( \Delta \) or \( \Delta + 1 \) .
## 5.1 Colouring maps and planar graphs
If any result in graph theory has a claim to be known to the world outside, it is the following four colour theorem (which implies that every map can be coloured with at most four colours):
Theorem 5.1.1. (Four Colour Theorem)
Every planar graph is 4-colourable.
Some remarks about the proof of the four colour theorem and its history can be found in the notes at the end of this chapter. Here, we prove the following weakening:
Proposition 5.1.2. (Five Colour Theorem)
Every planar graph is 5-colourable.
(4.1.1)
Proof. Let \( G \) be a plane graph with \( n \geq 6 \) vertices and \( m \) edges. We (4.2.10) assume inductively that every plane graph with fewer than \( n \) vertices
\( n, m \) can be 5-coloured. By Corollary 4.2.10,
\[
d\left( G\right) = {2m}/n \leq 2\left( {{3n} - 6}\right) /n < 6;
\]
\( v \) let \( v \in G \) be a vertex of degree at most 5 . By the induction hypothesis, \( H \) the graph \( H \mathrel{\text{:=}} G - v \) has a vertex colouring \( c : V\left( H\right) \rightarrow \{ 1,\ldots ,5\} \) . If \( c \) \( c \) uses at most 4 colours for the neighbours of \( v \), we can extend it to a 5- colouring of \( G \) . Let us assume, therefore, that \( v \) has exactly 5 neighbours, and that these have distinct colours.
Let \( D \) be an open disc around \( v \), so small that it meets only those five straight edge segments of \( G \) that contain \( v \) . Let us enumerate these
segments according to their cyclic position in \( D \) as \( {s}_{1},\ldots ,{s}_{5} \), and let \( {s}_{1},\ldots ,{s}_{5} \) \( v{v}_{i} \) be the edge containing \( {s}_{i}\left( {i = 1,\ldots ,5\text{; Fig. 5.1.1}}\right) \) . Without loss of \( {v}_{1},\ldots ,{v}_{5} \) generality we may assume that \( c\left( {v}_{i}\right) = i \) for each \( i \) .
Fig. 5.1.1. The proof of the five colour theorem
Let us show first that every \( {v}_{1} - {v}_{3} \) path \( P \subseteq H \) separates \( {v}_{2} \) from \( {v}_{4} \) in \( H \) . Clearly, this is the case if and only if the cycle \( C \mathrel{\text{:=}} v{v}_{1}P{v}_{3}v \) separates \( {v}_{2} \) from \( {v}_{4} \) in \( G \) . We prove this by showing that \( {v}_{2} \) and \( {v}_{4} \) lie in different faces of \( C \) .
Let us pick an inner point \( {x}_{2} \) of \( {s}_{2} \) in \( D \) and an inner point \( {x}_{4} \) of \( {s}_{4} \) in \( D \) . Then in \( D \smallsetminus \left( {{s}_{1} \cup {s}_{3}}\right) \subseteq {\mathbb{R}}^{2} \smallsetminus C \) every point can be linked by a polygonal arc to \( {x}_{2} \) or to \( {x}_{4} \) . This implies that \( {x}_{2} \) and \( {x}_{4} \) (and hence also \( {v}_{2} \) and \( {v}_{4} \) ) lie in different faces of \( C \) : otherwise \( D \) would meet only one of the two faces of \( C \), which would contradict the fact that \( v \) lies on the frontier of both these faces (Theorem 4.1.1).
Given \( i, j \in \{ 1,\ldots ,5\} \), let \( {H}_{i, j} \) be the subgraph of \( H \) induced by the vertices coloured \( i \) or \( j \) . We may assume that the component \( {C}_{1} \) of \( {H}_{1,3} \) containing \( {v}_{1} \) also contains \( {v}_{3} \) . Indeed, if we interchange the colours 1 and 3 at all the vertices of \( {C}_{1} \), we obtain another 5-colouring of \( H \) ; if \( {v}_{3} \notin {C}_{1} \), then \( {v}_{1} \) and \( {v}_{3} \) are both coloured 3 in this new colouring, and we may assign colour 1 to \( v \) . Thus, \( {H}_{1,3} \) contains a \( {v}_{1} - {v}_{3} \) path \( P \) . As shown above, \( P \) separates \( {v}_{2} \) from \( {v}_{4} \) in \( H \) . Since \( P \cap {H}_{2,4} = \varnothing \) , this means that \( {v}_{2} \) and \( {v}_{4} \) lie in different components of \( {H}_{2,4} \) . In the component containing \( {v}_{2} \), we now interchange the colours 2 and 4, thus recolouring \( {v}_{2} \) with colour 4 . Now \( v \) no longer has a neighbour coloured 2, and we may give it this colour.
As a backdrop to the two famous theorems above, let us cite another well-known result:
Theorem 5.1.3. (Grötzsch 1959)
Every planar graph not containing a triangle is 3-colourable.
## 5.2 Colouring vertices
How do we determine the chromatic number of a given graph? How can we find a vertex-colouring with as few colours as possible? How does the chromatic number relate to other graph invariants, such as average degree, connectivity or girth?
Straight from the definition of the chromatic number we may derive the following upper bound:
Proposition 5.2.1. Every graph \( G \) with \( m \) edges satisfies
\[
\chi \left( G\right) \leq \frac{1}{2} + \sqrt{{2m} + \frac{1}{4}}.
\]
Proof. Let \( c \) be a vertex colouring of \( G \) with \( k = \chi \left( G\right) \) colours. Then \( G \) has at least one edge between any two colour classes: if not, we could have used the same colour for both classes. Thus, \( m \geq \frac{1}{2}k\left( {k - 1}\right) \) . Solving this inequality for \( k \), we obtain the assertion claimed.
---
greedy algorithm
---
One obvious way to colour a graph \( G \) with not too many colours is the following greedy algorithm: starting from a fixed vertex enumeration \( {v}_{1},\ldots ,{v}_{n} \) of \( G \), we consider the vertices in turn and colour each \( {v}_{i} \) with the first available colour-e.g., with the smallest positive integer not already used to colour any neighbour of \( {v}_{i} \) among \( {v}_{1},\ldots ,{v}_{i - 1} \) . In this way, we never use more than \( \Delta \left( G\right) + 1 \) colours, even for unfavourable choices of the enumeration \( {v}_{1},\ldots ,{v}_{n} \) . If \( G \) is complete or an odd cycle, then this is even best possible.
In general, though, this upper bound of \( \Delta + 1 \) is rather generous, even for greedy colourings. Indeed, when we come to colour the vertex \( {v}_{i} \) in the above algorithm, we only need a supply of \( {d}_{G\left\lbrack {{v}_{1},\ldots ,{v}_{i}}\right\rbrack }\left( {v}_{i}\right) + 1 \) rather than \( {d}_{G}\left( {v}_{i}\right) + 1 \) colours to proceed; recall that, at this stage, the algorithm ignores any neighbours \( {v}_{j} \) of \( {v}_{i} \) with \( j > i \) . Hence in most graphs, there will be scope for an improvement of the \( \Delta + 1 \) bound by choosing a particularly suitable vertex ordering to start with: one that picks vertices of large degree early (when most neighbours are ignored) and vertices of small degree last. Locally, the number \( {d}_{G\left\lbrack {{v}_{1},\ldots ,{v}_{i}}\right\rbrack }\left( {v}_{i}\right) + 1 \) of colours required will be smallest if \( {v}_{i} \) has minimum degree in \( G\left\lbrack {{v}_{1},\ldots ,{v}_{i}}\right\rbrack \) . But this is easily achieved: we just choose \( {v}_{n} \) first, with \( d\left( {v}_{n}\right) = \delta \left( G\right) \), then choose as \( {v}_{n - 1} \) a vertex of minimum degree in \( G - {v}_{n} \), and so on.
The least number \( k \) such that \( G \) has a vertex enumeration in which
---
colouring number \( \operatorname{col}\left( G\right) \)
---
each vertex is preceded by fewer than \( k \) of its neighbours is called the colouring number \( \operatorname{col}\left( G\right) \) of \( G \) . The enumeration we just discussed shows that \( \operatorname{col}\left( G\right) \leq \mathop{\max }\limits_{{H \subseteq G}}\delta \left( H\right) + 1 \) . But for \( H \subseteq G \) clearly also \( \operatorname{col}\left( G\right) \geq \operatorname{col}\left( H\right) \) and \( \operatorname{col}\left( H\right) \geq \del |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 40 | {v}_{i}\right) + 1 \) of colours required will be smallest if \( {v}_{i} \) has minimum degree in \( G\left\lbrack {{v}_{1},\ldots ,{v}_{i}}\right\rbrack \) . But this is easily achieved: we just choose \( {v}_{n} \) first, with \( d\left( {v}_{n}\right) = \delta \left( G\right) \), then choose as \( {v}_{n - 1} \) a vertex of minimum degree in \( G - {v}_{n} \), and so on.
The least number \( k \) such that \( G \) has a vertex enumeration in which
---
colouring number \( \operatorname{col}\left( G\right) \)
---
each vertex is preceded by fewer than \( k \) of its neighbours is called the colouring number \( \operatorname{col}\left( G\right) \) of \( G \) . The enumeration we just discussed shows that \( \operatorname{col}\left( G\right) \leq \mathop{\max }\limits_{{H \subseteq G}}\delta \left( H\right) + 1 \) . But for \( H \subseteq G \) clearly also \( \operatorname{col}\left( G\right) \geq \operatorname{col}\left( H\right) \) and \( \operatorname{col}\left( H\right) \geq \delta \left( H\right) + 1 \), since the ’back-degree’ of the last vertex in any enumeration of \( H \) is just its ordinary degree in \( H \) , which is at least \( \delta \left( H\right) \) . So we have proved the following:
Proposition 5.2.2. Every graph \( G \) satisfies
\[
\chi \left( G\right) \leq \operatorname{col}\left( G\right) = \max \{ \delta \left( H\right) \mid H \subseteq G\} + 1.
\]
Corollary 5.2.3. Every graph \( G \) has a subgraph of minimum degree at least \( \chi \left( G\right) - 1 \) .
The colouring number of a graph is closely related to its arboricity; see the remark following Theorem 2.4.4.
As we have seen, every graph \( G \) satisfies \( \chi \left( G\right) \leq \Delta \left( G\right) + 1 \), with equality for complete graphs and odd cycles. In all other cases, this general bound can be improved a little:
Theorem 5.2.4. (Brooks 1941)
Let \( G \) be a connected graph. If \( G \) is neither complete nor an odd cycle, then
\[
\chi \left( G\right) \leq \Delta \left( G\right)
\]
Proof. We apply induction on \( \left| G\right| \) . If \( \Delta \left( G\right) \leq 2 \), then \( G \) is a path or a cycle, and the assertion is trivial. We therefore assume that \( \Delta \mathrel{\text{:=}} \) \( \Delta \left( G\right) \geq 3 \), and that the assertion holds for graphs of smaller order. Suppose that \( \chi \left( G\right) > \Delta \) .
Let \( v \in G \) be a vertex and \( H \mathrel{\text{:=}} G - v \) . Then \( \chi \left( H\right) \leq \Delta \) : by
\( v, H \)
induction, every component \( {H}^{\prime } \) of \( H \) satisfies \( \chi \left( {H}^{\prime }\right) \leq \Delta \left( {H}^{\prime }\right) \leq \Delta \) unless \( {H}^{\prime } \) is complete or an odd cycle, in which case \( \chi \left( {H}^{\prime }\right) = \Delta \left( {H}^{\prime }\right) + 1 \leq \Delta \) as every vertex of \( {H}^{\prime } \) has maximum degree in \( {H}^{\prime } \) and one such vertex is also adjacent to \( v \) in \( G \) .
Since \( H \) can be \( \Delta \) -coloured but \( G \) cannot, we have the following:
Every \( \Delta \) -colouring of \( H \) uses all the colours \( 1,\ldots ,\Delta \) on
(1)
the neighbours of \( v \) ; in particular, \( d\left( v\right) = \Delta \) .
Given any \( \Delta \) -colouring of \( H \), let us denote the neighbour of \( v \) coloured \( i \) by \( {v}_{i}, i = 1,\ldots ,\Delta \) . For all \( i \neq j \), let \( {H}_{i, j} \) denote the subgraph
---
\( {v}_{1},\ldots ,{v}_{\Delta } \) \( {H}_{i, j} \)
---
of \( H \) spanned by all the vertices coloured \( i \) or \( j \) .
For all \( i \neq j \), the vertices \( {v}_{i} \) and \( {v}_{j} \) lie in a common com-
ponent \( {C}_{i, j} \) of \( {H}_{i, j} \) .
(2)
Otherwise we could interchange the colours \( i \) and \( j \) in one of those components; then \( {v}_{i} \) and \( {v}_{j} \) would be coloured the same, contrary to (1).
\[
{C}_{i, j}\text{is always a}{v}_{i} - {v}_{j}\text{path.}
\]
(3)
Indeed, let \( P \) be a \( {v}_{i} - {v}_{j} \) path in \( {C}_{i, j} \) . As \( {d}_{H}\left( {v}_{i}\right) \leq \Delta - 1 \), the neighbours of \( {v}_{i} \) have pairwise different colours: otherwise we could recolour \( {v}_{i} \) , contrary to (1). Hence the neighbour of \( {v}_{i} \) on \( P \) is its only neighbour in \( {C}_{i, j} \), and similarly for \( {v}_{j} \) . Thus if \( {C}_{i, j} \neq P \), then \( P \) has an inner vertex with three identically coloured neighbours in \( H \) ; let \( u \) be the first such vertex on \( P \) (Fig. 5.2.1). Since at most \( \Delta - 2 \) colours are used on the neighbours of \( u \), we may recolour \( u \) . But this makes \( P\overset{ \circ }{u} \) into a component of \( {H}_{i, j} \), contradicting (2).
Fig. 5.2.1. The proof of (3) in Brooks's theorem
For distinct \( i, j, k \), the paths \( {C}_{i, j} \) and \( {C}_{i, k} \) meet only in \( {v}_{i} \) .
(4)
For if \( {v}_{i} \neq u \in {C}_{i, j} \cap {C}_{i, k} \), then \( u \) has two neighbours coloured \( j \) and two coloured \( k \), so we may recolour \( u \) . In the new colouring, \( {v}_{i} \) and \( {v}_{j} \) lie in different components of \( {H}_{i, j} \), contrary to (2).
The proof of the theorem now follows easily. If the neighbours of \( v \) are pairwise adjacent, then each has \( \Delta \) neighbours in \( N\left( v\right) \cup \{ v\} \) already, so \( G = G\left\lbrack {N\left( v\right) \cup \{ v\} }\right\rbrack = {K}^{\Delta + 1} \) . As \( G \) is complete, there is nothing to
---
\( {v}_{1},\ldots ,{v}_{\Delta } \) \( c \) \( u \) \( {c}^{\prime } \)
---
show. We may thus assume that \( {v}_{1}{v}_{2} \notin G \), where \( {v}_{1},\ldots ,{v}_{\Delta } \) derive their names from some fixed \( \Delta \) -colouring \( c \) of \( H \) . Let \( u \neq {v}_{2} \) be the neighbour of \( {v}_{1} \) on the path \( {C}_{1,2} \) ; then \( c\left( u\right) = 2 \) . Interchanging the colours 1 and 3 in \( {C}_{1,3} \), we obtain a new colouring \( {c}^{\prime } \) of \( H \) ; let \( {v}_{i}^{\prime },{H}_{i, j}^{\prime },{C}_{i, j}^{\prime } \) etc. be defined with respect to \( {c}^{\prime } \) in the obvious way. As a neighbour of \( {v}_{1} = {v}_{3}^{\prime } \), our vertex \( u \) now lies in \( {C}_{2,3}^{\prime } \), since \( {c}^{\prime }\left( u\right) = c\left( u\right) = 2 \) . By (4) for \( c \), however, the path \( {\overset{ \circ }{v}}_{1}{C}_{1,2} \) retained its original colouring, so \( u \in {\overset{ \circ }{v}}_{1}{C}_{1,2} \subseteq {C}_{1,2}^{\prime } \) . Hence \( u \in {C}_{2,3}^{\prime } \cap {C}_{1,2}^{\prime } \), contradicting (4) for \( {c}^{\prime } \) .
As we have seen, a graph \( G \) of large chromatic number must have large maximum degree: trivially at least \( \chi \left( G\right) - 1 \), and less trivially at least \( \chi \left( G\right) \) (in most cases). What more can we say about the structure of graphs with large chromatic number?
One obvious possible cause for \( \chi \left( G\right) \geq k \) is the presence of a \( {K}^{k} \) subgraph. This is a local property of \( G \), compatible with arbitrary values of global invariants such as \( \varepsilon \) and \( \kappa \) . Hence, the assumption of \( \chi \left( G\right) \geq k \) does not tell us anything about those invariants for \( G \) itself. It does, however, imply the existence of a subgraph where those invariants are large: by Corollary 5.2.3, \( G \) has a subgraph \( H \) with \( \delta \left( H\right) \geq k - 1 \), and hence by Theorem 1.4.3 a subgraph \( {H}^{\prime } \) with \( \kappa \left( {H}^{\prime }\right) \geq \left\lfloor {\frac{1}{4}\left( {k - 1}\right) }\right\rfloor \) .
But is, conversely, the somewhat higher density of those subgraphs in any sense the ’cause’ for \( \chi \) to be large? That is to say, do arbitrary graphs with such values of \( \delta \) and \( \kappa \) in turn have large chromatic number, say at least \( f\left( k\right) \) for some function \( f : \mathbb{N} \rightarrow \mathbb{N} \) tending to infinity (however slowly)? No, not at all: the graphs \( {K}_{n, n} \), for example, have a minimum degree and connectivity that exceeds any bound in terms of \( k \) as \( n \rightarrow \infty \) , but are only 2-chromatic. Thus, the sort of large (constant \( {}^{1} \) ) average or minimum degree that a high chromatic number can force in a suitable subgraph is itself not nearly large enough to force even \( \chi > 2 \) .
Yet even if local edge density is not by itself responsible for \( \chi \) to be large, it might still be the case that, somehow, a chromatic number of at least \( k \) forces the existence of one of finitely many ’canonical’ subgraphs of chromatic number at least, say, \( f\left( k\right) \) (with \( f \) as above). However, this is radically not the case: as soon as a graph \( H \) contains a cycle (which highly chromatic graphs clearly do), we cannot force an arbitrary graph \( G \) to contain a copy of \( H \) just by making \( \chi \left( G\right) \) large enough:
## Theorem 5.2.5. (Erdős 1959)
For every integer \( k \) there exists a graph \( G \) with girth \( g\left( G\right) > k \) and chromatic number \( \chi \left( G\right) > k \) .
Theorem 5.2.5 was first proved non-constructively using random graphs, and we shall give this proof in Chapter 11.2. Constructing graphs of large chromatic number and girth directly is not easy; cf. Exercise 23 for the simplest case.
The message of Erdős's theorem is that, contrary to our initial guess, large chromatic number can occur as a purely global phenomenon: note that locally, around each vertex, a graph of large girth looks just like a tree, and in particular is 2-colourable there. But what exactly can cause high chromaticity as a global phenomenon remains a mystery.
Nevertheless, there exists a simple though not always short-procedure to construct all the graphs of chromatic n |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 41 | not force an arbitrary graph \( G \) to contain a copy of \( H \) just by making \( \chi \left( G\right) \) large enough:
## Theorem 5.2.5. (Erdős 1959)
For every integer \( k \) there exists a graph \( G \) with girth \( g\left( G\right) > k \) and chromatic number \( \chi \left( G\right) > k \) .
Theorem 5.2.5 was first proved non-constructively using random graphs, and we shall give this proof in Chapter 11.2. Constructing graphs of large chromatic number and girth directly is not easy; cf. Exercise 23 for the simplest case.
The message of Erdős's theorem is that, contrary to our initial guess, large chromatic number can occur as a purely global phenomenon: note that locally, around each vertex, a graph of large girth looks just like a tree, and in particular is 2-colourable there. But what exactly can cause high chromaticity as a global phenomenon remains a mystery.
Nevertheless, there exists a simple though not always short-procedure to construct all the graphs of chromatic number at least \( k \) . For
---
\( k \) -constructible
---
each \( k \in \mathbb{N} \), let us define the class of \( k \) -constructible graphs recursively as follows:
(i) \( {K}^{k} \) is \( k \) -constructible.
(ii) If \( G \) is \( k \) -constructible and \( x, y \in V\left( G\right) \) are non-adjacent, then also \( \left( {G + {xy}}\right) /{xy} \) is \( k \) -constructible.
(iii) If \( {G}_{1},{G}_{2} \) are \( k \) -constructible and there are vertices \( x,{y}_{1},{y}_{2} \) such that \( {G}_{1} \cap {G}_{2} = \{ x\} \) and \( x{y}_{1} \in E\left( {G}_{1}\right) \) and \( x{y}_{2} \in E\left( {G}_{2}\right) \), then also \( \left( {{G}_{1} \cup {G}_{2}}\right) - x{y}_{1} - x{y}_{2} + {y}_{1}{y}_{2} \) is \( k \) -constructible (Fig. 5.2.2).
---
1 Which non-constant average degree will force the existence of a given subgraph will be a topic in Chapter 7.
---
Fig. 5.2.2. The Hajós construction (iii)
One easily checks inductively that all \( k \) -constructible graphs - and hence their supergraphs - are at least \( k \) -chromatic. Indeed, if \( \left( {G + {xy}}\right) /{xy} \) as in (ii) has a colouring with fewer than \( k \) colours, then this defines such a colouring also for \( G \), a contradiction. Similarly, in any colouring of the graph constructed in (iii), the vertices \( {y}_{1} \) and \( {y}_{2} \) do not both have the same colour as \( x \), so this colouring induces a colouring of either \( {G}_{1} \) or \( {G}_{2} \) and hence uses at least \( k \) colours.
It is remarkable, though, that the converse holds too:
Theorem 5.2.6. (Hajós 1961)
Let \( G \) be a graph and \( k \in \mathbb{N} \) . Then \( \chi \left( G\right) \geq k \) if and only if \( G \) has a \( k \) -constructible subgraph.
Proof. Let \( G \) be a graph with \( \chi \left( G\right) \geq k \) ; we show that \( G \) has a \( k \) - constructible subgraph. Suppose not; then \( k \geq 3 \) . Adding some edges if necessary, let us make \( G \) edge-maximal with the property that none of its subgraphs is \( k \) -constructible. Now \( G \) is not a complete \( r \) -partite graph for any \( r \) : for then \( \chi \left( G\right) \geq k \) would imply \( r \geq k \), and \( G \) would contain the \( k \) -constructible graph \( {K}^{k} \) .
Since \( G \) is not a complete multipartite graph, non-adjacency is not an equivalence relation on \( V\left( G\right) \) . So there are vertices \( {y}_{1}, x,{y}_{2} \) such that
\( x,{y}_{1},{y}_{2} \) \( {y}_{1}x, x{y}_{2} \notin E\left( G\right) \) but \( {y}_{1}{y}_{2} \in E\left( G\right) \) . Since \( G \) is edge-maximal without a \( k \) -constructible subgraph, each edge \( x{y}_{i} \) lies in some \( k \) -constructible
\( {H}_{1},{H}_{2} \) subgraph \( {H}_{i} \) of \( G + x{y}_{i}\left( {i = 1,2}\right) \) .
\( {H}_{2}^{\prime } \) Let \( {H}_{2}^{\prime } \) be an isomorphic copy of \( {H}_{2} \) that contains \( x \) and \( {H}_{2} - {H}_{1} \) \( {v}^{\prime } \) etc. but is otherwise disjoint from \( G \), together with an isomorphism \( v \mapsto {v}^{\prime } \) from \( {H}_{2} \) to \( {H}_{2}^{\prime } \) that fixes \( {H}_{2} \cap {H}_{2}^{\prime } \) pointwise. Then \( {H}_{1} \cap {H}_{2}^{\prime } = \{ x\} \), so
\[
H \mathrel{\text{:=}} \left( {{H}_{1} \cup {H}_{2}^{\prime }}\right) - x{y}_{1} - x{y}_{2}^{\prime } + {y}_{1}{y}_{2}^{\prime }
\]
is \( k \) -constructible by (iii). One vertex at a time, let us identify in \( H \) each vertex \( {v}^{\prime } \in {H}_{2}^{\prime } - G \) with its partner \( v \) ; since \( v{v}^{\prime } \) is never an edge of \( H \) , each of these identifications amounts to a construction step of type (ii). Eventually, we obtain the graph
\[
\left( {{H}_{1} \cup {H}_{2}}\right) - x{y}_{1} - x{y}_{2} + {y}_{1}{y}_{2} \subseteq G
\]
this is the desired \( k \) -constructible subgraph of \( G \) .
## 5.3 Colouring edges
Clearly, every graph \( G \) satisfies \( {\chi }^{\prime }\left( G\right) \geq \Delta \left( G\right) \) . For bipartite graphs, we have equality here:
Proposition 5.3.1. (König 1916)
\( \left\lbrack {5.4.5}\right\rbrack \)
Every bipartite graph \( G \) satisfies \( {\chi }^{\prime }\left( G\right) = \Delta \left( G\right) \) .
Proof. We apply induction on \( \parallel G\parallel \) . For \( \parallel G\parallel = 0 \) the assertion holds.
(1.6.1)
Now assume that \( \parallel G\parallel \geq 1 \), and that the assertion holds for graphs with fewer edges. Let \( \Delta \mathrel{\text{:=}} \Delta \left( G\right) \), pick an edge \( {xy} \in G \), and choose a \( \Delta \) -
\( \Delta ,{xy} \)
edge-colouring of \( G - {xy} \) by the induction hypothesis. Let us refer to the edges coloured \( \alpha \) as \( \alpha \) -edges, etc. \( \alpha \) -edge
In \( G - {xy} \), each of \( x \) and \( y \) is incident with at most \( \Delta - 1 \) edges. Hence there are \( \alpha ,\beta \in \{ 1,\ldots ,\Delta \} \) such that \( x \) is not incident with an \( \alpha ,\beta \) \( \alpha \) -edge and \( y \) is not incident with a \( \beta \) -edge. If \( \alpha = \beta \), we can colour the edge \( {xy} \) with this colour and are done; so we may assume that \( \alpha \neq \beta \) , and that \( x \) is incident with a \( \beta \) -edge.
Let us extend this edge to a maximal walk \( W \) from \( x \) whose edges are coloured \( \beta \) and \( \alpha \) alternately. Since no such walk contains a vertex twice (why not?), \( W \) exists and is a path. Moreover, \( W \) does not contain \( y \) : if it did, it would end in \( y \) on an \( \alpha \) -edge (by the choice of \( \beta \) ) and thus have even length, so \( W + {xy} \) would be an odd cycle in \( G \) (cf. Proposition 1.6.1). We now recolour all the edges on \( W \), swapping \( \alpha \) with \( \beta \) . By the choice of \( \alpha \) and the maximality of \( W \), adjacent edges of \( G - {xy} \) are still coloured differently. We have thus found a \( \Delta \) -edge-colouring of \( G - {xy} \) in which neither \( x \) nor \( y \) is incident with a \( \beta \) -edge. Colouring \( {xy} \) with \( \beta \) , we extend this colouring to a \( \Delta \) -edge-colouring of \( G \) .
Theorem 5.3.2. (Vizing 1964)
Every graph \( G \) satisfies
\[
\Delta \left( G\right) \leq {\chi }^{\prime }\left( G\right) \leq \Delta \left( G\right) + 1.
\]
Proof. We prove the second inequality by induction on \( \parallel G\parallel \) . For \( \parallel G\parallel = 0 \)
\( V, E \)
it is trivial. For the induction step let \( G = \left( {V, E}\right) \) with \( \Delta \mathrel{\text{:=}} \Delta \left( G\right) > 0 \) be given, and assume that the assertion holds for graphs with fewer edges. Instead of ’ \( \left( {\Delta + 1}\right) \) -edge-colouring’ let us just say ’colouring’. An edge
---
colouring \( \alpha \) -edge
---
coloured \( \alpha \) will again be called an \( \alpha \) -edge.
For every edge \( e \in G \) there exists a colouring of \( G - e \), by the induction hypothesis. In such a colouring, the edges at a given vertex \( v \) use at most \( d\left( v\right) \leq \Delta \) colours, so some colour \( \beta \in \{ 1,\ldots ,\Delta + 1\} \) is missing at \( v \) . For any other colour \( \alpha \), there is a unique maximal walk missing (possibly trivial) starting at \( v \), whose edges are coloured alternately \( \alpha \) and \( \beta \) . This walk is a path; we call it the \( \alpha /\beta \) -path from \( v \) .
\( \alpha /\beta \) -path
Suppose that \( G \) has no colouring. Then the following holds:
Given \( {xy} \in E \), and any colouring of \( G - {xy} \) in which the colour \( \alpha \) is missing at \( x \) and the colour \( \beta \) is missing at \( y \) ,
(1)
the \( \alpha /\beta \) -path from \( y \) ends in \( x \) .
Otherwise we could interchange the colours \( \alpha \) and \( \beta \) along this path and colour \( {xy} \) with \( \alpha \), obtaining a colouring of \( G \) (contradiction).
---
\( x{y}_{0} \)
\( {G}_{0},{c}_{0},\alpha \)
\( {y}_{1},\ldots ,{y}_{k} \)
---
Let \( x{y}_{0} \in G \) be an edge. By induction, \( {G}_{0} \mathrel{\text{:=}} G - x{y}_{0} \) has a colouring \( {c}_{0} \) . Let \( \alpha \) be a colour missing at \( x \) in this colouring. Further, let \( {y}_{0},{y}_{1},\ldots ,{y}_{k} \) be a maximal sequence of distinct neighbours of \( x \) in \( G \) , such that \( {c}_{0}\left( {x{y}_{i}}\right) \) is missing in \( {c}_{0} \) at \( {y}_{i - 1} \) for each \( i = 1,\ldots, k \) . For each \( {G}_{i} \) of the graphs \( {G}_{i} \mathrel{\text{:=}} G - x{y}_{i} \) we define a colouring \( {c}_{i} \), setting
\( {c}_{i} \)
\[
{c}_{i}\left( e\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} {c}_{0}\left( {x{y}_{j + 1}}\right) & \text{ for }e = x{y}_{j}\text{ with }j \in \{ 0,\ldots, i - 1\} \\ {c}_{0}\left( e\right) & \text{ otherwise; } \end{array}\right.
\]
note that in each of these |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 42 | obtaining a colouring of \( G \) (contradiction).
---
\( x{y}_{0} \)
\( {G}_{0},{c}_{0},\alpha \)
\( {y}_{1},\ldots ,{y}_{k} \)
---
Let \( x{y}_{0} \in G \) be an edge. By induction, \( {G}_{0} \mathrel{\text{:=}} G - x{y}_{0} \) has a colouring \( {c}_{0} \) . Let \( \alpha \) be a colour missing at \( x \) in this colouring. Further, let \( {y}_{0},{y}_{1},\ldots ,{y}_{k} \) be a maximal sequence of distinct neighbours of \( x \) in \( G \) , such that \( {c}_{0}\left( {x{y}_{i}}\right) \) is missing in \( {c}_{0} \) at \( {y}_{i - 1} \) for each \( i = 1,\ldots, k \) . For each \( {G}_{i} \) of the graphs \( {G}_{i} \mathrel{\text{:=}} G - x{y}_{i} \) we define a colouring \( {c}_{i} \), setting
\( {c}_{i} \)
\[
{c}_{i}\left( e\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} {c}_{0}\left( {x{y}_{j + 1}}\right) & \text{ for }e = x{y}_{j}\text{ with }j \in \{ 0,\ldots, i - 1\} \\ {c}_{0}\left( e\right) & \text{ otherwise; } \end{array}\right.
\]
note that in each of these colourings the same colours are missing at \( x \) as in \( {c}_{0} \) .
\( \beta \) Now let \( \beta \) be a colour missing at \( {y}_{k} \) in \( {c}_{0} \) . Clearly, \( \beta \) is still missing at \( {y}_{k} \) in \( {c}_{k} \) . If \( \beta \) were also missing at \( x \), we could colour \( x{y}_{k} \) with \( \beta \) and thus extend \( {c}_{k} \) to a colouring of \( G \) . Hence, \( x \) is incident with a \( \beta \) -edge (in every colouring). By the maximality of \( k \), therefore, there is an \( i \in \{ 1,\ldots, k - 1\} \) such that
\[
{c}_{0}\left( {x{y}_{i}}\right) = \beta
\]
(2)
Fig. 5.3.1. The \( \alpha /\beta \) -path \( P \) in \( {G}_{k} \)
Let \( P \) be the \( \alpha /\beta \) -path from \( {y}_{k} \) in \( {G}_{k} \) (with respect to \( {c}_{k} \) ; Fig. 5.3.1). By (1), \( P \) ends in \( x \), and it does so on a \( \beta \) -edge, since \( \alpha \) is missing at \( x \) . As \( \beta = {c}_{0}\left( {x{y}_{i}}\right) = {c}_{k}\left( {x{y}_{i - 1}}\right) \), this is the edge \( x{y}_{i - 1} \) . In \( {c}_{0} \), however, and hence also in \( {c}_{i - 1},\beta \) is missing at \( {y}_{i - 1} \) (by (2) and the choice of \( {y}_{i} \) ); let \( {P}^{\prime } \) be the \( \alpha /\beta \) -path from \( {y}_{i - 1} \) in \( {G}_{i - 1} \) (with respect to \( {c}_{i - 1} \) ). Since \( {P}^{\prime } \) is uniquely determined, it starts with \( {y}_{i - 1}P{y}_{k} \) ; note that the edges of \( P\mathring{x} \) are coloured the same in \( {c}_{i - 1} \) as in \( {c}_{k} \) . But in \( {c}_{0} \), and hence in \( {c}_{i - 1} \) , there is no \( \beta \) -edge at \( {y}_{k} \) (by the choice of \( \beta \) ). Therefore \( {P}^{\prime } \) ends in \( {y}_{k} \) , contradicting (1).
Vizing's theorem divides the finite graphs into two classes according to their chromatic index; graphs satisfying \( {\chi }^{\prime } = \Delta \) are called (imaginatively) class 1, those with \( {\chi }^{\prime } = \Delta + 1 \) are class 2 .
## 5.4 List colouring
In this section, we take a look at a relatively recent generalization of the concepts of colouring studied so far. This generalization may seem a little far-fetched at first glance, but it turns out to supply a fundamental link between the classical (vertex and edge) chromatic numbers of a graph and its other invariants.
Suppose we are given a graph \( G = \left( {V, E}\right) \), and for each vertex of \( G \) a list of colours permitted at that particular vertex: when can we colour \( G \) (in the usual sense) so that each vertex receives a colour from its list? More formally, let \( {\left( {S}_{v}\right) }_{v \in V} \) be a family of sets. We call a vertex colouring \( c \) of \( G \) with \( c\left( v\right) \in {S}_{v} \) for all \( v \in V \) a colouring from the
lists \( {S}_{v} \) . The graph \( G \) is called \( k \) -list-colourable, or \( k \) -choosable, if, for \( k \) -choosable every family \( {\left( {S}_{v}\right) }_{v \in V} \) with \( \left| {S}_{v}\right| = k \) for all \( v \), there is a vertex colouring
of \( G \) from the lists \( {S}_{v} \) . The least integer \( k \) for which \( G \) is \( k \) -choosable is choice the list-chromatic number, or choice number \( \operatorname{ch}\left( G\right) \) of \( G \) . number
List-colourings of edges are defined analogously. The least integer
\( \operatorname{ch}\left( G\right) \)
\( k \) such that \( G \) has an edge colouring from any family of lists of size \( k \) is the list-chromatic index \( {\operatorname{ch}}^{\prime }\left( G\right) \) of \( G \) ; formally, we just set \( {\operatorname{ch}}^{\prime }\left( G\right) \mathrel{\text{:=}} \)
\( {\operatorname{ch}}^{\prime }\left( G\right) \)
\( \operatorname{ch}\left( {L\left( G\right) }\right) \), where \( L\left( G\right) \) is the line graph of \( G \) .
In principle, showing that a given graph is \( k \) -choosable is more difficult than proving it to be \( k \) -colourable: the latter is just the special case of the former where all lists are equal to \( \{ 1,\ldots, k\} \) . Thus,
\[
\operatorname{ch}\left( G\right) \geq \chi \left( G\right) \;\text{ and }\;{\operatorname{ch}}^{\prime }\left( G\right) \geq {\chi }^{\prime }\left( G\right)
\]
for all graphs \( G \) .
In spite of these inequalities, many of the known upper bounds for the chromatic number have turned out to be valid for the choice number, too. Examples for this phenomenon include Brooks's theorem and Proposition 5.2.2; in particular, graphs of large choice number still have subgraphs of large minimum degree. On the other hand, it is easy to construct graphs for which the two invariants are wide apart (Exercise 25). Taken together, these two facts indicate a little how far those general upper bounds on the chromatic number may be from the truth.
The following theorem shows that, in terms of its relationship to other graph invariants, the choice number differs fundamentally from the chromatic number. As mentioned before, there are 2-chromatic graphs of arbitrarily large minimum degree, e.g. the graphs \( {K}_{n, n} \) . The choice number, however, will be forced up by large values of invariants like \( \delta ,\varepsilon \) or \( \kappa \) :
Theorem 5.4.1. (Alon 1993)
There exists a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that, given any integer \( k \), all graphs \( G \) with average degree \( d\left( G\right) \geq f\left( k\right) \) satisfy \( \operatorname{ch}\left( G\right) \geq k \) .
The proof of Theorem 5.4.1 uses probabilistic methods as introduced in Chapter 11.
Although statements of the form \( \operatorname{ch}\left( G\right) \leq k \) are formally stronger than the corresponding statement of \( \chi \left( G\right) \leq k \), they can be easier to prove. A pretty example is the list version of the five colour theorem: every planar graph is 5-choosable. The proof of this does not use the five colour theorem (or even Euler's formula, on which the proof of the five colour theorem is based). We thus reobtain the five colour theorem as a corollary, with a very different proof.
Theorem 5.4.2. (Thomassen 1994)
Every planar graph is 5-choosable.
(4.2.8)
Proof. We shall prove the following assertion for all plane graphs \( G \) with at least 3 vertices:
Suppose that every inner face of \( G \) is bounded by a triangle and its outer face by a cycle \( C = {v}_{1}\ldots {v}_{k}{v}_{1} \) . Suppose further that \( {v}_{1} \) has already been coloured with the colour 1, and \( {v}_{2} \) has been coloured 2 . Suppose finally that
\( \left( *\right) \)
with every other vertex of \( C \) a list of at least 3 colours is associated, and with every vertex of \( G - C \) a list of at least 5 colours. Then the colouring of \( {v}_{1} \) and \( {v}_{2} \) can be extended to a colouring of \( G \) from the given lists.
Let us check first that \( \left( *\right) \) implies the assertion of the theorem. Let any plane graph be given, together with a list of 5 colours for each vertex. Add edges to this graph until it is a maximal plane graph \( G \) . By Proposition 4.2.8, \( G \) is a plane triangulation; let \( {v}_{1}{v}_{2}{v}_{3}{v}_{1} \) be the boundary of its outer face. We now colour \( {v}_{1} \) and \( {v}_{2} \) (differently) from their lists, and extend this colouring by \( \left( *\right) \) to a colouring of \( G \) from the lists given.
Let us now prove \( \left( *\right) \), by induction on \( \left| G\right| \) . If \( \left| G\right| = 3 \), then \( G = \) \( C \) and the assertion is trivial. Now let \( \left| G\right| \geq 4 \), and assume \( \left( *\right) \) for smaller graphs. If \( C \) has a chord \( {vw} \), then \( {vw} \) lies on two unique cycles \( {C}_{1},{C}_{2} \subseteq C + {vw} \) with \( {v}_{1}{v}_{2} \in {C}_{1} \) and \( {v}_{1}{v}_{2} \notin {C}_{2} \) . For \( i = 1,2 \), let \( {G}_{i} \) denote the subgraph of \( G \) induced by the vertices lying on \( {C}_{i} \) or in its inner face (Fig. 5.4.1). Applying the induction hypothesis first to \( {G}_{1} \) and then with the colours now assigned to \( v \) and \( w \) -to \( {G}_{2} \) yields the desired colouring of \( G \) .
Fig. 5.4.1. The induction step with a chord \( {vw} \) ; here the case of \( w = {v}_{2} \)
If \( C \) has no chord, let \( {v}_{1},{u}_{1},\ldots ,{u}_{m},{v}_{k - 1} \) be the neighbours of \( {v}_{k} \) in \( {u}_{1},\ldots ,{u}_{m} \) their natural cyclic order order around \( {v}_{k}{;}^{2} \) by definition of \( C \), all those neighbours \( {u}_{i} \) lie in the inner face of \( C \) (Fig. 5.4.2). As the inner faces of \( C \) are bounded by triangles, \( P \mathrel{\text{:=}} {v}_{1}{u}_{1}\ldots {u}_{m}{v}_{k - 1} \) is a path in \( G \), and \( {C}^{\prime } \mathrel{\text{:=}} P \cup \left( {C - {v |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 43 | aph of \( G \) induced by the vertices lying on \( {C}_{i} \) or in its inner face (Fig. 5.4.1). Applying the induction hypothesis first to \( {G}_{1} \) and then with the colours now assigned to \( v \) and \( w \) -to \( {G}_{2} \) yields the desired colouring of \( G \) .
Fig. 5.4.1. The induction step with a chord \( {vw} \) ; here the case of \( w = {v}_{2} \)
If \( C \) has no chord, let \( {v}_{1},{u}_{1},\ldots ,{u}_{m},{v}_{k - 1} \) be the neighbours of \( {v}_{k} \) in \( {u}_{1},\ldots ,{u}_{m} \) their natural cyclic order order around \( {v}_{k}{;}^{2} \) by definition of \( C \), all those neighbours \( {u}_{i} \) lie in the inner face of \( C \) (Fig. 5.4.2). As the inner faces of \( C \) are bounded by triangles, \( P \mathrel{\text{:=}} {v}_{1}{u}_{1}\ldots {u}_{m}{v}_{k - 1} \) is a path in \( G \), and \( {C}^{\prime } \mathrel{\text{:=}} P \cup \left( {C - {v}_{k}}\right) \) a cycle.
Fig. 5.4.2. The induction step without a chord
We now choose two different colours \( j,\ell \neq 1 \) from the list of \( {v}_{k} \) and delete these colours from the lists of all the vertices \( {u}_{i} \) . Then every list of a vertex on \( {C}^{\prime } \) still has at least 3 colours, so by induction we may colour \( {C}^{\prime } \) and its interior, i.e. the graph \( G - {v}_{k} \) . At least one of the two colours \( j,\ell \) is not used for \( {v}_{k - 1} \), and we may assign that colour to \( {v}_{k} \) .
As is often the case with induction proofs, the key to the proof above lies in its delicately balanced strengthening of the assertion proved. Compared with ordinary colouring, the task of finding a suitable strengthening is helped greatly by the possibility to give different vertices lists of different lengths, and thus to tailor the colouring problem more fittingly to the structure of the graph. This suggests that maybe in other unsolved
---
2 as in the first proof of the five colour theorem
---
colouring problems too it might be of advantage to aim straight for their list version, i.e. to prove an assertion of the form \( \operatorname{ch}\left( G\right) \leq k \) instead of the formally weaker \( \chi \left( G\right) \leq k \) . Unfortunately, this approach fails for the four colour theorem: planar graphs are not in general 4-choosable.
As mentioned before, the chromatic number of a graph and its choice number may differ a lot. Surprisingly, however, no such examples are known for edge colourings. Indeed it has been conjectured that none exist:
## List Colouring Conjecture. Every graph \( G \) satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) = {\chi }^{\prime }\left( G\right) \) .
We shall prove the list colouring conjecture for bipartite graphs. As a tool we shall use orientations of graphs, defined in Chapter 1.10. If \( D \)
---
\( {N}^{ + }\left( v\right) \) \( {d}^{ + }\left( v\right) \)
---
is a directed graph and \( v \in V\left( D\right) \), we denote by \( {N}^{ + }\left( v\right) \) the set, and by \( {d}^{ + }\left( v\right) \) the number, of vertices \( w \) such that \( D \) contains an edge directed from \( v \) to \( w \) .
To see how orientations come into play in the context of colouring, recall the greedy algorithm from Section 5.2. This colours the vertices of a graph \( G \) in turn, following a previously fixed ordering \( \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) . This ordering defines an orientation of \( G \) if we orient every edge \( {v}_{i}{v}_{j} \) ’backwards’, that is, from \( {v}_{i} \) to \( {v}_{j} \) if \( i > j \) . Then to determine a colour for \( {v}_{i} \) the algorithm only looks at previously coloured neighbours of \( {v}_{i} \) , those to which \( {v}_{i} \) sends a directed edge. In particular, if \( {d}^{ + }\left( v\right) < k \) for all vertices \( v \), the algorithm will use at most \( k \) colours.
If we rewrite the proof of this fact (rather awkwardly) as a formal induction on \( k \), we notice that the essential property of the set \( U \) of vertices coloured 1 is that every vertex in \( G - U \) sends an edge to \( U \) : this ensures that \( {d}_{G - U}^{ + }\left( v\right) < {d}_{G}^{ + }\left( v\right) \) for all \( v \in G - U \), so we can colour \( G - U \) with the remaining \( k - 1 \) colours by the induction hypothesis.
The following lemma generalizes these observations to list colouring, and to orientations \( D \) of \( G \) that do not necessarily come from a vertex enumeration but may contain some directed cycles. Let us call an kernel independent set \( U \subseteq V\left( D\right) \) a kernel of \( D \) if, for every vertex \( v \in D - U \) , there is an edge in \( D \) directed from \( v \) to a vertex in \( U \) . Note that kernels of non-empty directed graphs are themselves non-empty.
Lemma 5.4.3. Let \( H \) be a graph and \( {\left( {S}_{v}\right) }_{v \in V\left( H\right) } \) a family of lists. If \( H \) has an orientation \( D \) with \( {d}^{ + }\left( v\right) < \left| {S}_{v}\right| \) for every \( v \), and such that every induced subgraph of \( D \) has a kernel, then \( H \) can be coloured from the lists \( {S}_{v} \) .
Proof. We apply induction on \( \left| H\right| \) . For \( \left| H\right| = 0 \) we take the empty colouring. For the induction step, let \( \left| H\right| > 0 \) . Let \( \alpha \) be a colour occurring in one of the lists \( {S}_{v} \), and let \( D \) be an orientation of \( H \) as stated. The vertices \( v \) with \( \alpha \in {S}_{v} \) span a non-empty subgraph \( {D}^{\prime } \) in \( D \) ; by assumption, \( {D}^{\prime } \) has a kernel \( U \neq \varnothing \) .
Let us colour the vertices in \( U \) with \( \alpha \), and remove \( \alpha \) from the lists of all the other vertices of \( {D}^{\prime } \) . Since each of those vertices sends an edge to \( U \), the modified lists \( {S}_{v}^{\prime } \) for \( v \in D - U \) again satisfy the condition \( {d}^{ + }\left( v\right) < \left| {S}_{v}^{\prime }\right| \) in \( D - U \) . Since \( D - U \) is an orientation of \( H - U \), we can thus colour \( H - U \) from those lists by the induction hypothesis. As none of these lists contains \( \alpha \), this extends our colouring \( U \rightarrow \{ \alpha \} \) to the desired list colouring of \( H \) .
In our proof of the list colouring conjecture for bipartite graphs we shall apply Lemma 5.4.3 only to colourings from lists of uniform length \( k \) . However, note that keeping list lengths variable is essential for the proof of the lemma itself: its simple induction could not be performed with uniform list lengths.
Theorem 5.4.4. (Galvin 1995)
Every bipartite graph \( G \) satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) = {\chi }^{\prime }\left( G\right) \) .
Proof. Let \( G = : \left( {X \cup Y, E}\right) \), where \( \{ X, Y\} \) is a vertex bipartition of \( G \) .
(2.1.4)
Let us say that two edges of \( G \) meet in \( X \) if they share an end in \( X \), and \( X, Y, E \) correspondingly for \( Y \) . Let \( {\chi }^{\prime }\left( G\right) = : k \), and let \( c \) be a \( k \) -edge-colouring of \( G \) .
Clearly, \( {\operatorname{ch}}^{\prime }\left( G\right) \geq k \) ; we prove that \( {\operatorname{ch}}^{\prime }\left( G\right) \leq k \) . Our plan is to use Lemma 5.4.3 to show that the line graph \( H \) of \( G \) is \( k \) -choosable. To apply the lemma, it suffices to find an orientation \( D \) of \( H \) with \( {d}^{ + }\left( e\right) < k \) for every vertex \( e \) of \( H \), and such that every induced subgraph of \( D \) has a kernel. To define \( D \), consider adjacent \( e,{e}^{\prime } \in E \), say with \( c\left( e\right) < c\left( {e}^{\prime }\right) \) . If \( e \) and \( {e}^{\prime } \) meet in \( X \), we orient the edge \( e{e}^{\prime } \in H \) from \( {e}^{\prime } \) towards \( e \) ; if \( e \) and \( {e}^{\prime } \) meet in \( Y \), we orient it from \( e \) to \( {e}^{\prime } \) (Fig 5.4.3).
Fig. 5.4.3. Orienting the line graph of \( G \)
Let us compute \( {d}^{ + }\left( e\right) \) for given \( e \in E = V\left( D\right) \) . If \( c\left( e\right) = i \), say, then every \( {e}^{\prime } \in {N}^{ + }\left( e\right) \) meeting \( e \) in \( X \) has its colour in \( \{ 1,\ldots, i - 1\} \) , and every \( {e}^{\prime } \in {N}^{ + }\left( e\right) \) meeting \( e \) in \( Y \) has its colour in \( \{ i + 1,\ldots, k\} \) . As any two neighbours \( {e}^{\prime } \) of \( e \) meeting \( e \) either both in \( X \) or both in \( Y \) are themselves adjacent and hence coloured differently, this implies \( {d}^{ + }\left( e\right) < k \) as desired.
It remains to show that every induced subgraph \( {D}^{\prime } \) of \( D \) has a kernel. This, however, is immediate by the stable marriage theorem (2.1.4) for \( G \) , if we interpret the directions in \( D \) as expressing preference. Indeed, given a vertex \( v \in X \cup Y \) and edges \( e,{e}^{\prime } \in V\left( {D}^{\prime }\right) \) at \( v \), write \( e{ < }_{v}{e}^{\prime } \) if the edge \( e{e}^{\prime } \) of \( H \) is directed from \( e \) to \( {e}^{\prime } \) in \( D \) . Then any stable matching in the graph \( \left( {X \cup Y, V\left( {D}^{\prime }\right) }\right) \) for this set of preferences is a kernel in \( {D}^{\prime } \) .
(5.3.1)
By Proposition 5.3.1, we now know the exact list-chromatic index of bipartite graphs:
Corollary 5.4.5. Every bipartite graph \( G \) satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) = \Delta \left( G\right) \) .
## 5.5 Perfect graphs
As discussed in Section 5.2, a high chromatic number may occur as a purely global phenomenon: even when a grap |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 44 | ced subgraph \( {D}^{\prime } \) of \( D \) has a kernel. This, however, is immediate by the stable marriage theorem (2.1.4) for \( G \) , if we interpret the directions in \( D \) as expressing preference. Indeed, given a vertex \( v \in X \cup Y \) and edges \( e,{e}^{\prime } \in V\left( {D}^{\prime }\right) \) at \( v \), write \( e{ < }_{v}{e}^{\prime } \) if the edge \( e{e}^{\prime } \) of \( H \) is directed from \( e \) to \( {e}^{\prime } \) in \( D \) . Then any stable matching in the graph \( \left( {X \cup Y, V\left( {D}^{\prime }\right) }\right) \) for this set of preferences is a kernel in \( {D}^{\prime } \) .
(5.3.1)
By Proposition 5.3.1, we now know the exact list-chromatic index of bipartite graphs:
Corollary 5.4.5. Every bipartite graph \( G \) satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) = \Delta \left( G\right) \) .
## 5.5 Perfect graphs
As discussed in Section 5.2, a high chromatic number may occur as a purely global phenomenon: even when a graph has large girth, and thus locally looks like a tree, its chromatic number may be arbitrarily high. Since such 'global dependence' is obviously difficult to deal with, one may become interested in graphs where this phenomenon does not occur, i.e. whose chromatic number is high only when there is a local reason for it.
Before we make this precise, let us note two definitions for a graph \( G \) .
\( \omega \left( G\right) \)
The greatest integer \( r \) such that \( {K}^{r} \subseteq G \) is the clique number \( \omega \left( G\right) \) of \( G \) , and the greatest integer \( r \) such that \( \overline{{K}^{r}} \subseteq G \) (induced) is the indepen-
\( \alpha \left( G\right) \)
dence number \( \alpha \left( G\right) \) of \( G \) . Clearly, \( \alpha \left( G\right) = \omega \left( \bar{G}\right) \) and \( \omega \left( G\right) = \alpha \left( \bar{G}\right) \) .
perfect A graph is called perfect if every induced subgraph \( H \subseteq G \) has chromatic number \( \chi \left( H\right) = \omega \left( H\right) \), i.e. if the trivial lower bound of \( \omega \left( H\right) \) colours always suffices to colour the vertices of \( H \) . Thus, while proving an assertion of the form \( \chi \left( G\right) > k \) may in general be difficult, even in principle, for a given graph \( G \), it can always be done for a perfect graph simply by exhibiting some \( {K}^{k + 1} \) subgraph as a ’certificate’ for non-colourability with \( k \) colours.
At first glance, the structure of the class of perfect graphs appears somewhat contrived: although it is closed under induced subgraphs (if only by explicit definition), it is not closed under taking general subgraphs or supergraphs, let alone minors (examples?). However, perfection is an important notion in graph theory: the fact that several fundamental classes of graphs are perfect (as if by fluke) may serve as a superficial indication of this. \( {}^{3} \)
What graphs, then, are perfect? Bipartite graphs are, for instance. Less trivially, the complements of bipartite graphs are perfect, too-a fact equivalent to König's duality theorem 2.1.1 (Exercise 36). The so-called comparability graphs are perfect, and so are the interval graphs (see the exercises); both these turn up in numerous applications.
In order to study at least one such example in some detail, we prove here that the chordal graphs are perfect: a graph is chordal (or chordal triangulated) if each of its cycles of length at least 4 has a chord, i.e. if it contains no induced cycles other than triangles.
To show that chordal graphs are perfect, we shall first characterize their structure. If \( G \) is a graph with induced subgraphs \( {G}_{1},{G}_{2} \) and \( S \) , such that \( G = {G}_{1} \cup {G}_{2} \) and \( S = {G}_{1} \cap {G}_{2} \), we say that \( G \) arises from \( {G}_{1} \) and \( {G}_{2} \) by pasting these graphs together along \( S \) . pasting
Proposition 5.5.1. A graph is chordal if and only if it can be con-
\( \left\lbrack {12.3.11}\right\rbrack \)
structed recursively by pasting along complete subgraphs, starting from complete graphs.
Proof. If \( G \) is obtained from two chordal graphs \( {G}_{1},{G}_{2} \) by pasting them together along a complete subgraph, then \( G \) is clearly again chordal: any induced cycle in \( G \) lies in either \( {G}_{1} \) or \( {G}_{2} \), and is hence a triangle by assumption. Since complete graphs are chordal, this proves that all graphs constructible as stated are chordal.
Conversely, let \( G \) be a chordal graph. We show by induction on \( \left| G\right| \) that \( G \) can be constructed as described. This is trivial if \( G \) is complete. We therefore assume that \( G \) is not complete, in particular that \( \left| G\right| > 1 \) , and that all smaller chordal graphs are constructible as stated. Let \( a, b \in \) \( a, b \) \( G \) be two non-adjacent vertices, and let \( X \subseteq V\left( G\right) \smallsetminus \{ a, b\} \) be a minimal \( a - b \) separator. Let \( C \) denote the component of \( G - X \) containing \( a \), and put \( {G}_{1} \mathrel{\text{:=}} G\left\lbrack {V\left( C\right) \cup X}\right\rbrack \) and \( {G}_{2} \mathrel{\text{:=}} G - C \) . Then \( G \) arises from \( {G}_{1} \) and \( {G}_{1},{G}_{2} \) \( {G}_{2} \) by pasting these graphs together along \( S \mathrel{\text{:=}} G\left\lbrack X\right\rbrack \) .
Since \( {G}_{1} \) and \( {G}_{2} \) are both chordal (being induced subgraphs of \( G \) ) and hence constructible by induction, it suffices to show that \( S \) is complete. Suppose, then, that \( s, t \in S \) are non-adjacent. By the minimality of \( X = V\left( S\right) \) as an \( a - b \) separator, both \( s \) and \( t \) have a neighbour in \( C \) . Hence, there is an \( X \) -path from \( s \) to \( t \) in \( {G}_{1} \) ; we let \( {P}_{1} \) be a shortest such path. Analogously, \( {G}_{2} \) contains a shortest \( X \) -path \( {P}_{2} \) from \( s \) to \( t \) . But then \( {P}_{1} \cup {P}_{2} \) is a chordless cycle of length \( \geq 4 \) (Fig. 5.5.1), contradicting our assumption that \( G \) is chordal.
3 The class of perfect graphs has duality properties with deep connections to optimization and complexity theory, which are far from understood. Theorem 5.5.6 shows the tip of an iceberg here; for more, the reader is referred to Lovász's survey cited in the notes.
Fig. 5.5.1. If \( {G}_{1} \) and \( {G}_{2} \) are chordal, then so is \( G \)
## Proposition 5.5.2. Every chordal graph is perfect.
Proof. Since complete graphs are perfect, it suffices by Proposition 5.5.1 to show that any graph \( G \) obtained from perfect graphs \( {G}_{1},{G}_{2} \) by pasting them together along a complete subgraph \( S \) is again perfect. So let \( H \subseteq G \) be an induced subgraph; we show that \( \chi \left( H\right) \leq \omega \left( H\right) \) .
Let \( {H}_{i} \mathrel{\text{:=}} H \cap {G}_{i} \) for \( i = 1,2 \), and let \( T \mathrel{\text{:=}} H \cap S \) . Then \( T \) is again complete, and \( H \) arises from \( {H}_{1} \) and \( {H}_{2} \) by pasting along \( T \) . As an induced subgraph of \( {G}_{i} \), each \( {H}_{i} \) can be coloured with \( \omega \left( {H}_{i}\right) \) colours. Since \( T \) is complete and hence coloured injectively, two such colourings, one of \( {H}_{1} \) and one of \( {H}_{2} \), may be combined into a colouring of \( H \) with \( \max \left\{ {\omega \left( {H}_{1}\right) ,\omega \left( {H}_{2}\right) }\right\} \leq \omega \left( H\right) \) colours - if necessary by permuting the colours in one of the \( {H}_{i} \) .
By definition, every induced subgraph of a perfect graph is again perfect. The property of perfection can therefore be characterized by forbidden induced subgraphs: there exists a set \( \mathcal{H} \) of imperfect graphs such that any graph is perfect if and only if it has no induced subgraph isomorphic to an element of \( \mathcal{H} \) . (For example, we may choose as \( \mathcal{H} \) the set of all imperfect graphs with vertices in \( \mathbb{N} \) .)
Naturally, one would like to keep \( \mathcal{H} \) as small as possible. It is one of the deepest results in graph theory that \( \mathcal{H} \) need only contain two types of graph: the odd cycles of length \( \geq 5 \) and their complements. (Neither of these are perfect; cf. Theorem 5.5.4 below.) This fact, the famous strong perfect graph conjecture of Berge (1963), was proved only very
---
strong perfect graph theorem
---
recently:
Theorem 5.5.3. (Chudnovsky, Robertson, Seymour & Thomas 2002) A graph \( G \) is perfect if and only if neither \( G \) nor \( \bar{G} \) contains an odd cycle of length at least 5 as an induced subgraph.
The proof of the strong perfect graph theorem is long and technical, and it would not be too illuminating to attempt to sketch it. To shed more light on the notion of perfection, we instead give two direct proofs of its most important consequence: the perfect graph theorem, formerly Berge's weak perfect graph conjecture:
Theorem 5.5.4. (Lovász 1972)
---
perfect graph theorem
---
A graph is perfect if and only if its complement is perfect.
The first proof we give for Theorem 5.5.4 is Lovász's original proof, which is still unsurpassed in its clarity and the amount of 'feel' for the problem it conveys. Our second proof, due to Gasparian (1996), is an elegant linear algebra proof of another theorem of Lovász's (Theorem 5.5.6), which easily implies Theorem 5.5.4.
Let us prepare our first proof of Theorem 5.5.4 by a lemma. Let \( G \) be a graph and \( x \in G \) a vertex, and let \( {G}^{\prime } \) be obtained from \( G \) by adding a vertex \( {x}^{\prime } \) and joining it to \( x \) and all the neighbours of \( x \) . We
---
expanding a vertex
---
say that \( {G}^{\prime } \) is obtained from \( G \) by expanding th |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 45 | ight on the notion of perfection, we instead give two direct proofs of its most important consequence: the perfect graph theorem, formerly Berge's weak perfect graph conjecture:
Theorem 5.5.4. (Lovász 1972)
---
perfect graph theorem
---
A graph is perfect if and only if its complement is perfect.
The first proof we give for Theorem 5.5.4 is Lovász's original proof, which is still unsurpassed in its clarity and the amount of 'feel' for the problem it conveys. Our second proof, due to Gasparian (1996), is an elegant linear algebra proof of another theorem of Lovász's (Theorem 5.5.6), which easily implies Theorem 5.5.4.
Let us prepare our first proof of Theorem 5.5.4 by a lemma. Let \( G \) be a graph and \( x \in G \) a vertex, and let \( {G}^{\prime } \) be obtained from \( G \) by adding a vertex \( {x}^{\prime } \) and joining it to \( x \) and all the neighbours of \( x \) . We
---
expanding a vertex
---
say that \( {G}^{\prime } \) is obtained from \( G \) by expanding the vertex \( x \) to an edge \( x{x}^{\prime } \) (Fig. 5.5.2).
Fig. 5.5.2. Expanding the vertex \( x \) in the proof of Lemma 5.5.5
Lemma 5.5.5. Any graph obtained from a perfect graph by expanding a vertex is again perfect.
Proof. We use induction on the order of the perfect graph considered. Expanding the vertex of \( {K}^{1} \) yields \( {K}^{2} \), which is perfect. For the induction step, let \( G \) be a non-trivial perfect graph, and let \( {G}^{\prime } \) be obtained from \( G \) by expanding a vertex \( x \in G \) to an edge \( x{x}^{\prime } \) . For our proof that \( {G}^{\prime } \) is perfect it suffices to show \( \chi \left( {G}^{\prime }\right) \leq \omega \left( {G}^{\prime }\right) \) : every proper induced subgraph \( H \) of \( {G}^{\prime } \) is either isomorphic to an induced subgraph of \( G \) or obtained from a proper induced subgraph of \( G \) by expanding \( x \) ; in either case, \( H \) is perfect by assumption and the induction hypothesis, and can hence be coloured with \( \omega \left( H\right) \) colours.
Let \( \omega \left( G\right) = : \omega \) ; then \( \omega \left( {G}^{\prime }\right) \in \{ \omega ,\omega + 1\} \) . If \( \omega \left( {G}^{\prime }\right) = \omega + 1 \), then
\[
\chi \left( {G}^{\prime }\right) \leq \chi \left( G\right) + 1 = \omega + 1 = \omega \left( {G}^{\prime }\right)
\]
and we are done. So let us assume that \( \omega \left( {G}^{\prime }\right) = \omega \) . Then \( x \) lies in no \( {K}^{\omega } \subseteq G \) : together with \( {x}^{\prime } \), this would yield a \( {K}^{\omega + 1} \) in \( {G}^{\prime } \) . Let us colour \( G \) with \( \omega \) colours. Since every \( {K}^{\omega } \subseteq G \) meets the colour class \( X \) of \( x \) but not \( x \) itself, the graph \( H \mathrel{\text{:=}} G - \left( {X\smallsetminus \{ x\} }\right) \) has clique number \( \omega \left( H\right) < \omega \) (Fig. 5.5.2). Since \( G \) is perfect, we may thus colour \( H \) with \( \omega - 1 \) colours. Now \( X \) is independent, so the set \( \left( {X\smallsetminus \{ x\} }\right) \cup \left\{ {x}^{\prime }\right\} = V\left( {{G}^{\prime } - H}\right) \) is also independent. We can therefore extend our \( \left( {\omega - 1}\right) \) -colouring of \( H \) to an \( \omega \) -colouring of \( {G}^{\prime } \), showing that \( \chi \left( {G}^{\prime }\right) \leq \omega = \omega \left( {G}^{\prime }\right) \) as desired.
Proof of Theorem 5.5.4. Applying induction on \( \left| G\right| \), we show that
\( G = \left( {V, E}\right) \)
the complement \( \bar{G} \) of any perfect graph \( G = \left( {V, E}\right) \) is again perfect. For \( \mathcal{K} \) \( \left| G\right| = 1 \) this is trivial, so let \( \left| G\right| \geq 2 \) for the induction step. Let \( \mathcal{K} \) denote \( \alpha \) the set of all vertex sets of complete subgraphs of \( G \) . Put \( \alpha \left( G\right) = : \alpha \) , \( \mathcal{A} \) and let \( \mathcal{A} \) be the set of all independent vertex sets \( A \) in \( G \) with \( \left| A\right| = \alpha \) .
Every proper induced subgraph of \( \bar{G} \) is the complement of a proper induced subgraph of \( G \), and is hence perfect by induction. For the perfection of \( \bar{G} \) it thus suffices to prove \( \chi \left( \bar{G}\right) \leq \omega \left( \bar{G}\right) \left( { = \alpha }\right) \) . To this end, we shall find a set \( K \in \mathcal{K} \) such that \( K \cap A \neq \varnothing \) for all \( A \in \mathcal{A} \) ; then
\[
\omega \left( {\bar{G} - K}\right) = \alpha \left( {G - K}\right) < \alpha = \omega \left( \bar{G}\right) ,
\]
so by the induction hypothesis
\[
\chi \left( \bar{G}\right) \leq \chi \left( {\bar{G} - K}\right) + 1 = \omega \left( {\bar{G} - K}\right) + 1 \leq \omega \left( \bar{G}\right)
\]
as desired.
Suppose there is no such \( K \) ; thus, for every \( K \in \mathcal{K} \) there exists a \( {A}_{K} \) set \( {A}_{K} \in \mathcal{A} \) with \( K \cap {A}_{K} = \varnothing \) . Let us replace in \( G \) every vertex \( x \) by a \( {G}_{x} \) complete graph \( {G}_{x} \) of order
\( k\left( x\right) \)
\[
k\left( x\right) \mathrel{\text{:=}} \left| \left\{ {K \in \mathcal{K} \mid x \in {A}_{K}}\right\} \right|
\]
joining all the vertices of \( {G}_{x} \) to all the vertices of \( {G}_{y} \) whenever \( x \) and \( y \) are \( {G}^{\prime } \) adjacent in \( G \) . The graph \( {G}^{\prime } \) thus obtained has vertex set \( \mathop{\bigcup }\limits_{{x \in V}}V\left( {G}_{x}\right) \) , and two vertices \( v \in {G}_{x} \) and \( w \in {G}_{y} \) are adjacent in \( {G}^{\prime } \) if and only if \( x = y \) or \( {xy} \in E \) . Moreover, \( {G}^{\prime } \) can be obtained by repeated vertex expansion from the graph \( G\left\lbrack {\{ x \in V \mid k\left( x\right) > 0\} }\right\rbrack \) . Being an induced subgraph of \( G \), this latter graph is perfect by assumption, so \( {G}^{\prime } \) is perfect by Lemma 5.5.5. In particular,
\[
\chi \left( {G}^{\prime }\right) \leq \omega \left( {G}^{\prime }\right)
\]
(1)
In order to obtain a contradiction to (1), we now compute in turn the actual values of \( \omega \left( {G}^{\prime }\right) \) and \( \chi \left( {G}^{\prime }\right) \) . By construction of \( {G}^{\prime } \), every maximal complete subgraph of \( {G}^{\prime } \) has the form \( {G}^{\prime }\left\lbrack {\mathop{\bigcup }\limits_{{x \in X}}{G}_{x}}\right\rbrack \) for some \( X \in \mathcal{K} \) . So there exists a set \( X \in \mathcal{K} \) such that
\[
\omega \left( {G}^{\prime }\right) = \mathop{\sum }\limits_{{x \in X}}k\left( x\right)
\]
\[
= \left| \left\{ {\left( {x, K}\right) : x \in X, K \in \mathcal{K}, x \in {A}_{K}}\right\} \right|
\]
\[
= \mathop{\sum }\limits_{{K \in \mathcal{K}}}\left| {X \cap {A}_{K}}\right|
\]
\[
\leq \left| \mathcal{K}\right| - 1
\]
(2)
the last inequality follows from the fact that \( \left| {X \cap {A}_{K}}\right| \leq 1 \) for all \( K \) (since \( {A}_{K} \) is independent but \( G\left\lbrack X\right\rbrack \) is complete), and \( \left| {X \cap {A}_{X}}\right| = 0 \) (by the choice of \( {A}_{X} \) ). On the other hand,
\[
\left| {G}^{\prime }\right| = \mathop{\sum }\limits_{{x \in V}}k\left( x\right)
\]
\[
= \left| \left\{ {\left( {x, K}\right) : x \in V, K \in \mathcal{K}, x \in {A}_{K}}\right\} \right|
\]
\[
= \mathop{\sum }\limits_{{K \in \mathcal{K}}}\left| {A}_{K}\right|
\]
\[
= \left| \mathcal{K}\right| \cdot \alpha \text{.}
\]
As \( \alpha \left( {G}^{\prime }\right) \leq \alpha \) by construction of \( {G}^{\prime } \), this implies
\[
\chi \left( {G}^{\prime }\right) \geq \frac{\left| {G}^{\prime }\right| }{\alpha \left( {G}^{\prime }\right) } \geq \frac{\left| {G}^{\prime }\right| }{\alpha } = \left| \mathcal{K}\right|
\]
(3)
Putting (2) and (3) together we obtain
\[
\chi \left( {G}^{\prime }\right) \geq \left| \mathcal{K}\right| > \left| \mathcal{K}\right| - 1 \geq \omega \left( {G}^{\prime }\right)
\]
a contradiction to (1).
At first reading, the proof of Theorem 5.5.4 appears magical: it starts with an unmotivated lemma about expanding a vertex, shifts the problem to a strange graph \( {G}^{\prime } \) obtained in this way, performs some double counting - and finished. With hindsight, however, we can understand it a little better. The proof is completely natural up to the point where we assume that for every \( K \in \mathcal{K} \) there is an \( {A}_{K} \in \mathcal{A} \) such that \( K \cap {A}_{K} = \varnothing \) . To show that this contradicts our assumption that \( G \) is perfect, we would like to show next that its subgraph \( \widetilde{G} \) induced by all the \( {A}_{K} \) has a chromatic number that is too large, larger than its clique number. And, as always when we try to bound the chromatic number from below, our only hope is to bound \( \left| \widetilde{G}\right| /\alpha \) instead, i.e. to show that this is larger than \( \omega \left( \widetilde{G}\right) \) .
But is the bound of \( \left| \widetilde{G}\right| /\alpha \) likely to reflect the true value of \( \chi \left( \widetilde{G}\right) \) ? In one special case it is: if the sets \( {A}_{K} \) happen to be disjoint, we have \( \left| \widetilde{G}\right| = \left| \mathcal{K}\right| \cdot \alpha \) and \( \chi \left( \widetilde{G}\right) = \left| \mathcal{K}\right| \), with the \( {A}_{K} \) as colour classes. Of course, the sets \( {A}_{K} \) will not in general be disjoint. But we can make them so: by replacing every vertex \( x \) with \( k\left( x\right) \) vertices, where \( k\left( x\right) \) is the number of sets \( {A}_{K} \) it lives in! This is the idea behind \( {G}^{\prime } \) . What remains is to endow \( {G}^{\prime } \) with the right set of edges to make it perfect (assuming that \( G \) is perfect)-which leads straight to the definition of vertex expansion and Lemma 5.5.5.
Since t |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 46 | stead, i.e. to show that this is larger than \( \omega \left( \widetilde{G}\right) \) .
But is the bound of \( \left| \widetilde{G}\right| /\alpha \) likely to reflect the true value of \( \chi \left( \widetilde{G}\right) \) ? In one special case it is: if the sets \( {A}_{K} \) happen to be disjoint, we have \( \left| \widetilde{G}\right| = \left| \mathcal{K}\right| \cdot \alpha \) and \( \chi \left( \widetilde{G}\right) = \left| \mathcal{K}\right| \), with the \( {A}_{K} \) as colour classes. Of course, the sets \( {A}_{K} \) will not in general be disjoint. But we can make them so: by replacing every vertex \( x \) with \( k\left( x\right) \) vertices, where \( k\left( x\right) \) is the number of sets \( {A}_{K} \) it lives in! This is the idea behind \( {G}^{\prime } \) . What remains is to endow \( {G}^{\prime } \) with the right set of edges to make it perfect (assuming that \( G \) is perfect)-which leads straight to the definition of vertex expansion and Lemma 5.5.5.
Since the following characterization of perfection is symmetrical in \( G \) and \( \bar{G} \), it clearly implies Theorem 5.5.4. As our proof of Theorem 5.5.6 will again be from first principles, we thus obtain a second and independent proof of Theorem 5.5.4.
Theorem 5.5.6. (Lovász 1972)
A graph \( G \) is perfect if and only if
\[
\left| H\right| \leq \alpha \left( H\right) \cdot \omega \left( H\right)
\]
\( \left( *\right) \)
for all induced subgraphs \( H \subseteq G \) .
\( V,{v}_{i}, n \) Proof. Let us write \( V\left( G\right) = : V = : \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \), and put \( \alpha \mathrel{\text{:=}} \alpha \left( G\right) \) \( \alpha ,\omega \) and \( \omega \mathrel{\text{:=}} \omega \left( G\right) \) . The necessity of \( \left( *\right) \) is immediate: if \( G \) is perfect, then every induced subgraph \( H \) of \( G \) can be partitioned into at most \( \omega \left( H\right) \) colour classes each containing at most \( \alpha \left( H\right) \) vertices, and \( \left( *\right) \) follows.
To prove sufficiency, we apply induction on \( n = \left| G\right| \) . Assume that every induced subgraph \( H \) of \( G \) satisfies \( \left( *\right) \), and suppose that \( G \) is not perfect. By the induction hypothesis, every proper induced subgraph of \( G \) is perfect. Hence, every non-empty independent set \( U \subseteq V \) satisfies
\[
\chi \left( {G - U}\right) = \omega \left( {G - U}\right) = \omega .
\]
(1)
Indeed, while the first equality is immediate from the perfection of \( G - U \) , the second is easy: ’ \( \leq \) ’ is obvious, while \( \chi \left( {G - U}\right) < \omega \) would imply \( \chi \left( G\right) \leq \omega \), so \( G \) would be perfect contrary to our assumption.
Let us apply (1) to a singleton \( U = \{ u\} \) and consider an \( \omega \) -colouring of \( G - u \) . Let \( K \) be the vertex set of any \( {K}^{\omega } \) in \( G \) . Clearly,
\[
\text{if}u \notin K\text{then}K\text{meets every colour class of}G - u\text{;}
\]
(2)
if \( u \in K \) then \( K \) meets all but exactly one colour class of \( G - u \) .
(3)
Let \( {A}_{0} = \left\{ {{u}_{1},\ldots ,{u}_{\alpha }}\right\} \) be an independent set in \( G \) of size \( \alpha \) . Let \( {A}_{1},\ldots ,{A}_{\omega } \) be the colour classes of an \( \omega \) -colouring of \( G - {u}_{1} \), let \( {A}_{\omega + 1},\ldots ,{A}_{2\omega } \) be the colour classes of an \( \omega \) -colouring of \( G - {u}_{2} \), and so on; altogether, this gives us \( {\alpha \omega } + 1 \) independent sets \( {A}_{0},{A}_{1},\ldots ,{A}_{\alpha \omega } \) in \( G \) . For each \( i = 0,\ldots ,{\alpha \omega } \), there exists by (1) a \( {K}^{\omega } \subseteq G - {A}_{i} \) ; we denote its vertex set by \( {K}_{i} \) .
Note that if \( K \) is the vertex set of any \( {K}^{\omega } \) in \( G \), then
\[
K \cap {A}_{i} = \varnothing \text{for exactly one}i \in \{ 0,\ldots ,{\alpha \omega }\} \text{.}
\]
(4)
Indeed, if \( K \cap {A}_{0} = \varnothing \) then \( K \cap {A}_{i} \neq \varnothing \) for all \( i \neq 0 \), by definition of \( {A}_{i} \) and (2). Similarly if \( K \cap {A}_{0} \neq \varnothing \), then \( \left| {K \cap {A}_{0}}\right| = 1 \), so \( K \cap {A}_{i} = \varnothing \) for exactly one \( i \neq 0 \) : apply (3) to the unique vertex \( u \in K \cap {A}_{0} \), and (2) to all the other vertices \( u \in {A}_{0} \) .
Let \( J \) be the real \( \left( {{\alpha \omega } + 1}\right) \times \left( {{\alpha \omega } + 1}\right) \) matrix with zero entries in the main diagonal and all other entries 1 . Let \( A \) be the real \( \left( {{\alpha \omega } + 1}\right) \times n \) matrix whose rows are the incidence vectors of the subsets \( {A}_{i} \subseteq V \) : if \( {a}_{i1},\ldots ,{a}_{in} \) denote the entries of the \( i \) th row of \( A \), then \( {a}_{ij} = 1 \) if \( {v}_{j} \in {A}_{i} \) , and \( {a}_{ij} = 0 \) otherwise. Similarly, let \( B \) denote the real \( n \times \left( {{\alpha \omega } + 1}\right) \) matrix whose columns are the incidence vectors of the subsets \( {K}_{i} \subseteq V \) . Now while \( \left| {{K}_{i} \cap {A}_{i}}\right| = 0 \) for all \( i \) by the choice of \( {K}_{i} \), we have \( {K}_{i} \cap {A}_{j} \neq \varnothing \) and hence \( \left| {{K}_{i} \cap {A}_{j}}\right| = 1 \) whenever \( i \neq j \), by (4). Thus,
\[
{AB} = J\text{.}
\]
Since \( J \) is non-singular, this implies that \( A \) has rank \( {\alpha \omega } + 1 \) . In particular, \( n \geq {\alpha \omega } + 1 \), which contradicts \( \left( *\right) \) for \( H \mathrel{\text{:=}} G \) .
## Exercises
1. \( {}^{ - } \) Show that the four colour theorem does indeed solve the map colouring problem stated in the first sentence of the chapter. Conversely, does the 4-colourability of every map imply the four colour theorem?
2. \( {}^{ - } \) Show that, for the map colouring problem above, it suffices to consider maps such that no point lies on the boundary of more than three countries. How does this affect the proof of the four colour theorem?
3. Try to turn the proof of the five colour theorem into one of the four colour theorem, as follows. Defining \( v \) and \( H \) as before, assume inductively that \( H \) has a 4-colouring; then proceed as before. Where does the proof fail?
4. Calculate the chromatic number of a graph in terms of the chromatic numbers of its blocks.
5. \( {}^{ - } \) Show that every graph \( G \) has a vertex ordering for which the greedy algorithm uses only \( \chi \left( G\right) \) colours.
6. For every \( n > 1 \), find a bipartite graph on \( {2n} \) vertices, ordered in such a way that the greedy algorithm uses \( n \) rather than 2 colours.
7. Consider the following approach to vertex colouring. First, find a maximal independent set of vertices and colour these with colour 1 ; then find a maximal independent set of vertices in the remaining graph and colour those 2, and so on. Compare this algorithm with the greedy algorithm: which is better?
8. Show that the bound of Proposition 5.2.2 is always at least as sharp as that of Proposition 5.2.1.
9. Find a lower bound for the colouring number in terms of average degree.
10. \( {}^{ - } \) A \( k \) -chromatic graph is called critically \( k \) -chromatic, or just critical, if \( \chi \left( {G - v}\right) < k \) for every \( v \in V\left( G\right) \) . Show that every \( k \) -chromatic graph has a critical \( k \) -chromatic induced subgraph, and that any such subgraph has minimum degree at least \( k - 1 \) .
11. Determine the critical 3-chromatic graphs.
12. \( {}^{ + } \) Show that every critical \( k \) -chromatic graph is \( \left( {k - 1}\right) \) - edge-connected.
13. Given \( k \in \mathbb{N} \), find a constant \( {c}_{k} > 0 \) such that every large enough graph \( G \) with \( \alpha \left( G\right) \leq k \) contains a cycle of length at least \( {c}_{k}\left| G\right| \) .
14. \( {}^{ - } \) Find a graph \( G \) for which Brooks’s theorem yields a significantly weaker bound on \( \chi \left( G\right) \) than Proposition 5.2.2.
15. \( {}^{ + } \) Show that, in order to prove Brooks’s theorem for a graph \( G = \left( {V, E}\right) \) , we may assume that \( \kappa \left( G\right) \geq 2 \) and \( \Delta \left( G\right) \geq 3 \) . Prove the theorem under these assumptions, showing first the following two lemmas.
(i) Let \( {v}_{1},\ldots ,{v}_{n} \) be an enumeration of \( V \) . If every \( {v}_{i}\left( {i < n}\right) \) has a neighbour \( {v}_{j} \) with \( j > i \), and if \( {v}_{1}{v}_{n},{v}_{2}{v}_{n} \in E \) but \( {v}_{1}{v}_{2} \notin E \) , then the greedy algorithm uses at most \( \Delta \left( G\right) \) colours.
(ii) If \( G \) is not complete and \( {v}_{n} \) has maximum degree in \( G \), then \( {v}_{n} \) has neighbours \( {v}_{1},{v}_{2} \) as in (i).
16. \( {}^{ + } \) Show that the following statements are equivalent for a graph \( G \) :
(i) \( \chi \left( G\right) \leq k \) ;
(ii) \( G \) has an orientation without directed paths of length \( k - 1 \) ;
(iii) \( G \) has an acyclic such orientation (one without directed cycles).
17. Given a graph \( G \) and \( k \in \mathbb{N} \), let \( {P}_{G}\left( k\right) \) denote the number of vertex colourings \( V\left( G\right) \rightarrow \{ 1,\ldots, k\} \) . Show that \( {P}_{G} \) is a polynomial in \( k \) of degree \( n \mathrel{\text{:=}} \left| G\right| \), in which the coefficient of \( {k}^{n} \) is 1 and the coefficient of \( {k}^{n - 1} \) is \( - \parallel G\parallel \) . ( \( {P}_{G} \) is called the chromatic polynomial of \( G \) .)
(Hint. Apply induction on \( \parallel G\parallel \) .)
18. \( {}^{ + } \) Determine the class of all graphs \( G \) for which \( {P}_{G}\left( k\right) = k{\left( k - 1\right) }^{n - 1} \) . (As in the previous exercise, let \( n \mathrel{\text{:=}} \left| G\right| \), and let \( {P}_{G} \) denote the chromatic polynomial of \( G \) .)
19. In the |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 47 | ht) \leq k \) ;
(ii) \( G \) has an orientation without directed paths of length \( k - 1 \) ;
(iii) \( G \) has an acyclic such orientation (one without directed cycles).
17. Given a graph \( G \) and \( k \in \mathbb{N} \), let \( {P}_{G}\left( k\right) \) denote the number of vertex colourings \( V\left( G\right) \rightarrow \{ 1,\ldots, k\} \) . Show that \( {P}_{G} \) is a polynomial in \( k \) of degree \( n \mathrel{\text{:=}} \left| G\right| \), in which the coefficient of \( {k}^{n} \) is 1 and the coefficient of \( {k}^{n - 1} \) is \( - \parallel G\parallel \) . ( \( {P}_{G} \) is called the chromatic polynomial of \( G \) .)
(Hint. Apply induction on \( \parallel G\parallel \) .)
18. \( {}^{ + } \) Determine the class of all graphs \( G \) for which \( {P}_{G}\left( k\right) = k{\left( k - 1\right) }^{n - 1} \) . (As in the previous exercise, let \( n \mathrel{\text{:=}} \left| G\right| \), and let \( {P}_{G} \) denote the chromatic polynomial of \( G \) .)
19. In the definition of \( k \) -constructible graphs, replace the axiom (ii) by
(ii)' Every supergraph of a \( k \) -constructible graph is \( k \) -constructible;
and the axiom (iii) by
(iii) \( {}^{\prime } \) If \( G \) is a graph with vertices \( x,{y}_{1},{y}_{2} \) such that \( {y}_{1}{y}_{2} \in E\left( G\right) \) but \( x{y}_{1}, x{y}_{2} \notin E\left( G\right) \), and if both \( G + x{y}_{1} \) and \( G + x{y}_{2} \) are \( k \) - constructible, then \( G \) is \( k \) -constructible.
Show that a graph is \( k \) -constructible with respect to this new definition if and only if its chromatic number is at least \( k \) .
20. \( {}^{ - } \) An \( n \times n \) -matrix with entries from \( \{ 1,\ldots, n\} \) is called a Latin square if every element of \( \{ 1,\ldots, n\} \) appears exactly once in each column and exactly once in each row. Recast the problem of constructing Latin squares as a colouring problem.
21. Without using Proposition 5.3.1, show that \( {\chi }^{\prime }\left( G\right) = k \) for every \( k \) - regular bipartite graph \( G \) .
22. Prove Proposition 5.3.1 from the statement of the previous exercise.
23. \( {}^{ + } \) For every \( k \in \mathbb{N} \), construct a triangle-free \( k \) -chromatic graph.
24. \( {}^{ - } \) Without using Theorem 5.4.2, show that every plane graph is 6-list-colourable.
25. For every integer \( k \), find a 2-chromatic graph whose choice number is at least \( k \) .
26. \( {}^{ - } \) Find a general upper bound for \( {\operatorname{ch}}^{\prime }\left( G\right) \) in terms of \( {\chi }^{\prime }\left( G\right) \) .
27. Compare the choice number of a graph with its colouring number: which is greater? Can you prove the analogue of Theorem 5.4.1 for the colouring number?
28. \( {}^{ + } \) Prove that the choice number of \( {K}_{2}^{r} \) is \( r \) .
29. The total chromatic number \( {\chi }^{\prime \prime }\left( G\right) \) of a graph \( G = \left( {V, E}\right) \) is the least number of colours needed to colour the vertices and edges of \( G \) simultaneously so that any adjacent or incident elements of \( V \cup E \) are coloured differently. The total colouring conjecture says that \( {\chi }^{\prime \prime }\left( G\right) \leq \Delta \left( G\right) + 2 \) . Bound the total chromatic number from above in terms of the list-chromatic index, and use this bound to deduce a weakening of the total colouring conjecture from the list colouring conjecture.
30. \( {}^{ - } \) Does every oriented graph have a kernel? If not, does every graph admit an orientation in which every induced subgraph has a kernel? If not, does every graph admit an orientation that has a kernel?
31. \( {}^{ + } \) Prove Richardson’s theorem: every directed graph without odd directed cycles has a kernel.
32. Show that every bipartite planar graph is 3-list-colourable.
(Hint. Apply the previous exercise and Lemma 5.4.3.)
33. \( {}^{ - } \) Show that perfection is closed neither under edge deletion nor under edge contraction.
34. \( {}^{ - } \) Deduce Theorem 5.5.6 from the strong perfect graph theorem.
35. Let \( {\mathcal{H}}_{1} \) and \( {\mathcal{H}}_{2} \) be two sets of imperfect graphs, each minimal with the property that a graph is perfect if and only if it has no induced subgraph in \( {\mathcal{H}}_{i}\left( {i = 1,2}\right) \) . Do \( {\mathcal{H}}_{1} \) and \( {\mathcal{H}}_{2} \) contain the same graphs, up to isomorphism?
36. Use König's Theorem 2.1.1 to show that the complement of any bipartite graph is perfect.
37. Using the results of this chapter, find a one-line proof of the following theorem of König, the dual of Theorem 2.1.1: in any bipartite graph without isolated vertices, the minimum number of edges meeting all vertices equals the maximum number of independent vertices.
38. A graph is called a comparability graph if there exists a partial ordering of its vertex set such that two vertices are adjacent if and only if they are comparable. Show that every comparability graph is perfect.
39. A graph \( G \) is called an interval graph if there exists a set \( \left\{ {{I}_{v} \mid v \in V\left( G\right) }\right\} \) of real intervals such that \( {I}_{u} \cap {I}_{v} \neq \varnothing \) if and only if \( {uv} \in E\left( G\right) \) .
(i) Show that every interval graph is chordal.
(ii) Show that the complement of any interval graph is a comparability graph.
(Conversely, a chordal graph is an interval graph if its complement is a comparability graph; this is a theorem of Gilmore and Hoffman (1964).)
40. Show that \( \chi \left( H\right) \in \{ \omega \left( H\right) ,\omega \left( H\right) + 1\} \) for every line graph \( H \) .
41. \( {}^{ + } \) Characterize the graphs whose line graphs are perfect.
42. Show that a graph \( G \) is perfect if and only if every non-empty induced subgraph \( H \) of \( G \) contains an independent set \( A \subseteq V\left( H\right) \) such that \( \omega \left( {H - A}\right) < \omega \left( H\right) \) .
43. \( {}^{ + } \) Consider the graphs \( G \) for which every induced subgraph \( H \) has the property that every maximal complete subgraph of \( H \) meets every maximal independent vertex set in \( H \) .
(i) Show that these graphs \( G \) are perfect.
(ii) Show that these graphs \( G \) are precisely the graphs not containing an induced copy of \( {P}^{3} \) .
44. \( {}^{ + } \) Show that in every perfect graph \( G \) one can find a set \( \mathcal{A} \) of independent vertex sets and a set \( \mathcal{O} \) of vertex sets of complete subgraphs such that \( \bigcup \mathcal{A} = V\left( G\right) = \bigcup \mathcal{O} \) and every set in \( \mathcal{A} \) meets every set in \( \mathcal{O} \) .
(Hint. Lemma 5.5.5.)
45. \( {}^{ + } \) Let \( G \) be a perfect graph. As in the proof of Theorem 5.5.4, replace every vertex \( x \) of \( G \) with a perfect graph \( {G}_{x} \) (not necessarily complete). Show that the resulting graph \( {G}^{\prime } \) is again perfect.
## Notes
The authoritative reference work on all questions of graph colouring is T.R. Jensen & B. Toft, Graph Coloring Problems, Wiley 1995. Starting with a brief survey of the most important results and areas of research in the field, this monograph gives a detailed account of over 200 open colouring problems, complete with extensive background surveys and references. Most of the remarks below are discussed comprehensively in this book, and all the references for this chapter can be found there.
The four colour problem, whether every map can be coloured with four colours so that adjacent countries are shown in different colours, was raised by a certain Francis Guthrie in 1852. He put the question to his brother Frederick, who was then a mathematics undergraduate in Cambridge. The problem was first brought to the attention of a wider public when Cayley presented it to the London Mathematical Society in 1878. A year later, Kempe published an incorrect proof, which was in 1890 modified by Heawood into a proof of the five colour theorem. In 1880, Tait announced 'further proofs' of the four colour conjecture, which never materialized; see the notes for Chapter 10.
The first generally accepted proof of the four colour theorem was published by Appel and Haken in 1977. The proof builds on ideas that can be traced back as far as Kempe's paper, and were developed largely by Birkhoff and Heesch. Very roughly, the proof sets out first to show that every plane triangulation must contain at least one of 1482 certain 'unavoidable configurations'. In a second step, a computer is used to show that each of those configurations is 'reducible', i.e., that any plane triangulation containing such a configuration can be 4-coloured by piecing together 4-colourings of smaller plane triangulations. Taken together, these two steps amount to an inductive proof that all plane triangulations, and hence all planar graphs, can be 4- coloured.
Appel & Haken's proof has not been immune to criticism, not only because of their use of a computer. The authors responded with a 741 page long algorithmic version of their proof, which addresses the various criticisms and corrects a number of errors (e.g. by adding more configurations to the 'unavoidable' list): K. Appel & W. Haken, Every Planar Map is Four Colorable, American Mathematical Society 1989. A much shorter proof, which is based on the same ideas (and, in particular, uses a computer in the same way) but can be more readily verified both in its verbal and its computer part, has been given by N. Robertson, D. Sanders, P.D. Seymour & R. Thomas, The four-colour theorem, J. Combin. Theory B 70 (1997), 2-44.
A relatively short proof of Grötzsch's theorem was found by C. Thomassen, A short list color proof of Grötzsch's theorem, J. Combin. Theory B 88 (2003), 189-192. Although not touched upon in this chapter, colouring problems for graphs embedded in surfaces other than the plane form a substantial and interesting part of colour |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 48 | ism, not only because of their use of a computer. The authors responded with a 741 page long algorithmic version of their proof, which addresses the various criticisms and corrects a number of errors (e.g. by adding more configurations to the 'unavoidable' list): K. Appel & W. Haken, Every Planar Map is Four Colorable, American Mathematical Society 1989. A much shorter proof, which is based on the same ideas (and, in particular, uses a computer in the same way) but can be more readily verified both in its verbal and its computer part, has been given by N. Robertson, D. Sanders, P.D. Seymour & R. Thomas, The four-colour theorem, J. Combin. Theory B 70 (1997), 2-44.
A relatively short proof of Grötzsch's theorem was found by C. Thomassen, A short list color proof of Grötzsch's theorem, J. Combin. Theory B 88 (2003), 189-192. Although not touched upon in this chapter, colouring problems for graphs embedded in surfaces other than the plane form a substantial and interesting part of colouring theory; see B. Mohar & C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press 2001.
The proof of Brooks's theorem indicated in Exercise 15, where the greedy algorithm is applied to a carefully chosen vertex ordering, is due to Lovász (1973). Lovász (1968) was also the first to construct graphs of arbitrarily large girth and chromatic number, graphs whose existence Erdős had proved by probabilistic methods ten years earlier.
A. Urquhart, The graph constructions of Hajós and Ore, J. Graph Theory 26 (1997), 211-215, showed that not only do the graphs of chromatic number at least \( k \) each contain a \( k \) -constructible graph (as by Hajós’s theorem); they are in fact all themselves \( k \) -constructible. Algebraic tools for showing that the chromatic number of a graph is large have been developed by Kleitman & Lovász (1982), and by Alon & Tarsi (1992); see Alon's paper cited below.
List colourings were first introduced in 1976 by Vizing. Among other things, Vizing proved the list-colouring equivalent of Brooks's theorem. Voigt (1993) constructed a plane graph of order 238 that is not 4-choosable; thus, Thomassen's list version of the five colour theorem is best possible. A stimulating survey on the list-chromatic number and how it relates to the more classical graph invariants (including a proof of Theorem 5.4.1) is given by N. Alon, Restricted colorings of graphs, in (K. Walker, ed.) Surveys in Combinatorics, LMS Lecture Notes 187, Cambridge University Press 1993. Both the list colouring conjecture and Galvin's proof of the bipartite case are originally stated for multigraphs. Kahn (1994) proved that the conjecture is asymptotically correct, as follows: given any \( \epsilon > 0 \), every graph \( G \) with large enough maximum degree satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) \leq \left( {1 + \epsilon }\right) \Delta \left( G\right) \) .
The total colouring conjecture was proposed around 1965 by Vizing and by Behzad; see Jensen & Toft for details.
A gentle introduction to the basic facts about perfect graphs and their applications is given by M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press 1980. A more comprehensive treatment is given in A. Schrijver, Combinatorial optimization, Springer 2003. Surveys on various aspects of perfect graphs are included in Perfect Graphs by J. Ramirez-Alfonsin & B. Reed (eds.), Wiley 2001. Our first proof of the perfect graph theorem, Theorem 5.5.4, follows L. Lovász's survey on perfect graphs in (L.W. Beineke and R.J. Wilson, eds.) Selected Topics in Graph Theory 2, Academic Press 1983. Our second proof, the proof of Theorem 5.5.6, is due to G.S. Gasparian, Minimal imperfect graphs: a simple approach, Combinatori-ca 16 (1996), 209-212. Theorem 5.5.3 was proved by Chudnovsky, Robertson, Seymour and Thomas, The strong perfect graph theorem, Ann. of Math. (to appear). Chudnovsky, Cornuejols, Liu, Seymour and Vušković, Recognizing Berge graphs, Combinatorica \( \mathbf{{25}} \) (2005),143-186, constructed an \( O\left( {n}^{9}\right) \) algorithm testing for 'holes' (induced odd cycles of length at least 5) and 'antiholes' (their induced complements), and thus by the theorem for perfection.
## Flows
Let us view a graph as a network: its edges carry some kind of flow-of water, electricity, data or similar. How could we model this precisely?
For a start, we ought to know how much flow passes through each edge \( e = {xy} \), and in which direction. In our model, we could assign a positive integer \( k \) to the pair \( \left( {x, y}\right) \) to express that a flow of \( k \) units passes through \( e \) from \( x \) to \( y \), or assign \( - k \) to \( \left( {x, y}\right) \) to express that \( k \) units of flow pass through \( e \) the other way, from \( y \) to \( x \) . For such an assignment \( f : {V}^{2} \rightarrow \mathbb{Z} \) we would thus have \( f\left( {x, y}\right) = - f\left( {y, x}\right) \) whenever \( x \) and \( y \) are adjacent vertices of \( G \) .
Typically, a network will have only a few nodes where flow enters or leaves the network; at all other nodes, the total amount of flow into that node will equal the total amount of flow out of it. For our model this means that, at most nodes \( x \), the function \( f \) will satisfy Kirchhoff’s law
---
Kirchhoff's
law
---
\[
\mathop{\sum }\limits_{{y \in N\left( x\right) }}f\left( {x, y}\right) = 0
\]
In this chapter, we call any map \( f : {V}^{2} \rightarrow \mathbb{Z} \) with the above two properties a ’flow’ on \( G \) . Sometimes, we shall replace \( \mathbb{Z} \) with another group, and as a rule we consider multigraphs rather than graphs. \( {}^{1} \) As it turns out, the theory of those 'flows' is not only useful as a model for real flows: it blends so well with other parts of graph theory that some deep and surprising connections become visible, connections particularly with connectivity and colouring problems.
---
1 For consistency, we shall phrase some of our proposition for graphs only: those whose proofs rely on assertions proved (for graphs) earlier in the book. However, all those results remain true for multigraphs.
---
## 6.1 Circulations
In the context of flows, we have to be able to speak about the 'directions'
\( G = \left( {V, E}\right) \)
of an edge. Since, in a multigraph \( G = \left( {V, E}\right) \), an edge \( e = {xy} \) is not identified uniquely by the pair \( \left( {x, y}\right) \) or \( \left( {y, x}\right) \), we define directed edges as triples:
\( \overrightarrow{E} \)
\[
\overrightarrow{E} \mathrel{\text{:=}} \{ \left( {e, x, y}\right) \mid e \in E;x, y \in V;e = {xy}\} .
\]
direction Thus, an edge \( e = {xy} \) with \( x \neq y \) has the two directions \( \left( {e, x, y}\right) \) and \( \left( {e, x, y}\right) \) \( \left( {e, y, x}\right) \) ; a loop \( e = {xx} \) has only one direction, the triple \( \left( {e, x, x}\right) \) . For \( \overleftarrow{e} \) given \( \overrightarrow{e} = \left( {e, x, y}\right) \in \overrightarrow{E} \), we set \( \overleftarrow{e} \mathrel{\text{:=}} \left( {e, y, x}\right) \), and for an arbitrary set \( \overrightarrow{F} \subseteq \overrightarrow{E} \) of edge directions we put
\( \overleftarrow{F} \)
\[
\overleftarrow{F} \mathrel{\text{:=}} \{ \overleftarrow{e} \mid \overrightarrow{e} \in \overrightarrow{F}\}
\]
Note that \( \overrightarrow{E} \) itself is symmetrical: \( \bar{E} = \overrightarrow{E} \) . For \( X, Y \subseteq V \) and \( \overrightarrow{F} \subseteq \overrightarrow{E} \) , define
\( \overrightarrow{F}\left( {X, Y}\right) \)
\[
\overrightarrow{F}\left( {X, Y}\right) \mathrel{\text{:=}} \{ \left( {e, x, y}\right) \in \overrightarrow{F} \mid x \in X;y \in Y;x \neq y\} ,
\]
\( \overrightarrow{F}\left( {x, Y}\right) \) abbreviate \( \overrightarrow{F}\left( {\{ x\}, Y}\right) \) to \( \overrightarrow{F}\left( {x, Y}\right) \) etc., and write
\( \overrightarrow{F}\left( x\right) \)
\[
\overrightarrow{F}\left( x\right) \mathrel{\text{:=}} \overrightarrow{F}\left( {x, V}\right) = \overrightarrow{F}\left( {\{ x\} ,\overline{\{ x\} }}\right) .
\]
\( \bar{X} \) Here, as below, \( \bar{X} \) denotes the complement \( V \smallsetminus X \) of a vertex set \( X \subseteq V \) . Note that any loops at vertices \( x \in X \cap Y \) are disregarded in the definitions of \( \overrightarrow{F}\left( {X, Y}\right) \) and \( \overrightarrow{F}\left( x\right) \) .
Let \( H \) be an abelian semigroup, \( {}^{2} \) written additively with zero 0 . Given vertex sets \( X, Y \subseteq V \) and a function \( f : \overrightarrow{E} \rightarrow H \), let
\( f\left( {X, Y}\right) \)
\[
f\left( {X, Y}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\overrightarrow{e} \in \overrightarrow{E}\left( {X, Y}\right) }}f\left( \overrightarrow{e}\right)
\]
\( f\left( {x, Y}\right) \)
Instead of \( f\left( {\{ x\}, Y}\right) \) we again write \( f\left( {x, Y}\right) \), etc.
circulation From now on, we assume that \( H \) is a group. We call \( f \) a circulation on \( G \) (with values in \( H \) ), or an \( H \) -circulation, if \( f \) satisfies the following two conditions:
F1) \( f\left( {e, x, y}\right) = - f\left( {e, y, x}\right) \) for all \( \left( {e, x, y}\right) \in \overrightarrow{E} \) with \( x \neq y \) ;
(F2) \( f\left( {v, V}\right) = 0 \) for all \( v \in V \) .
---
2 This chapter contains no group theory. The only semigroups we ever consider for \( H \) are the natural numbers, the integers, the reals, the cyclic groups \( {\mathbb{Z}}_{k} \), and (once) the Klein four-group.
---
If \( f \) satisfies (F1), then
\[
f\left( {X, X}\right) = 0
\]
for all \( X \subseteq V \) . If \( f \) satisfies (F2), then
\[
f\left( {X, V}\right) = \mathop{\sum }\limits_{{x \in X}}f\left( {x, V}\right) = 0.
\]
Together, these two basic observations imply that, in a circulation, the net flow across any cut is zero:
Proposition 6.1.1. If \( f \) is a circulation, |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 49 | on, we assume that \( H \) is a group. We call \( f \) a circulation on \( G \) (with values in \( H \) ), or an \( H \) -circulation, if \( f \) satisfies the following two conditions:
F1) \( f\left( {e, x, y}\right) = - f\left( {e, y, x}\right) \) for all \( \left( {e, x, y}\right) \in \overrightarrow{E} \) with \( x \neq y \) ;
(F2) \( f\left( {v, V}\right) = 0 \) for all \( v \in V \) .
---
2 This chapter contains no group theory. The only semigroups we ever consider for \( H \) are the natural numbers, the integers, the reals, the cyclic groups \( {\mathbb{Z}}_{k} \), and (once) the Klein four-group.
---
If \( f \) satisfies (F1), then
\[
f\left( {X, X}\right) = 0
\]
for all \( X \subseteq V \) . If \( f \) satisfies (F2), then
\[
f\left( {X, V}\right) = \mathop{\sum }\limits_{{x \in X}}f\left( {x, V}\right) = 0.
\]
Together, these two basic observations imply that, in a circulation, the net flow across any cut is zero:
Proposition 6.1.1. If \( f \) is a circulation, then \( f\left( {X,\bar{X}}\right) = 0 \) for every set \( X \subseteq V \) .
Proof. \( f\left( {X,\bar{X}}\right) = f\left( {X, V}\right) - f\left( {X, X}\right) = 0 - 0 = 0 \) .
Since bridges form cuts by themselves, Proposition 6.1.1 implies that circulations are always zero on bridges:
Corollary 6.1.2. If \( f \) is a circulation and \( e = {xy} \) is a bridge in \( G \), then \( f\left( {e, x, y}\right) = 0 \) .
## 6.2 Flows in networks
In this section we give a brief introduction to the kind of network flow theory that is now a standard proof technique in areas such as matching and connectivity. By way of example, we shall prove a classic result of this theory, the so-called max-flow min-cut theorem of Ford and Fulk-erson. This theorem alone implies Menger's theorem without much difficulty (Exercise 3), which indicates some of the natural power lying in this approach.
Consider the task of modelling a network with one source \( s \) and one sink \( t \), in which the amount of flow through a given link between two nodes is subject to a certain capacity of that link. Our aim is to determine the maximum net amount of flow through the network from \( s \) to \( t \) . Somehow, this will depend both on the structure of the network and on the various capacities of its connections - how exactly, is what we wish to find out.
Let \( G = \left( {V, E}\right) \) be a multigraph, \( s, t \in V \) two fixed vertices, and
---
\( G = \left( {V, E}\right) \) \( s, t, c, N \) network flow
---
\( c : \overrightarrow{E} \rightarrow \mathbb{N} \) a map; we call \( c \) a capacity function on \( G \), and the tuple \( N \mathrel{\text{:=}} \left( {G, s, t, c}\right) \) a network. Note that \( c \) is defined independently for the two directions of an edge. A function \( f : \overrightarrow{E} \rightarrow \mathbb{R} \) is a flow in \( N \) if it satisfies the following three conditions (Fig. 6.2.1):
(F1) \( f\left( {e, x, y}\right) = - f\left( {e, y, x}\right) \) for all \( \left( {e, x, y}\right) \in \overrightarrow{E} \) with \( x \neq y \) ;
\( \left( {\mathrm{F}{2}^{\prime }}\right) f\left( {v, V}\right) = 0 \) for all \( v \in V \smallsetminus \{ s, t\} \)
(F3) \( f\left( \overrightarrow{e}\right) \leq c\left( \overrightarrow{e}\right) \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) .
integral We call \( f \) integral if all its values are integers.
Fig. 6.2.1. A network flow in short notation: all values refer to
the direction indicated (capacities are not shown)
\( f \) Let \( f \) be a flow in \( N \) . If \( S \subseteq V \) is such that \( s \in S \) and \( t \in \bar{S} \), we call
cut in \( N \) the pair \( \left( {S,\bar{S}}\right) \) a cut in \( N \), and \( c\left( {S,\bar{S}}\right) \) the capacity of this cut.
capacity Since \( f \) now has to satisfy only \( \left( {\mathrm{F}{2}^{\prime }}\right) \) rather than \( \left( {\mathrm{F}2}\right) \), we no longer have \( f\left( {X,\bar{X}}\right) = 0 \) for all \( X \subseteq V \) (as in Proposition 6.1.1). However, the value is the same for all cuts:
Proposition 6.2.1. Every cut \( \left( {S,\bar{S}}\right) \) in \( N \) satisfies \( f\left( {S,\bar{S}}\right) = f\left( {s, V}\right) \) .
Proof. As in the proof of Proposition 6.1.1, we have
\[
f\left( {S,\bar{S}}\right) = f\left( {S, V}\right) - f\left( {S, S}\right)
\]
\[
\underset{\left( \mathrm{F}1\right) }{ = }f\left( {s, V}\right) + \mathop{\sum }\limits_{{v \in S\smallsetminus \{ s\} }}f\left( {v, V}\right) - 0
\]
\[
\underset{\left( {\mathrm{{F2}}}^{\prime }\right) }{ = }f\left( {s, V}\right) \text{.}
\]
---
total value \( \left| f\right| \)
---
The common value of \( f\left( {S,\bar{S}}\right) \) in Proposition 6.2.1 will be called the total value of \( f \) and denoted by \( \left| f\right| ;{}^{3} \) the flow shown in Figure 6.2.1 has total value 3.
By (F3), we have
\[
\left| f\right| = f\left( {S,\bar{S}}\right) \leq c\left( {S,\bar{S}}\right)
\]
for every cut \( \left( {S,\bar{S}}\right) \) in \( N \) . Hence the total value of a flow in \( N \) is never larger than the smallest capacity of a cut. The following max-flow min-cut theorem states that this upper bound is always attained by some flow:
---
3 Thus, formally, \( \left| f\right| \) may be negative. In practice, however, we can change the sign of \( \left| f\right| \) simply by swapping the roles of \( s \) and \( t \) .
---
Theorem 6.2.2. (Ford & Fulkerson 1956)
---
max-flow min-cut theorem
---
In every network, the maximum total value of a flow equals the minimum capacity of a cut.
Proof. Let \( N = \left( {G, s, t, c}\right) \) be a network, and \( G = : \left( {V, E}\right) \) . We shall define a sequence \( {f}_{0},{f}_{1},{f}_{2},\ldots \) of integral flows in \( N \) of strictly increasing total value, i.e. with
\[
\left| {f}_{0}\right| < \left| {f}_{1}\right| < \left| {f}_{2}\right| < \ldots
\]
Clearly, the total value of an integral flow is again an integer, so in fact \( \left| {f}_{n + 1}\right| \geq \left| {f}_{n}\right| + 1 \) for all \( n \) . Since all these numbers are bounded above by the capacity of any cut in \( N \), our sequence will terminate with some flow \( {f}_{n} \) . Corresponding to this flow, we shall find a cut of capacity \( {c}_{n} = \left| {f}_{n}\right| \) . Since no flow can have a total value greater than \( {c}_{n} \), and no cut can have a capacity less than \( \left| {f}_{n}\right| \), this number is simultaneously the maximum and the minimum referred to in the theorem.
For \( {f}_{0} \), we set \( {f}_{0}\left( \overrightarrow{e}\right) \mathrel{\text{:=}} 0 \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) . Having defined an integral flow \( {f}_{n} \) in \( N \) for some \( n \in \mathbb{N} \), we denote by \( {S}_{n} \) the set of all vertices \( v \) such that \( G \) contains an \( s - v \) walk \( {x}_{0}{e}_{0}\ldots {e}_{\ell - 1}{x}_{\ell } \) with
\[
{f}_{n}\left( \overrightarrow{{e}_{i}}\right) < c\left( \overrightarrow{{e}_{i}}\right)
\]
for all \( i < \ell \) ; here, \( \overrightarrow{{e}_{i}} \mathrel{\text{:=}} \left( {{e}_{i},{x}_{i},{x}_{i + 1}}\right) \) (and, of course, \( {x}_{0} = s \) and \( {x}_{\ell } = v \) ).
If \( t \in {S}_{n} \), let \( W = {x}_{0}{e}_{0}\ldots {e}_{\ell - 1}{x}_{\ell } \) be the corresponding \( s - t \) walk; without loss of generality we may assume that \( W \) does not repeat any vertices. Let
\[
\epsilon \mathrel{\text{:=}} \min \left\{ {c\left( \overrightarrow{{e}_{i}}\right) - {f}_{n}\left( \overrightarrow{{e}_{i}}\right) \mid i < \ell }\right\} .
\]
Then \( \epsilon > 0 \), and since \( {f}_{n} \) (like \( c \) ) is integral by assumption, \( \epsilon \) is an integer. Let
\[
{f}_{n + 1} : \overrightarrow{e} \mapsto \left\{ \begin{array}{ll} {f}_{n}\left( \overrightarrow{e}\right) + \epsilon & \text{ for }\overrightarrow{e} = \overrightarrow{{e}_{i}}, i = 0,\ldots ,\ell - 1; \\ {f}_{n}\left( \overrightarrow{e}\right) - \epsilon & \text{ for }\overrightarrow{e} = \dot{{e}_{i}}, i = 0,\ldots ,\ell - 1; \\ {f}_{n}\left( \overrightarrow{e}\right) & \text{ for }e \notin W. \end{array}\right.
\]
Intuitively, \( {f}_{n + 1} \) is obtained from \( {f}_{n} \) by sending additional flow of value \( \epsilon \) along \( W \) from \( s \) to \( t \) (Fig. 6.2.2).
Fig. 6.2.2. An ’augmenting path’ \( W \) with increment \( \epsilon = 2 \), for constant flow \( {f}_{n} = 0 \) and capacities \( c = 3 \)
Clearly, \( {f}_{n + 1} \) is again an integral flow in \( N \) . Let us compute its total value \( \left| {f}_{n + 1}\right| = {f}_{n + 1}\left( {s, V}\right) \) . Since \( W \) contains the vertex \( s \) only once, \( {\bar{e}}_{0} \) is the only triple \( \left( {e, x, y}\right) \) with \( x = s \) and \( y \in V \) whose \( f \) -value was changed. This value, and hence that of \( {f}_{n + 1}\left( {s, V}\right) \) was raised. Therefore \( \left| {f}_{n + 1}\right| > \left| {f}_{n}\right| \) as desired.
If \( t \notin {S}_{n} \), then \( \left( {{S}_{n},\overline{{S}_{n}}}\right) \) is a cut in \( N \) . By (F3) for \( {f}_{n} \), and the definition of \( {S}_{n} \), we have
\[
{f}_{n}\left( \overrightarrow{e}\right) = c\left( \overrightarrow{e}\right)
\]
for all \( \overrightarrow{e} \in \overrightarrow{E}\left( {{S}_{n},\overline{{S}_{n}}}\right) \), so
\[
\left| {f}_{n}\right| = {f}_{n}\left( {{S}_{n},\overline{{S}_{n}}}\right) = c\left( {{S}_{n},\overline{{S}_{n}}}\right)
\]
as desired.
Since the flow constructed in the proof of Theorem 6.2.2 is integral, we have also proved the following:
Corollary 6.2.3. In every network (with integral capacity function) there exists an integral flow of maximum total value.
## 6.3 Group-valued flows
Let \( G = \left( {V, E}\right) \) be a multig |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 50 | \) whose \( f \) -value was changed. This value, and hence that of \( {f}_{n + 1}\left( {s, V}\right) \) was raised. Therefore \( \left| {f}_{n + 1}\right| > \left| {f}_{n}\right| \) as desired.
If \( t \notin {S}_{n} \), then \( \left( {{S}_{n},\overline{{S}_{n}}}\right) \) is a cut in \( N \) . By (F3) for \( {f}_{n} \), and the definition of \( {S}_{n} \), we have
\[
{f}_{n}\left( \overrightarrow{e}\right) = c\left( \overrightarrow{e}\right)
\]
for all \( \overrightarrow{e} \in \overrightarrow{E}\left( {{S}_{n},\overline{{S}_{n}}}\right) \), so
\[
\left| {f}_{n}\right| = {f}_{n}\left( {{S}_{n},\overline{{S}_{n}}}\right) = c\left( {{S}_{n},\overline{{S}_{n}}}\right)
\]
as desired.
Since the flow constructed in the proof of Theorem 6.2.2 is integral, we have also proved the following:
Corollary 6.2.3. In every network (with integral capacity function) there exists an integral flow of maximum total value.
## 6.3 Group-valued flows
Let \( G = \left( {V, E}\right) \) be a multigraph and \( H \) an abelian group. If \( f \) and \( f + g \) \( g \) are two \( H \) -circulations then, clearly, \( \left( {f + g}\right) : \overrightarrow{e} \mapsto f\left( \overrightarrow{e}\right) + g\left( \overrightarrow{e}\right) \) and \( - f \) \( - f : \overrightarrow{e} \mapsto - f\left( \overrightarrow{e}\right) \) are again \( H \) -circulations. The \( H \) -circulations on \( G \) thus form a group in a natural way.
nowhere A function \( f : \overrightarrow{E} \rightarrow H \) is nowhere zero if \( f\left( \overrightarrow{e}\right) \neq 0 \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) . An zero \( H \) -circulation that is nowhere zero is called an \( H \) -flow. \( {}^{4} \) Note that the
\( H \) -flow set of \( H \) -flows on \( G \) is not closed under addition: if two \( H \) -flows add up to zero on some edge \( \overrightarrow{e} \), then their sum is no longer an \( H \) -flow. By Corollary 6.1.2, a graph with an \( H \) -flow cannot have a bridge.
For finite groups \( H \), the number of \( H \) -flows on \( G \) - and, in particular, their existence surprisingly depends only on the order of \( H \), not on \( H \) itself:
Theorem 6.3.1. (Tutte 1954)
For every multigraph \( G \) there exists a polynomial \( P \) such that, for any finite abelian group \( H \), the number of \( H \) -flows on \( G \) is \( P\left( {\left| H\right| - 1}\right) \) .
---
4 This terminology seems simplest for our purposes but is not standard; see the footnote in the notes.
---
Proof. Let \( G = : \left( {V, E}\right) \) ; we use induction on \( m \mathrel{\text{:=}} \left| E\right| \) . Let us assume
(6.1.1)
first that all the edges of \( G \) are loops. Then, given any finite abelian group \( H \), every map \( \overrightarrow{E} \rightarrow H \smallsetminus \{ 0\} \) is an \( H \) -flow on \( G \) . Since \( \left| \overrightarrow{E}\right| = \left| E\right| \) when all edges are loops, there are \( {\left( \left| H\right| - 1\right) }^{m} \) such maps, and \( P \mathrel{\text{:=}} {x}^{m} \) is the polynomial sought.
Now assume there is an edge \( {e}_{0} = {xy} \in E \) that is not a loop; let \( {e}_{0} = {xy} \) \( \overrightarrow{{e}_{0}} \mathrel{\text{:=}} \left( {{e}_{0}, x, y}\right) \) and \( {E}^{\prime } \mathrel{\text{:=}} E \smallsetminus \left\{ {e}_{0}\right\} \) . We consider the multigraphs
\[
{G}_{1} \mathrel{\text{:=}} G - {e}_{0}\;\text{ and }\;{G}_{2} \mathrel{\text{:=}} G/{e}_{0}.
\]
By the induction hypothesis, there are polynomials \( {P}_{i} \) for \( i = 1,2 \) such \( {P}_{1},{P}_{2} \) that, for any finite abelian group \( H \) and \( k \mathrel{\text{:=}} \left| H\right| - 1 \), the number of \( H \) -flows on \( {G}_{i} \) is \( {P}_{i}\left( k\right) \) . We shall prove that the number of \( H \) -flows on \( G \) equals \( {P}_{2}\left( k\right) - {P}_{1}\left( k\right) \) ; then \( P \mathrel{\text{:=}} {P}_{2} - {P}_{1} \) is the desired polynomial.
Let \( H \) be given, and denote the set of all \( H \) -flows on \( G \) by \( F \) . We are trying to show that
\[
\left| F\right| = {P}_{2}\left( k\right) - {P}_{1}\left( k\right)
\]
(1)
The \( H \) -flows on \( {G}_{1} \) are precisely the restrictions to \( \overrightarrow{{E}^{\prime }} \) of those \( H \) -circulations on \( G \) that are zero on \( {e}_{0} \) but nowhere else. Let us denote the set of these circulations on \( G \) by \( {F}_{1} \) ; then
\[
{P}_{1}\left( k\right) = \left| {F}_{1}\right|
\]
Our aim is to show that, likewise, the \( H \) -flows on \( {G}_{2} \) correspond bijectively to those \( H \) -circulations on \( G \) that are nowhere zero except possibly on \( {e}_{0} \) . The set \( {F}_{2} \) of those circulations on \( G \) then satisfies
\[
{P}_{2}\left( k\right) = \left| {F}_{2}\right|
\]
and \( {F}_{2} \) is the disjoint union of \( {F}_{1} \) and \( F \) . This will prove (1), and hence the theorem.
Fig. 6.3.1. Contracting the edge \( {e}_{0} \)
In \( {G}_{2} \), let \( {v}_{0} \mathrel{\text{:=}} {v}_{{e}_{0}} \) be the vertex contracted from \( {e}_{0} \) (Fig. 6.3.1; see Chapter 1.10). We are looking for a bijection \( f \mapsto g \) between \( {F}_{2} \) and the set of \( H \) -flows on \( {G}_{2} \) . Given \( f \), let \( g \) be the restriction of \( f \) to \( \overrightarrow{{E}^{\prime }} \smallsetminus \overrightarrow{{E}^{\prime }}\left( {y, x}\right) \) . (As the \( x - y \) edges \( e \in {E}^{\prime } \) become loops in \( {G}_{2} \) , they have only the one direction \( \left( {e,{v}_{0},{v}_{0}}\right) \) there; as its \( g \) -value, we choose \( f\left( {e, x, y}\right) \) .) Then \( g \) is indeed an \( H \) -flow on \( {G}_{2} \) ; note that (F2) holds at \( {v}_{0} \) by Proposition 6.1.1 for \( G \), with \( X \mathrel{\text{:=}} \{ x, y\} \) .
It remains to show that the map \( f \mapsto g \) is a bijection. If we are given an \( H \) -flow \( g \) on \( {G}_{2} \) and try to find an \( f \in {F}_{2} \) with \( f \mapsto g \), then \( f\left( \overrightarrow{e}\right) \) is already determined as \( f\left( \overrightarrow{e}\right) = g\left( \overrightarrow{e}\right) \) for all \( \overrightarrow{e} \in \overrightarrow{{E}^{\prime }} \smallsetminus \overrightarrow{{E}^{\prime }}\left( {y, x}\right) \) ; by (F1), we further have \( f\left( \overrightarrow{e}\right) = - f\left( \bar{e}\right) \) for all \( \overrightarrow{e} \in {\overrightarrow{E}}^{\prime }\left( {y, x}\right) \) . Thus our map \( f \mapsto g \) is bijective if and only if for given \( g \) there is always a unique way to define the remaining values of \( f\left( \overrightarrow{{e}_{0}}\right) \) and \( f\left( \overleftarrow{{e}_{0}}\right) \) so that \( f \) satisfies (F1) in \( {e}_{0} \) and (F2) in \( x \) and \( y \) .
Now \( f\left( \overrightarrow{{e}_{0}}\right) \) is already determined by (F2) for \( x \) and the known values of \( f\left( \overrightarrow{e}\right) \) for edges \( e \) at \( x \), while \( f\left( \overleftarrow{{e}_{0}}\right) \) is already determined by (F2) for \( y \) and the known values of \( f\left( \overrightarrow{e}\right) \) for edges \( e \) at \( y \) . Indeed, with
\[
h \mathrel{\text{:=}} \mathop{\sum }\limits_{{\overrightarrow{e} \in \overrightarrow{{E}^{\prime }}\left( {x, y}\right) }}f\left( \overrightarrow{e}\right) \;\left( { = \mathop{\sum }\limits_{{e \in {E}^{\prime }\left( {x, y}\right) }}g\left( {e,{v}_{0},{v}_{0}}\right) }\right)
\]
and \( {V}^{\prime } \mathrel{\text{:=}} V \smallsetminus \{ x, y\} \) ,(F2) will hold for \( f \) if and only if
\[
0 = f\left( {x, V}\right) = f\left( \overrightarrow{{e}_{0}}\right) + h + f\left( {x,{V}^{\prime }}\right)
\]
and
\[
0 = f\left( {y, V}\right) = f\left( \bar{{e}_{0}}\right) - h + f\left( {y,{V}^{\prime }}\right) ,
\]
that is, if and only if we set
\[
f\left( \overrightarrow{{e}_{0}}\right) \mathrel{\text{:=}} - f\left( {x,{V}^{\prime }}\right) - h\;\text{ and }\;f\left( \overleftarrow{{e}_{0}}\right) \mathrel{\text{:=}} - f\left( {y,{V}^{\prime }}\right) + h.
\]
Fortunately, defining \( f\left( \overrightarrow{{e}_{0}}\right) \) and \( f\left( \overleftarrow{{e}_{0}}\right) \) in this way also satisfies (F1) for \( f \) , as
\[
f\left( \overrightarrow{{e}_{0}}\right) + f\left( \overleftarrow{{e}_{0}}\right) = - f\left( {x,{V}^{\prime }}\right) - f\left( {y,{V}^{\prime }}\right) = - g\left( {{v}_{0},{V}^{\prime }}\right) = 0
\]
by (F2) for \( g \) at \( {v}_{0} \) .
flow The polynomial \( P \) of Theorem 6.3.1 is known as the flow polynomial polynomial of \( G \) .
\( \left\lbrack {6.4.5}\right\rbrack \)
Corollary 6.3.2. If \( H \) and \( {H}^{\prime } \) are two finite abelian groups of equal order, then \( G \) has an \( H \) -flow if and only if \( G \) has an \( {H}^{\prime } \) -flow.
Corollary 6.3.2 has fundamental implications for the theory of algebraic flows: it indicates that crucial difficulties in existence proofs of \( H \) -flows are unlikely to be of a group-theoretic nature. On the other hand, being able to choose a convenient group can be quite helpful; we shall see a pretty example for this in Proposition 6.4.5.
Let \( k \geq 1 \) be an integer and \( G = \left( {V, E}\right) \) a multigraph. A \( \mathbb{Z} \) -flow \( f \) \( k \) on \( G \) such that \( 0 < \left| {f\left( \overrightarrow{e}\right) }\right| < k \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) is called a \( k \) -flow. Clearly, \( k \) -flow any \( k \) -flow is also an \( \ell \) -flow for all \( \ell > k \) . Thus, we may ask which is the least integer \( k \) such that \( G \) admits a \( k \) -flow-assuming that such a \( k \)
---
flow number \( \varphi \left( G\right) \)
---
exists. We call this least \( k \) the flow number of \( G \) and denote it by \( \varphi \left( G\right) \) ; if \( G \) has no \( k \) -flow for any \( k \), we put \( \varphi \left( G |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 51 | t crucial difficulties in existence proofs of \( H \) -flows are unlikely to be of a group-theoretic nature. On the other hand, being able to choose a convenient group can be quite helpful; we shall see a pretty example for this in Proposition 6.4.5.
Let \( k \geq 1 \) be an integer and \( G = \left( {V, E}\right) \) a multigraph. A \( \mathbb{Z} \) -flow \( f \) \( k \) on \( G \) such that \( 0 < \left| {f\left( \overrightarrow{e}\right) }\right| < k \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) is called a \( k \) -flow. Clearly, \( k \) -flow any \( k \) -flow is also an \( \ell \) -flow for all \( \ell > k \) . Thus, we may ask which is the least integer \( k \) such that \( G \) admits a \( k \) -flow-assuming that such a \( k \)
---
flow number \( \varphi \left( G\right) \)
---
exists. We call this least \( k \) the flow number of \( G \) and denote it by \( \varphi \left( G\right) \) ; if \( G \) has no \( k \) -flow for any \( k \), we put \( \varphi \left( G\right) \mathrel{\text{:=}} \infty \) .
The task of determining flow numbers quickly leads to some of the deepest open problems in graph theory. We shall consider these later in the chapter. First, however, let us see how \( k \) -flows are related to the more general concept of \( H \) -flows.
There is an intimate connection between \( k \) -flows and \( {\mathbb{Z}}_{k} \) -flows. Let \( {\sigma }_{k} \) denote the natural homomorphism \( i \mapsto \bar{i} \) from \( \mathbb{Z} \) to \( {\mathbb{Z}}_{k} \) . By compo- \( {\sigma }_{k} \) sition with \( {\sigma }_{k} \), every \( k \) -flow defines a \( {\mathbb{Z}}_{k} \) -flow. As the following theorem shows, the converse holds too: from every \( {\mathbb{Z}}_{k} \) -flow on \( G \) we can construct a \( k \) -flow on \( G \) . In view of Corollary 6.3.2, this means that the general question about the existence of \( H \) -flows for arbitrary groups \( H \) reduces to the corresponding question for \( k \) -flows.
## Theorem 6.3.3. (Tutte 1950)
A multigraph admits a \( k \) -flow if and only if it admits a \( {\mathbb{Z}}_{k} \) -flow. \( \begin{array}{l} \text{ [6.4.1] } \\ \text{ [6.4.2] } \\ \text{ [6.4.3] } \\ \text{ [6.4.5] } \end{array} \)
Proof. Let \( g \) be a \( {\mathbb{Z}}_{k} \) -flow on a multigraph \( G = \left( {V, E}\right) \) ; we construct a \( k \) -flow \( f \) on \( G \) . We may assume without loss of generality that \( G \) has no loops. Let \( F \) be the set of all functions \( f : \overrightarrow{E} \rightarrow \mathbb{Z} \) that satisfy (F1), \( \left| {f\left( \overrightarrow{e}\right) }\right| < k \) for all \( \overrightarrow{e} \in \overrightarrow{E} \), and \( {\sigma }_{k} \circ f = g \) ; note that, like \( g \), any \( f \in F \) is nowhere zero.
Let us show first that \( F \neq \varnothing \) . Since we can express every value \( g\left( \overrightarrow{e}\right) \in {\mathbb{Z}}_{k} \) as \( \bar{i} \) with \( \left| i\right| < k \) and then put \( f\left( \overrightarrow{e}\right) \mathrel{\text{:=}} i \), there is clearly a map \( f : \overrightarrow{E} \rightarrow \mathbb{Z} \) such that \( \left| {f\left( \overrightarrow{e}\right) }\right| < k \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) and \( {\sigma }_{k} \circ f = g \) . For each edge \( e \in E \), let us choose one of its two directions and denote this by \( \overrightarrow{e} \) . We may then define \( {f}^{\prime } : \overrightarrow{E} \rightarrow \mathbb{Z} \) by setting \( {f}^{\prime }\left( \overrightarrow{e}\right) \mathrel{\text{:=}} f\left( \overrightarrow{e}\right) \) and \( {f}^{\prime }\left( \overrightarrow{e}\right) \mathrel{\text{:=}} - f\left( \overrightarrow{e}\right) \) for every \( e \in E \) . Then \( {f}^{\prime } \) is a function satisfying (F1) and with values in the desired range; it remains to show that \( {\sigma }_{k} \circ {f}^{\prime } \) and \( g \) agree not only on the chosen directions \( \overrightarrow{e} \) but also on their inverses \( \bar{e} \) . Since \( {\sigma }_{k} \) is a homomorphism, this is indeed so:
\[
\left( {{\sigma }_{k} \circ {f}^{\prime }}\right) \left( \bar{e}\right) = {\sigma }_{k}\left( {-f\left( \overrightarrow{e}\right) }\right) = - \left( {{\sigma }_{k} \circ f}\right) \left( \overrightarrow{e}\right) = - g\left( \overrightarrow{e}\right) = g\left( \bar{e}\right) .
\]
Hence \( {f}^{\prime } \in F \), so \( F \) is indeed non-empty.
Our aim is to find an \( f \in F \) that satisfies Kirchhoff’s law (F2), and is thus a \( k \) -flow. As a candidate, let us consider an \( f \in F \) for which the sum
\[
K\left( f\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{x \in V}}\left| {f\left( {x, V}\right) }\right|
\]
of all deviations from Kirchhoff's law is least possible. We shall prove that \( K\left( f\right) = 0 \) ; then, clearly, \( f\left( {x, V}\right) = 0 \) for every \( x \), as desired.
Suppose \( K\left( f\right) \neq 0 \) . Since \( f \) satisfies \( \left( {\mathrm{F}1}\right) \), and hence \( \mathop{\sum }\limits_{{x \in V}}f\left( {x, V}\right) = \) \( f\left( {V, V}\right) = 0 \), there exists a vertex \( x \) with
\[
f\left( {x, V}\right) > 0\text{.}
\]
(1)
Let \( X \subseteq V \) be the set of all vertices \( {x}^{\prime } \) for which \( G \) contains a walk \( {x}_{0}{e}_{0}\ldots {e}_{\ell - 1}{x}_{\ell } \) from \( x \) to \( {x}^{\prime } \) such that \( f\left( {{e}_{i},{x}_{i},{x}_{i + 1}}\right) > 0 \) for all \( i < \ell \) ; furthermore, let \( {X}^{\prime } \mathrel{\text{:=}} X \smallsetminus \{ x\} \) .
We first show that \( {X}^{\prime } \) contains a vertex \( {x}^{\prime } \) with \( f\left( {{x}^{\prime }, V}\right) < 0 \) . By definition of \( X \), we have \( f\left( {e,{x}^{\prime }, y}\right) \leq 0 \) for all edges \( e = {x}^{\prime }y \) such that \( {x}^{\prime } \in X \) and \( y \in \bar{X} \) . In particular, this holds for \( {x}^{\prime } = x \) . Thus,(1) implies \( f\left( {x,{X}^{\prime }}\right) > 0 \) . Then \( f\left( {{X}^{\prime }, x}\right) < 0 \) by (F1), as well as \( f\left( {{X}^{\prime },{X}^{\prime }}\right) = 0 \) . Therefore
\[
\mathop{\sum }\limits_{{{x}^{\prime } \in {X}^{\prime }}}f\left( {{x}^{\prime }, V}\right) = f\left( {{X}^{\prime }, V}\right) = f\left( {{X}^{\prime },\bar{X}}\right) + f\left( {{X}^{\prime }, x}\right) + f\left( {{X}^{\prime },{X}^{\prime }}\right) < 0,
\]
so some \( {x}^{\prime } \in {X}^{\prime } \) must indeed satisfy
\[
f\left( {{x}^{\prime }, V}\right) < 0.
\]
(2)
As \( {x}^{\prime } \in X \), there is an \( x - {x}^{\prime } \) walk \( W = {x}_{0}{e}_{0}\ldots {e}_{\ell - 1}{x}_{\ell } \) such that \( f\left( {{e}_{i},{x}_{i},{x}_{i + 1}}\right) > 0 \) for all \( i < \ell \) . We now modify \( f \) by sending some flow back along \( W \), letting \( {f}^{\prime } : \overrightarrow{E} \rightarrow \mathbb{Z} \) be given by
\[
{f}^{\prime } : \overrightarrow{e} \mapsto \left\{ \begin{array}{ll} f\left( \overrightarrow{e}\right) - k & \text{ for }\overrightarrow{e} = \left( {{e}_{i},{x}_{i},{x}_{i + 1}}\right), i = 0,\ldots ,\ell - 1; \\ f\left( \overrightarrow{e}\right) + k & \text{ for }\overrightarrow{e} = \left( {{e}_{i},{x}_{i + 1},{x}_{i}}\right), i = 0,\ldots ,\ell - 1; \\ f\left( \overrightarrow{e}\right) & \text{ for }e \notin W. \end{array}\right.
\]
By definition of \( W \), we have \( \left| {{f}^{\prime }\left( \overrightarrow{e}\right) }\right| < k \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) . Hence \( {f}^{\prime } \), like \( f \) , lies in \( F \) .
How does the modification of \( f \) affect \( K \) ? At all inner vertices \( v \) of \( W \), as well as outside \( W \), the deviation from Kirchhoff’s law remains unchanged:
\[
{f}^{\prime }\left( {v, V}\right) = f\left( {v, V}\right) \;\text{ for all }v \in V \smallsetminus \left\{ {x,{x}^{\prime }}\right\} .
\]
(3)
For \( x \) and \( {x}^{\prime } \), on the other hand, we have
\[
{f}^{\prime }\left( {x, V}\right) = f\left( {x, V}\right) - k\;\text{ and }\;{f}^{\prime }\left( {{x}^{\prime }, V}\right) = f\left( {{x}^{\prime }, V}\right) + k.
\]
(4)
Since \( g \) is a \( {\mathbb{Z}}_{k} \) -flow and hence
\[
{\sigma }_{k}\left( {f\left( {x, V}\right) }\right) = g\left( {x, V}\right) = \overline{0} \in {\mathbb{Z}}_{k}
\]
and
\[
{\sigma }_{k}\left( {f\left( {{x}^{\prime }, V}\right) }\right) = g\left( {{x}^{\prime }, V}\right) = \overline{0} \in {\mathbb{Z}}_{k},
\]
\( f\left( {x, V}\right) \) and \( f\left( {{x}^{\prime }, V}\right) \) are both multiples of \( k \) . Thus \( f\left( {x, V}\right) \geq k \) and \( f\left( {{x}^{\prime }, V}\right) \leq - k \), by (1) and (2). But then (4) implies that
\[
\left| {{f}^{\prime }\left( {x, V}\right) }\right| < \left| {f\left( {x, V}\right) }\right| \text{ and }\left| {{f}^{\prime }\left( {{x}^{\prime }, V}\right) }\right| < \left| {f\left( {{x}^{\prime }, V}\right) }\right| .
\]
Together with (3), this gives \( K\left( {f}^{\prime }\right) < K\left( f\right) \), a contradiction to the choice of \( f \) .
Therefore \( K\left( f\right) = 0 \) as claimed, and \( f \) is indeed a \( k \) -flow.
Since the sum of two \( {\mathbb{Z}}_{k} \) -circulations is always another \( {\mathbb{Z}}_{k} \) -circulation, \( {\mathbb{Z}}_{k} \) -flows are often easier to construct (by summing over suitable partial flows) than \( k \) -flows. In this way, Theorem 6.3.3 may be of considerable help in determining whether or not some given graph has a \( k \) -flow. In the following sections we shall meet a number of examples for this.
Although Theorem 6.3.3 tells us whether a given multigraph admits a \( k \) -flow (assuming we know the value of its flow-polynomial for \( k - 1 \) ), it does not say anything about the number of such flows. By a recent result of Kochol, this number is also a polynomial in \( k \), whose values can be bounded above and below by the corresponding values of the flow polynomial. See the notes for details.
## 6.4 \( k \) -Flows for small \( k \)
|
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 52 | tion to the choice of \( f \) .
Therefore \( K\left( f\right) = 0 \) as claimed, and \( f \) is indeed a \( k \) -flow.
Since the sum of two \( {\mathbb{Z}}_{k} \) -circulations is always another \( {\mathbb{Z}}_{k} \) -circulation, \( {\mathbb{Z}}_{k} \) -flows are often easier to construct (by summing over suitable partial flows) than \( k \) -flows. In this way, Theorem 6.3.3 may be of considerable help in determining whether or not some given graph has a \( k \) -flow. In the following sections we shall meet a number of examples for this.
Although Theorem 6.3.3 tells us whether a given multigraph admits a \( k \) -flow (assuming we know the value of its flow-polynomial for \( k - 1 \) ), it does not say anything about the number of such flows. By a recent result of Kochol, this number is also a polynomial in \( k \), whose values can be bounded above and below by the corresponding values of the flow polynomial. See the notes for details.
## 6.4 \( k \) -Flows for small \( k \)
Trivially, a graph has a 1-flow (the empty set) if and only if it has no edges. In this section we collect a few simple examples of sufficient conditions under which a graph has a 2-, 3- or 4-flow. More examples can be found in the exercises.
Proposition 6.4.1. A graph has a 2-flow if and only if all its degrees
\( \left\lbrack {6.6.1}\right\rbrack \)
are even.
Proof. By Theorem 6.3.3, a graph \( G = \left( {V, E}\right) \) has a 2-flow if and only if
(6.3.3)
it has a \( {\mathbb{Z}}_{2} \) -flow, i.e. if and only if the constant map \( \overrightarrow{E} \rightarrow {\mathbb{Z}}_{2} \) with value \( \overline{1} \) satisfies (F2). This is the case if and only if all degrees are even.
---
even graph
---
For the remainder of this chapter, let us call a graph even if all its vertex degrees are even.
Proposition 6.4.2. A cubic graph has a 3-flow if and only if it is bipartite.
Proof. Let \( G = \left( {V, E}\right) \) be a cubic graph. Let us assume first that \( G \) has a 3-flow, and hence also a \( {\mathbb{Z}}_{3} \) -flow \( f \) . We show that any cycle \( C = {x}_{0}\ldots {x}_{\ell }{x}_{0} \) in \( G \) has even length (cf. Proposition 1.6.1). Consider two consecutive edges on \( C \), say \( {e}_{i - 1} \mathrel{\text{:=}} {x}_{i - 1}{x}_{i} \) and \( {e}_{i} \mathrel{\text{:=}} {x}_{i}{x}_{i + 1} \) . If \( f \) assigned the same value to these edges in the direction of the forward orientation of \( C \), i.e. if \( f\left( {{e}_{i - 1},{x}_{i - 1},{x}_{i}}\right) = f\left( {{e}_{i},{x}_{i},{x}_{i + 1}}\right) \), then \( f \) could not satisfy (F2) at \( {x}_{i} \) for any non-zero value of the third edge at \( {x}_{i} \) . Therefore \( f \) assigns the values \( \overline{1} \) and \( \overline{2} \) to the edges of \( C \) alternately, and in particular \( C \) has even length.
Conversely, let \( G \) be bipartite, with vertex bipartition \( \{ X, Y\} \) . Since \( G \) is cubic, the map \( \overrightarrow{E} \rightarrow {\mathbb{Z}}_{3} \) defined by \( f\left( {e, x, y}\right) \mathrel{\text{:=}} \overline{1} \) and \( f\left( {e, y, x}\right) \mathrel{\text{:=}} \overline{2} \) for all edges \( e = {xy} \) with \( x \in X \) and \( y \in Y \) is a \( {\mathbb{Z}}_{3} \) - flow on \( G \) . By Theorem 6.3.3, then, \( G \) has a 3-flow.
What are the flow numbers of the complete graphs \( {K}^{n} \) ? For odd \( n > 1 \), we have \( \varphi \left( {K}^{n}\right) = 2 \) by Proposition 6.4.1. Moreover, \( \varphi \left( {K}^{2}\right) = \infty \) , and \( \varphi \left( {K}^{4}\right) = 4 \) ; this is easy to see directly (and it follows from Propositions 6.4.2 and 6.4.5). Interestingly, \( {K}^{4} \) is the only complete graph with flow number 4:
Proposition 6.4.3. For all even \( n > 4,\varphi \left( {K}^{n}\right) = 3 \) .
\( \left( {6.3.3}\right) \)
Proof. Proposition 6.4.1 implies that \( \varphi \left( {K}^{n}\right) \geq 3 \) for even \( n \) . We show, by induction on \( n \), that every \( G = {K}^{n} \) with even \( n > 4 \) has a 3-flow.
For the induction start, let \( n = 6 \) . Then \( G \) is the edge-disjoint union of three graphs \( {G}_{1},{G}_{2},{G}_{3} \), with \( {G}_{1},{G}_{2} = {K}^{3} \) and \( {G}_{3} = {K}_{3,3} \) . Clearly \( {G}_{1} \) and \( {G}_{2} \) each have a 2-flow, while \( {G}_{3} \) has a 3-flow by Proposition 6.4.2. The union of all these flows is a 3-flow on \( G \) .
Now let \( n > 6 \), and assume the assertion holds for \( n - 2 \) . Clearly, \( G \) is the edge-disjoint union of a \( {K}^{n - 2} \) and a graph \( {G}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \) with \( {G}^{\prime } = \) \( \overline{{K}^{n - 2}} * {K}^{2} \) . The \( {K}^{n - 2} \) has a 3-flow by induction. By Theorem 6.3.3, it thus suffices to find a \( {\mathbb{Z}}_{3} \) -flow on \( {G}^{\prime } \) . For every vertex \( z \) of the \( \overline{{K}^{n - 2}} \subseteq {G}^{\prime } \) , let \( {f}_{z} \) be a \( {\mathbb{Z}}_{3} \) -flow on the triangle \( {zxyz} \subseteq {G}^{\prime } \), where \( e = {xy} \) is the edge of the \( {K}^{2} \) in \( {G}^{\prime } \) . Let \( f : {\overrightarrow{E}}^{\prime } \rightarrow {\mathbb{Z}}_{3} \) be the sum of these flows. Clearly, \( f \) is nowhere zero, except possibly in \( \left( {e, x, y}\right) \) and \( \left( {e, y, x}\right) \) . If \( f\left( {e, x, y}\right) \neq \overline{0} \) , then \( f \) is the desired \( {\mathbb{Z}}_{3} \) -flow on \( {G}^{\prime } \) . If \( f\left( {e, x, y}\right) = \overline{0} \), then \( f + {f}_{z} \) (for any \( z \) ) is a \( {\mathbb{Z}}_{3} \) -flow on \( {G}^{\prime } \) .
Proposition 6.4.4. Every 4-edge-connected graph has a 4-flow.
Proof. Let \( G \) be a 4-edge-connected graph. By Corollary 2.4.2, \( G \) has
(2.4.2)
two edge-disjoint spanning trees \( {T}_{i}, i = 1,2 \) . For each edge \( e \notin {T}_{i} \), let \( {C}_{i, e} \) be the unique cycle in \( {T}_{i} + e \), and let \( {f}_{i, e} \) be a \( {\mathbb{Z}}_{4} \) -flow of value \( \bar{i} \) \( {f}_{1, e},{f}_{2, e} \) around \( {C}_{i, e} \) -more precisely: a \( {\mathbb{Z}}_{4} \) -circulation on \( G \) with values \( \bar{i} \) and \( - \bar{i} \) on the edges of \( {C}_{i, e} \) and zero otherwise.
Let \( {f}_{1} \mathrel{\text{:=}} \mathop{\sum }\limits_{{e \notin {T}_{1}}}{f}_{1, e} \) . Since each \( e \notin {T}_{1} \) lies on only one cycle \( {C}_{1,{e}^{\prime }} \) (namely, for \( e = {e}^{\prime } \) ), \( {f}_{1} \) takes only the values \( \overline{1} \) and \( - \overline{1}\left( { = \overline{3}}\right) \) outside \( {T}_{1} \) . Let
\[
F \mathrel{\text{:=}} \left\{ {e \in E\left( {T}_{1}\right) \mid {f}_{1}\left( e\right) = \overline{0}}\right\}
\]
and \( {f}_{2} \mathrel{\text{:=}} \mathop{\sum }\limits_{{e \in F}}{f}_{2, e} \) . As above, \( {f}_{2}\left( e\right) = \overline{2} = - \overline{2} \) for all \( e \in F \) . Now \( f \mathrel{\text{:=}} {f}_{1} + {f}_{2} \) is the sum of \( {\mathbb{Z}}_{4} \) -circulations, and hence itself a \( {\mathbb{Z}}_{4} \) -circulation. Moreover, \( f \) is nowhere zero: on edges in \( F \) it takes the value \( \overline{2} \), on edges of \( {T}_{1} - F \) it agrees with \( {f}_{1} \) (and is hence non-zero by the choice of \( F \) ), and on all edges outside \( {T}_{1} \) it takes one of the values \( \overline{1} \) or \( \overline{3} \) . Hence, \( f \) is a \( {\mathbb{Z}}_{4} \) -flow on \( G \), and the assertion follows by Theorem 6.3.3.
The following proposition describes the graphs with a 4-flow in terms of those with a 2-flow:
## Proposition 6.4.5.
(i) A graph has a 4-flow if and only if it is the union of two even subgraphs.
(ii) A cubic graph has a 4-flow if and only if it is 3-edge-colourable.
Proof. Let \( {\mathbb{Z}}_{2}^{2} = {\mathbb{Z}}_{2} \times {\mathbb{Z}}_{2} \) be the Klein four-group. (Thus, the elements of
(6.3.3)
\( {\mathbb{Z}}_{2}^{2} \) are the pairs \( \left( {a, b}\right) \) with \( a, b \in {\mathbb{Z}}_{2} \), and \( \left( {a, b}\right) + \left( {{a}^{\prime },{b}^{\prime }}\right) = \left( {a + {a}^{\prime }, b + {b}^{\prime }}\right) \) .) By Corollary 6.3.2 and Theorem 6.3.3, a graph has a 4-flow if and only if it has a \( {\mathbb{Z}}_{2}^{2} \) -flow.
(i) now follows directly from Proposition 6.4.1.
(ii) Let \( G = \left( {V, E}\right) \) be a cubic graph. We assume first that \( G \) has a \( {\mathbb{Z}}_{2}^{2} \) -flow \( f \), and define an edge colouring \( E \rightarrow {\mathbb{Z}}_{2}^{2} \smallsetminus \{ 0\} \) . As \( a = - a \) for all \( a \in {\mathbb{Z}}_{2}^{2} \), we have \( f\left( \overrightarrow{e}\right) = f\left( \bar{e}\right) \) for every \( \overrightarrow{e} \in \bar{E} \) ; let us colour the edge \( e \) with this colour \( f\left( \overrightarrow{e}\right) \) . Now if two edges with a common end \( v \) had the same colour, then these two values of \( f \) would sum to zero; by (F2), \( f \) would then assign zero to the third edge at \( v \) . As this contradicts the definition of \( f \), our edge colouring is correct.
Conversely, since the three non-zero elements of \( {\mathbb{Z}}_{2}^{2} \) sum to zero, every 3-edge-colouring \( c : E \rightarrow {\mathbb{Z}}_{2}^{2} \smallsetminus \{ 0\} \) defines a \( {\mathbb{Z}}_{2}^{2} \) -flow on \( G \) by letting \( f\left( \overrightarrow{e}\right) = f\left( \bar{e}\right) = c\left( e\right) \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) .
Corollary 6.4.6. Every cubic 3-edge-colourable graph is bridgeless.
## 6.5 Flow-colouring duality
In this section we shall see a surprising connection between flows and colouring: every \( k \) -flow on a plane multigraph gives rise to a \( k \) -vertex-colouring of its dual, and vice versa. In this way, the investigation of \( k \) -flows appears as a natural generalization of the familiar map colouring problems in the plane.
---
\( G = \left( {V, E}\right) \) \( {G}^{ * } \)
---
Let \( G = \left( {V, E}\right) \) and \( {G}^{ * } = \left( {{V}^{ * },{E}^{ * |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 53 | \( f \), our edge colouring is correct.
Conversely, since the three non-zero elements of \( {\mathbb{Z}}_{2}^{2} \) sum to zero, every 3-edge-colouring \( c : E \rightarrow {\mathbb{Z}}_{2}^{2} \smallsetminus \{ 0\} \) defines a \( {\mathbb{Z}}_{2}^{2} \) -flow on \( G \) by letting \( f\left( \overrightarrow{e}\right) = f\left( \bar{e}\right) = c\left( e\right) \) for all \( \overrightarrow{e} \in \overrightarrow{E} \) .
Corollary 6.4.6. Every cubic 3-edge-colourable graph is bridgeless.
## 6.5 Flow-colouring duality
In this section we shall see a surprising connection between flows and colouring: every \( k \) -flow on a plane multigraph gives rise to a \( k \) -vertex-colouring of its dual, and vice versa. In this way, the investigation of \( k \) -flows appears as a natural generalization of the familiar map colouring problems in the plane.
---
\( G = \left( {V, E}\right) \) \( {G}^{ * } \)
---
Let \( G = \left( {V, E}\right) \) and \( {G}^{ * } = \left( {{V}^{ * },{E}^{ * }}\right) \) be dual plane multigraphs. For simplicity, let us assume that \( G \) and \( {G}^{ * } \) have neither bridges nor loops and are non-trivial. For edge sets \( F \subseteq E \), let us write
\( {F}^{ * } \)
\[
{F}^{ * } \mathrel{\text{:=}} \left\{ {{e}^{ * } \in {E}^{ * } \mid e \in F}\right\} .
\]
Conversely, if a subset of \( {E}^{ * } \) is given, we shall usually write it immediately in the form \( {F}^{ * } \), and thus let \( F \subseteq E \) be defined implicitly via the bijection \( e \mapsto {e}^{ * } \) .
Suppose we are given a circulation \( g \) on \( {G}^{ * } \) : how can we employ the duality between \( G \) and \( {G}^{ * } \) to derive from \( g \) some information about \( G \) ? The most general property of all circulations is Proposition 6.1.1, which says that \( g\left( {X,\bar{X}}\right) = 0 \) for all \( X \subseteq {V}^{ * } \) . By Proposition 4.6.1, the minimal cuts \( {E}^{ * }\left( {X,\bar{X}}\right) \) in \( {G}^{ * } \) correspond precisely to the cycles in \( G \) . Thus if we take the composition \( f \) of the maps \( e \mapsto {e}^{ * } \) and \( g \), and sum its values over the edges of a cycle in \( G \), then this sum should again be zero.
Of course, there is still a technical hitch: since \( g \) takes its arguments not in \( {E}^{ * } \) but in \( {E}^{ * } \), we cannot simply define \( f \) as above: we first have to refine the bijection \( e \mapsto {e}^{ * } \) into one from \( \overrightarrow{E} \) to \( {\overrightarrow{E}}^{ * } \), i.e. assign to every \( \overrightarrow{e} \in \overrightarrow{E} \) canonically one of the two directions of \( {e}^{ * } \) . This will be the purpose of our first lemma. After that, we shall show that \( f \) does indeed sum to zero along any cycle in \( G \) .
If \( C = {v}_{0}\ldots {v}_{\ell - 1}{v}_{0} \) is a cycle with edges \( {e}_{i} = {v}_{i}{v}_{i + 1} \) (and \( {v}_{\ell } \mathrel{\text{:=}} {v}_{0} \) ), we shall call
---
\( \overrightarrow{C} \)
cycle with orientation
---
\[
\overrightarrow{C} \mathrel{\text{:=}} \left\{ {\left( {{e}_{i},{v}_{i},{v}_{i + 1}}\right) \mid i < \ell }\right\}
\]
a cycle with orientation. Note that this definition of \( \overrightarrow{C} \) depends on the vertex enumeration chosen to denote \( C \) : every cycle has two orientations. Conversely, of course, \( C \) can be reconstructed from the set \( \overrightarrow{C} \) . In practice, we shall therefore speak about \( C \) freely even when, formally, only \( \overrightarrow{C} \) has been defined.
Lemma 6.5.1. There exists a bijection \( {}^{ * } : \overrightarrow{e} \mapsto {\overrightarrow{e}}^{ * } \) from \( \overrightarrow{E} \) to \( \overrightarrow{{E}^{ * }} \) with the following properties:
(i) The underlying edge of \( {\overrightarrow{e}}^{ * } \) is always \( {e}^{ * } \), i.e. \( {\overrightarrow{e}}^{ * } \) is one of the two directions \( \overrightarrow{{e}^{ * }},\overleftarrow{{e}^{ * }} \) of \( {e}^{ * } \) ;
(ii) If \( C \subseteq G \) is a cycle, \( F \mathrel{\text{:=}} E\left( C\right) \), and if \( X \subseteq {V}^{ * } \) is such that \( {F}^{ * } = {E}^{ * }\left( {X,\bar{X}}\right) \), then there exists an orientation \( \overrightarrow{C} \) of \( C \) with \( \left\{ {{\overrightarrow{e}}^{ * } \mid \overrightarrow{e} \in \overrightarrow{C}}\right\} = {\overrightarrow{E}}^{ * }\left( {X,\bar{X}}\right) . \)
The proof of Lemma 6.5.1 is not entirely trivial: it is based on the so-called orientability of the plane, and we cannot give it here. Still, the assertion of the lemma is intuitively plausible. Indeed if we define for \( e = {vw} \) and \( {e}^{ * } = {xy} \) the assignment \( \left( {e, v, w}\right) \mapsto {\left( e, v, w\right) }^{ * } \in \) \( \left\{ {\left( {{e}^{ * }, x, y}\right) ,\left( {{e}^{ * }, y, x}\right) }\right\} \) simply by turning \( e \) and its ends clockwise onto \( {e}^{ * } \) (Fig. 6.5.1), then the resulting map \( \overrightarrow{e} \mapsto {\overrightarrow{e}}^{ * } \) satisfies the two assertions of the lemma.
Fig. 6.5.1. Oriented cycle-cut duality
Given an abelian group \( H \), let \( f : \overrightarrow{E} \rightarrow H \) and \( g : \overrightarrow{{E}^{ * }} \rightarrow H \) be two maps \( f, g \) such that
\[
f\left( \overrightarrow{e}\right) = g\left( {\overrightarrow{e}}^{ * }\right)
\]
for all \( \overrightarrow{e} \in \overrightarrow{E} \) . For \( \overrightarrow{F} \subseteq \overrightarrow{E} \), we set
\[
f\left( \overrightarrow{F}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\overrightarrow{e} \in \overrightarrow{F}}}f\left( \overrightarrow{e}\right)
\]
\( f\left( \overrightarrow{C}\right) \) etc.
Lemma 6.5.2.
(i) The map \( g \) satisfies (F1) if and only if \( f \) does.
(ii) The map \( g \) is a circulation on \( {G}^{ * } \) if and only if \( f \) satisfies (F1) and \( f\left( \overrightarrow{C}\right) = 0 \) for every cycle \( \overrightarrow{C} \) with orientation.
Proof. (See also Exercise 17.) Assertion (i) follows from Lemma 6.5.1 (i)
\( \left( {4.6.1}\right) \left( {6.1.1}\right) \)
and the fact that \( \overrightarrow{e} \mapsto {\overrightarrow{e}}^{ * } \) is bijective.
For the forward implication of (ii), let us assume that \( g \) is a circulation on \( {G}^{ * } \), and consider a cycle \( C \subseteq G \) with some given orientation. Let \( F \mathrel{\text{:=}} E\left( C\right) \) . By Proposition 4.6.1, \( {F}^{ * } \) is a minimal cut in \( {G}^{ * } \), i.e. \( {F}^{ * } = {E}^{ * }\left( {X,\bar{X}}\right) \) for some suitable \( X \subseteq {V}^{ * } \) . By definition of \( f \) and \( g \) , Lemma 6.5.1 (ii) and Proposition 6.1.1 give
\[
f\left( \overrightarrow{C}\right) = \mathop{\sum }\limits_{{\overrightarrow{e} \in \overrightarrow{C}}}f\left( \overrightarrow{e}\right) = \mathop{\sum }\limits_{{\overrightarrow{d} \in {E}^{ * }\left( {X,\bar{X}}\right) }}g\left( \overrightarrow{d}\right) = g\left( {X,\bar{X}}\right) = 0
\]
for one of the two orientations \( \overrightarrow{C} \) of \( C \) . Then, by \( f\left( \bar{C}\right) = - f\left( \overrightarrow{C}\right) \), also the corresponding value for our given orientation of \( C \) must be zero.
For the backward implication it suffices by (i) to show that \( g \) satisfies (F2), i.e. that \( g\left( {x,{V}^{ * }}\right) = 0 \) for every \( x \in {V}^{ * } \) . We shall prove that \( g\left( {x, V\left( B\right) }\right) = 0 \) for every block \( B \) of \( {G}^{ * } \) containing \( x \) ; since every edge of \( {G}^{ * } \) at \( x \) lies in exactly one such block, this will imply \( g\left( {x,{V}^{ * }}\right) = 0 \) .
\( B \) So let \( x \in {V}^{ * } \) be given, and let \( B \) be any block of \( {G}^{ * } \) containing \( x \) . Since \( {G}^{ * } \) is a non-trivial plane dual, and hence connected, we \( {F}^{ * }, F \) have \( B - x \neq \varnothing \) . Let \( {F}^{ * } \) be the set of all edges of \( B \) at \( x \) (Fig. 6.5.2),
Fig. 6.5.2. The cut \( {F}^{ * } \) in \( {G}^{ * } \)
and let \( X \) be the vertex set of the component of \( {G}^{ * } - {F}^{ * } \) containing \( x \) . Then \( \varnothing \neq V\left( {B - x}\right) \subseteq \bar{X} \), by the maximality of \( B \) as a cutvertex-free subgraph. Hence
\[
{F}^{ * } = {E}^{ * }\left( {X,\bar{X}}\right)
\]
(1)
by definition of \( X \), i.e. \( {F}^{ * } \) is a cut in \( {G}^{ * } \) . As a dual, \( {G}^{ * } \) is connected, so \( {G}^{ * }\left\lbrack \bar{X}\right\rbrack \) too is connected. Indeed, every vertex of \( \bar{X} \) is linked to \( x \) by a path \( P \subseteq {G}^{ * } \) whose last edge lies in \( {F}^{ * } \) . Then \( P - x \) is a path in \( {G}^{ * }\left\lbrack \bar{X}\right\rbrack \) meeting \( B \) . Since \( x \) does not separate \( B \), this shows that \( {G}^{ * }\left\lbrack \bar{X}\right\rbrack \) is connected.
Thus, \( X \) and \( \bar{X} \) are both connected in \( {G}^{ * } \), so \( {F}^{ * } \) is even a minimal cut in \( {G}^{ * } \) . Let \( C \subseteq G \) be the cycle with \( E\left( C\right) = F \) that exists by Proposition 4.6.1. By Lemma 6.5.1 (ii), \( C \) has an orientation \( \overrightarrow{C} \) such that \( \left\{ {{\overrightarrow{e}}^{ * } \mid \overrightarrow{e} \in \overrightarrow{C}}\right\} = {\overrightarrow{E}}^{ * }\left( {X,\bar{X}}\right) \) . By (1), however, \( {\overrightarrow{E}}^{ * }\left( {X,\bar{X}}\right) = {\overrightarrow{E}}^{ * }\left( {x, V\left( B\right) }\right) \) , so
\[
g\left( {x, V\left( B\right) }\right) = g\left( {X,\bar{X}}\right) = f\left( \overrightarrow{C}\right) = 0
\]
by definition of \( f \) and \( g \) .
With the help of Lemma 6.5.2, we can now prove our colouring-flow duality theorem for plane multigraphs. If \( P = {v}_{0}\ldots {v}_{\ell } \) is a path with edge |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 54 | \( {G}^{ * }\left\lbrack \bar{X}\right\rbrack \) is connected.
Thus, \( X \) and \( \bar{X} \) are both connected in \( {G}^{ * } \), so \( {F}^{ * } \) is even a minimal cut in \( {G}^{ * } \) . Let \( C \subseteq G \) be the cycle with \( E\left( C\right) = F \) that exists by Proposition 4.6.1. By Lemma 6.5.1 (ii), \( C \) has an orientation \( \overrightarrow{C} \) such that \( \left\{ {{\overrightarrow{e}}^{ * } \mid \overrightarrow{e} \in \overrightarrow{C}}\right\} = {\overrightarrow{E}}^{ * }\left( {X,\bar{X}}\right) \) . By (1), however, \( {\overrightarrow{E}}^{ * }\left( {X,\bar{X}}\right) = {\overrightarrow{E}}^{ * }\left( {x, V\left( B\right) }\right) \) , so
\[
g\left( {x, V\left( B\right) }\right) = g\left( {X,\bar{X}}\right) = f\left( \overrightarrow{C}\right) = 0
\]
by definition of \( f \) and \( g \) .
With the help of Lemma 6.5.2, we can now prove our colouring-flow duality theorem for plane multigraphs. If \( P = {v}_{0}\ldots {v}_{\ell } \) is a path with edges \( {e}_{i} = {v}_{i}{v}_{i + 1}\left( {i < \ell }\right) \), we set (depending on our vertex enumeration of \( P \) )
\( \overrightarrow{P} \)
\[
\overrightarrow{P} \mathrel{\text{:=}} \left\{ {\left( {{e}_{i},{v}_{i},{v}_{i + 1}}\right) \mid i < \ell }\right\}
\]
\( {v}_{0} \rightarrow {v}_{\ell } \) and call \( \overrightarrow{P} \) a \( {v}_{0} \rightarrow {v}_{\ell } \) path. Again, \( P \) may be given implicitly by \( \overrightarrow{P} \) . path
Theorem 6.5.3. (Tutte 1954)
For every dual pair \( G,{G}^{ * } \) of plane multigraphs,
\[
\chi \left( G\right) = \varphi \left( {G}^{ * }\right)
\]
Proof. Let \( G = : \left( {V, E}\right) \) and \( {G}^{ * } = : \left( {{V}^{ * },{E}^{ * }}\right) \) . For \( \left| G\right| \in \{ 1,2\} \) the
---
(1.5.6) \( V, E \) \( {V}^{ * },{E}^{ * } \)
---
assertion is easily checked; we shall assume that \( \left| G\right| \geq 3 \), and apply induction on the number of bridges in \( G \) . If \( e \in G \) is a bridge then \( {e}^{ * } \) is a loop, and \( {G}^{ * } - {e}^{ * } \) is a plane dual of \( G/e \) (why?). Hence, by the induction hypothesis,
\[
\chi \left( G\right) = \chi \left( {G/e}\right) = \varphi \left( {{G}^{ * } - {e}^{ * }}\right) = \varphi \left( {G}^{ * }\right)
\]
for the first and the last equality we use that, by \( \left| G\right| \geq 3, e \) is not the only edge of \( G \) .
So all that remains to be checked is the induction start: let us assume that \( G \) has no bridge. If \( G \) has a loop, then \( {G}^{ * } \) has a bridge, and \( \chi \left( G\right) = \infty = \varphi \left( {G}^{ * }\right) \) by convention. So we may also assume that \( G \) has no loop. Then \( \chi \left( G\right) \) is finite; we shall prove for given \( k \geq 2 \) that \( G \) is \( k \) -colourable if and only if \( {G}^{ * } \) has a \( k \) -flow. As \( G \) - and hence \( {G}^{ * } \) - has neither loops nor bridges, we may apply Lemmas 6.5.1 and 6.5.2 to \( G \) and \( {G}^{ * } \) . Let \( \overrightarrow{e} \mapsto {\overrightarrow{e}}^{ * } \) be the bijection between \( \overrightarrow{E} \) and \( \overrightarrow{{E}^{ * }} \) from Lemma 6.5.1.
We first assume that \( {G}^{ * } \) has a \( k \) -flow. Then \( {G}^{ * } \) also has a \( {\mathbb{Z}}_{k} \) -flow \( g \) . As before, let \( f : \overrightarrow{E} \rightarrow {\mathbb{Z}}_{k} \) be defined by \( f\left( \overrightarrow{e}\right) \mathrel{\text{:=}} g\left( {\overrightarrow{e}}^{ * }\right) \) . We shall use \( f \) to define a vertex colouring \( c : V \rightarrow {\mathbb{Z}}_{k} \) of \( G \) .
Let \( T \) be a normal spanning tree of \( G \), with root \( r \), say. Put \( c\left( r\right) \mathrel{\text{:=}} \overline{0} \) . For every other vertex \( v \in V \) let \( c\left( v\right) \mathrel{\text{:=}} f\left( \overrightarrow{P}\right) \), where \( \overrightarrow{P} \) is the \( r \rightarrow v \) path in \( T \) . To check that this is a proper colouring, consider an edge \( e = {vw} \in E \) . As \( T \) is normal, we may assume that \( v < w \) in the tree-order of \( T \) . If \( e \) is an edge of \( T \) then \( c\left( w\right) - c\left( v\right) = f\left( {e, v, w}\right) \) by definition of \( c \) , so \( c\left( v\right) \neq c\left( w\right) \) since \( g \) (and hence \( f \) ) is nowhere zero. If \( e \notin T \), let \( \overrightarrow{P} \) denote the \( v \rightarrow w \) path in \( T \) . Then
\[
c\left( w\right) - c\left( v\right) = f\left( \overrightarrow{P}\right) = - f\left( {e, w, v}\right) \neq \bar{0}
\]
by Lemma 6.5.2 (ii).
Conversely, we now assume that \( G \) has a \( k \) -colouring \( c \) . Let us define \( f : \overrightarrow{E} \rightarrow \mathbb{Z} \) by
\[
f\left( {e, v, w}\right) \mathrel{\text{:=}} c\left( w\right) - c\left( v\right)
\]
and \( g : \overrightarrow{{E}^{ * }} \rightarrow \mathbb{Z} \) by \( g\left( {\overrightarrow{e}}^{ * }\right) \mathrel{\text{:=}} f\left( \overrightarrow{e}\right) \) . Clearly, \( f \) satisfies (F1) and takes values in \( \{ \pm 1,\ldots , \pm \left( {k - 1}\right) \} \), so by Lemma 6.5.2 (i) the same holds for \( g \) . By definition of \( f \), we further have \( f\left( \overrightarrow{C}\right) = 0 \) for every cycle \( \overrightarrow{C} \) with orientation. By Lemma 6.5.2 (ii), therefore, \( g \) is a \( k \) -flow.
## 6.6 Tutte's flow conjectures
How can we determine the flow number of a graph? Indeed, does every (bridgeless) graph have a flow number, a \( k \) -flow for some \( k \) ? Can flow numbers, like chromatic numbers, become arbitrarily large? Can we characterize the graphs admitting a \( k \) -flow, for given \( k \) ?
Of these four questions, we shall answer the second and third in this section: we prove that every bridgeless graph has a 6-flow. In particular, a graph has a flow number if and only if it has no bridge. The question asking for a characterization of the graphs with a \( k \) -flow remains interesting for \( k = 3,4,5 \) . Partial answers are suggested by the following three conjectures of Tutte, who initiated algebraic flow theory.
The oldest and best known of the Tutte conjectures is his 5-flow conjecture:
## Five-Flow Conjecture. (Tutte 1954)
Every bridgeless multigraph has a 5-flow.
Which graphs have a 4-flow? By Proposition 6.4.4, the 4-edge-connected graphs are among them. The Petersen graph (Fig. 6.6.1), on the other hand, is an example of a bridgeless graph without a 4-flow: since it is cubic but not 3-edge-colourable, it cannot have a 4-flow by Proposition 6.4.5 (ii).
Fig. 6.6.1. The Petersen graph
Tutte's 4-flow conjecture states that the Petersen graph must be present in every graph without a 4-flow:
## Four-Flow Conjecture. (Tutte 1966)
Every bridgeless multigraph not containing the Petersen graph as a minor has a 4-flow.
By Proposition 1.7.2, we may replace the word 'minor' in the 4-flow conjecture by 'topological minor'.
Even if true, the 4-flow conjecture will not be best possible: a \( {K}^{11} \) , for example, contains the Petersen graph as a minor but has a 4-flow, even a 2-flow. The conjecture appears more natural for sparser graphs; a proof for cubic graphs was announced in 1998 by Robertson, Sanders, Seymour and Thomas.
A cubic bridgeless graph or multigraph without a 4-flow (equivalently, without a 3-edge-colouring) is called a snark. The 4-flow conjec- snark ture for cubic graphs says that every snark contains the Petersen graph as a minor; in this sense, the Petersen graph has thus been shown to be the smallest snark. Snarks form the hard core both of the four colour theorem and of the 5-flow conjecture: the four colour theorem is equivalent to the assertion that no snark is planar (exercise), and it is not difficult to reduce the 5-flow conjecture to the case of snarks. \( {}^{5} \) However, although the snarks form a very special class of graphs, none of the problems mentioned seems to become much easier by this reduction. \( {}^{6} \)
## Three-Flow Conjecture. (Tutte 1972)
Every multigraph without a cut consisting of exactly one or exactly three edges has a 3-flow.
Again, the 3-flow conjecture will not be best possible: it is easy to construct graphs with three-edge cuts that have a 3-flow (exercise).
By our duality theorem (6.5.3), all three flow conjectures are true for planar graphs and thus motivated: the 3-flow conjecture translates to Grötzsch's theorem (5.1.3), the 4-flow conjecture to the four colour theorem (since the Petersen graph is not planar, it is not a minor of a planar graph), the 5-flow conjecture to the five colour theorem.
We finish this section with the main result of the chapter:
Theorem 6.6.1. (Seymour 1981)
Every bridgeless graph has a 6-flow.
Proof. Let \( G = \left( {V, E}\right) \) be a bridgeless graph. Since 6-flows on the
---
(3.3.6)
(6.1.1)
(6.4.1)
---
components of \( G \) will add up to a 6 -flow on \( G \), we may assume that \( G \) is connected; as \( G \) is bridgeless, it is then 2-edge-connected. Note that any two vertices in a 2-edge-connected graph lie in some common even connected subgraph - for example, in the union of two edge-disjoint paths linking these vertices by Menger’s theorem (3.3.6(ii)). We shall use this fact repeatedly.
---
5 The same applies to another well-known conjecture, the cycle double cover conjecture; see Exercise 13.
\( {}^{6} \) That snarks are elusive has been known to mathematicians for some time; cf. Lewis Carroll, The Hunting of the Snark, Macmillan 1876.
---
---
\( {H}_{0},\ldots ,{H}_{n} \)
\( {F}_{1},\ldots ,{F}_{n} \)
---
We shall construct a sequence \( {H}_{0},\ldots ,{H}_{n} \) of disjoint connected and even subgraphs of \( G \), together with a sequence \( {F}_{1},\ldots ,{F}_{n} \) of non-empty sets of edges between them. The sets \( {F}_{i} \) will each contain only one or \( {V}_{i},{E}_{i} \) two edges, between \( {H}_{ |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 55 | n \( G \), we may assume that \( G \) is connected; as \( G \) is bridgeless, it is then 2-edge-connected. Note that any two vertices in a 2-edge-connected graph lie in some common even connected subgraph - for example, in the union of two edge-disjoint paths linking these vertices by Menger’s theorem (3.3.6(ii)). We shall use this fact repeatedly.
---
5 The same applies to another well-known conjecture, the cycle double cover conjecture; see Exercise 13.
\( {}^{6} \) That snarks are elusive has been known to mathematicians for some time; cf. Lewis Carroll, The Hunting of the Snark, Macmillan 1876.
---
---
\( {H}_{0},\ldots ,{H}_{n} \)
\( {F}_{1},\ldots ,{F}_{n} \)
---
We shall construct a sequence \( {H}_{0},\ldots ,{H}_{n} \) of disjoint connected and even subgraphs of \( G \), together with a sequence \( {F}_{1},\ldots ,{F}_{n} \) of non-empty sets of edges between them. The sets \( {F}_{i} \) will each contain only one or \( {V}_{i},{E}_{i} \) two edges, between \( {H}_{i} \) and \( {H}_{0} \cup \ldots \cup {H}_{i - 1} \) . We write \( {H}_{i} = : \left( {{V}_{i},{E}_{i}}\right) \) ,
\( {H}^{i} \)
\[
{H}^{i} \mathrel{\text{:=}} \left( {{H}_{0} \cup \ldots \cup {H}_{i}}\right) + \left( {{F}_{1} \cup \ldots \cup {F}_{i}}\right)
\]
\( {V}^{i},{E}^{i} \) and \( {H}^{i} = : \left( {{V}^{i},{E}^{i}}\right) \) . Note that each \( {H}^{i} = \left( {{H}^{i - 1} \cup {H}_{i}}\right) + {F}_{i} \) is connected (induction on \( i \) ). Our assumption that \( {H}_{i} \) is even implies by Proposition 6.4.1 (or directly by Proposition 1.2.1) that \( {H}_{i} \) has no bridge.
As \( {H}_{0} \) we choose any \( {K}^{1} \) in \( G \) . Now assume that \( {H}_{0},\ldots ,{H}_{i - 1} \) and \( {F}_{1},\ldots ,{F}_{i - 1} \) have been defined for some \( i > 0 \) . If \( {V}^{i - 1} = V \), we terminate \( n \) the construction and set \( i - 1 = : n \) . Otherwise, we let \( {X}_{i} \subseteq \overline{{V}^{i - 1}} \) be \( {X}_{i} \) minimal such that \( {X}_{i} \neq \varnothing \) and
\[
\left| {E\left( {{X}_{i},\overline{{V}^{i - 1}} \smallsetminus {X}_{i}}\right) }\right| \leq 1
\]
(1)
(Fig. 6.6.2); such an \( {X}_{i} \) exists, because \( \overline{{V}^{i - 1}} \) is a candidate. Since \( G \) is 2-edge-connected,(1) implies that \( E\left( {{X}_{i},{V}^{i - 1}}\right) \neq \varnothing \) . By the mini-mality of \( {X}_{i} \), the graph \( G\left\lbrack {X}_{i}\right\rbrack \) is connected and bridgeless, i.e. 2-edge- \( {F}_{i} \) connected or a \( {K}^{1} \) . As the elements of \( {F}_{i} \) we pick one or two edges from \( E\left( {{X}_{i},{V}^{i - 1}}\right) \), if possible two. As \( {H}_{i} \) we choose any connected even subgraph of \( G\left\lbrack {X}_{i}\right\rbrack \) containing the ends in \( {X}_{i} \) of the edges in \( {F}_{i} \) .
Fig. 6.6.2. Constructing the \( {H}_{i} \) and \( {F}_{i} \)
\( H \) When our construction is complete, we set \( {H}^{n} = : H \) and \( {E}^{\prime } \mathrel{\text{:=}} \) \( {E}^{\prime } \) \( E \smallsetminus E\left( H\right) \) . By definition of \( n, H \) is a spanning connected subgraph of \( G \) .
\( {f}_{n},\ldots ,{f}_{0} \) We now define, by ’reverse’ induction, a sequence \( {f}_{n},\ldots ,{f}_{0} \) of \( {\mathbb{Z}}_{3} \) - \( \overrightarrow{{C}_{e}} \) circulations on \( G \) . For every edge \( e \in {E}^{\prime } \), let \( {C}_{e} \) be a cycle (with orientation) in \( H + e \) containing \( e \), and \( {f}_{e} \) a positive flow around \( \overrightarrow{{C}_{e}} \) ; formally, we let \( {f}_{e} \) be a \( {\mathbb{Z}}_{3} \) -circulation on \( G \) such that \( {f}_{e}^{-1}\left( \overline{0}\right) = \overrightarrow{E} \smallsetminus \left( {\overrightarrow{{C}_{e}} \cup \overleftarrow{{C}_{e}}}\right) \) . Let \( {f}_{n} \) be the sum of all these \( {f}_{e} \) . Since each \( {e}^{\prime } \in {E}^{\prime } \) lies on just one of the cycles \( {C}_{e} \) (namely, on \( {C}_{{e}^{\prime }} \) ), we have \( {f}_{n}\left( \overrightarrow{e}\right) \neq \overline{0} \) for all \( \overrightarrow{e} \in {\overrightarrow{E}}^{\prime } \) .
Assume now that \( {\mathbb{Z}}_{3} \) -circulations \( {f}_{n},\ldots ,{f}_{i} \) on \( G \) have been defined for some \( i \leq n \), and that
\[
{f}_{i}\left( \overrightarrow{e}\right) \neq \overline{0}\text{ for all }\overrightarrow{e} \in {\overrightarrow{E}}^{\prime } \cup \mathop{\bigcup }\limits_{{j > i}}{\overrightarrow{F}}_{j},
\]
(2)
where \( {\overrightarrow{F}}_{j} \mathrel{\text{:=}} \left\{ {\overrightarrow{e} \in \overrightarrow{E} \mid e \in {F}_{j}}\right\} \) . Our aim is to define \( {f}_{i - 1} \) in such a way that (2) also holds for \( i - 1 \) .
We first consider the case that \( \left| {F}_{i}\right| = 1 \), say \( {F}_{i} = \{ e\} \) . We then let \( {f}_{i - 1} \mathrel{\text{:=}} {f}_{i} \), and thus have to show that \( {f}_{i} \) is non-zero on (the two directions of) \( e \) . Our assumption of \( \left| {F}_{i}\right| = 1 \) implies by the choice of \( {F}_{i} \) that \( G \) contains no \( {X}_{i} - {V}^{i - 1} \) edge other than \( e \) . Since \( G \) is 2-edge-connected, it therefore has at least-and thus, by (1), exactly-one edge \( {e}^{\prime } \) between \( {X}_{i} \) and \( \overline{{V}^{i - 1}} \smallsetminus {X}_{i} \) . We show that \( {f}_{i} \) is non-zero on \( {e}^{\prime } \) ; as \( \left\{ {e,{e}^{\prime }}\right\} \) is a cut in \( G \), this implies by Proposition 6.1.1 that \( {f}_{i} \) is also non-zero on \( e \) .
To show that \( {f}_{i} \) is non-zero on \( {e}^{\prime } \), we use (2): we show that \( {e}^{\prime } \in \) \( {E}^{\prime } \cup \mathop{\bigcup }\limits_{{j > i}}{F}_{j} \), i.e. that \( {e}^{\prime } \) lies in no \( {H}_{k} \) and in no \( {F}_{j} \) with \( j \leq i \) . Since \( {e}^{\prime } \) has both ends in \( \overline{{V}^{i - 1}} \), it clearly lies in no \( {F}_{j} \) with \( j \leq i \) and in no \( {H}_{k} \) with \( k < i \) . But every \( {H}_{k} \) with \( k \geq i \) is a subgraph of \( G\left\lbrack \overline{{V}^{i - 1}}\right\rbrack \) . Since \( {e}^{\prime } \) is a bridge of \( G\left\lbrack \overline{{V}^{i - 1}}\right\rbrack \) but \( {H}_{k} \) has no bridge, this means that \( {e}^{\prime } \notin {H}_{k} \) . Hence, \( {f}_{i - 1} \) does indeed satisfy (2) for \( i - 1 \) in the case considered.
It remains to consider the case that \( \left| {F}_{i}\right| = 2 \), say \( {F}_{i} = \left\{ {{e}_{1},{e}_{2}}\right\} \) . Since \( {H}_{i} \) and \( {H}^{i - 1} \) are both connected, we can find a cycle \( C \) in \( {H}^{i} = \) \( \left( {{H}_{i} \cup {H}^{i - 1}}\right) + {F}_{i} \) that contains \( {e}_{1} \) and \( {e}_{2} \) . If \( {f}_{i} \) is non-zero on both these edges, we again let \( {f}_{i - 1} \mathrel{\text{:=}} {f}_{i} \) . Otherwise, there are directions \( \overrightarrow{{e}_{1}} \) and \( \overrightarrow{{e}_{2}} \) of \( {e}_{1} \) and \( {e}_{2} \) such that, without loss of generality, \( {f}_{i}\left( \overrightarrow{{e}_{1}}\right) = \overline{0} \) and \( {f}_{i}\left( \overrightarrow{{e}_{2}}\right) \in \{ \overline{0},\overline{1}\} \) . Let \( \overrightarrow{C} \) be the orientation of \( C \) with \( \overrightarrow{{e}_{2}} \in \overrightarrow{C} \), and let \( g \) be a flow of value \( \overline{1} \) around \( \overrightarrow{C} \) (formally: let \( g \) be a \( {\mathbb{Z}}_{3} \) -circulation on \( G \) such that \( g\left( \overrightarrow{{e}_{2}}\right) = \overline{1} \) and \( {g}^{-1}\left( \overline{0}\right) = \overrightarrow{E} \smallsetminus \left( {\overrightarrow{C} \cup \widetilde{C}}\right) ) \) . We then let \( {f}_{i - 1} \mathrel{\text{:=}} {f}_{i} + g \) . By choice of the directions \( \overrightarrow{{e}_{1}} \) and \( \overrightarrow{{e}_{2}},{f}_{i - 1} \) is non-zero on both edges. Since \( {f}_{i - 1} \) agrees with \( {f}_{i} \) on all of \( \overrightarrow{{E}^{\prime }} \cup \mathop{\bigcup }\limits_{{j > i}}\overrightarrow{{F}_{j}} \) and (2) holds for \( i \), we again have (2) also for \( i - 1 \) .
Eventually, \( {f}_{0} \) will be a \( {\mathbb{Z}}_{3} \) -circulation on \( G \) that is nowhere zero except possibly on edges of \( {H}_{0} \cup \ldots \cup {H}_{n} \) . Composing \( {f}_{0} \) with the map \( \bar{h} \mapsto \overline{2h} \) from \( {\mathbb{Z}}_{3} \) to \( {\mathbb{Z}}_{6}\left( {h \in \{ 1,2\} }\right) \), we obtain a \( {\mathbb{Z}}_{6} \) -circulation \( f \) on \( G \) with values in \( \{ \overline{0},\overline{2},\overline{4}\} \) for all edges lying in some \( {H}_{i} \), and with values in \( \{ \overline{2},\overline{4}\} \) for all other edges. Adding to \( f \) a 2-flow on each \( {H}_{i} \) (formally: a \( {\mathbb{Z}}_{6} \) -circulation on \( G \) with values in \( \{ \overline{1}, - \overline{1}\} \) on the edges of \( {H}_{i} \) and \( \overline{0} \) otherwise; this exists by Proposition 6.4.1), we obtain a \( {\mathbb{Z}}_{6} \) -circulation on \( G \) that is nowhere zero. Hence, \( G \) has a 6 -flow by Theorem 6.3.3.
## Exercises
1. \( {}^{ - } \) Prove Proposition 6.2.1 by induction on \( \left| S\right| \) .
2. (i) \( {}^{ - } \) Given \( n \in \mathbb{N} \), find a capacity function for the network below such that the algorithm from the proof of the max-flow min-cut theorem will need more than \( n \) augmenting paths \( W \) if these are badly chosen.
\( {\text{(ii)}}^{ + } \) Show that, if all augmenting paths are chosen as short as possible, their number is bounded by a function of the size of the network.
3. \( {}^{ + } \) Derive Menger’s Theorem 3.3.5 from the max-flow min-cut theorem.
(Hint. The edge version is easy. For the vertex version, apply the edge version to a suitable auxiliary graph.)
4. \( {}^{ - } \) Let \( f \) be an \( H \) -circulation on \( G \) |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 56 | n a \( {\mathbb{Z}}_{6} \) -circulation on \( G \) that is nowhere zero. Hence, \( G \) has a 6 -flow by Theorem 6.3.3.
## Exercises
1. \( {}^{ - } \) Prove Proposition 6.2.1 by induction on \( \left| S\right| \) .
2. (i) \( {}^{ - } \) Given \( n \in \mathbb{N} \), find a capacity function for the network below such that the algorithm from the proof of the max-flow min-cut theorem will need more than \( n \) augmenting paths \( W \) if these are badly chosen.
\( {\text{(ii)}}^{ + } \) Show that, if all augmenting paths are chosen as short as possible, their number is bounded by a function of the size of the network.
3. \( {}^{ + } \) Derive Menger’s Theorem 3.3.5 from the max-flow min-cut theorem.
(Hint. The edge version is easy. For the vertex version, apply the edge version to a suitable auxiliary graph.)
4. \( {}^{ - } \) Let \( f \) be an \( H \) -circulation on \( G \) and \( g : H \rightarrow {H}^{\prime } \) a group homomorphism. Show that \( g \circ f \) is an \( {H}^{\prime } \) -circulation on \( G \) . Is \( g \circ f \) an \( {H}^{\prime } \) -flow if \( f \) is an \( H \) -flow?
5. \( {}^{ - } \) Given \( k \geq 1 \), show that a graph has a \( k \) -flow if and only if each of its blocks has a \( k \) -flow.
6. \( {}^{ - } \) Show that \( \varphi \left( {G/e}\right) \leq \varphi \left( G\right) \) whenever \( G \) is a multigraph and \( e \) an edge of \( G \) . Does this imply that, for every \( k \), the class of all multigraphs admitting a \( k \) -flow is closed under taking minors?
7. \( {}^{ - } \) Work out the flow number of \( {K}^{4} \) directly, without using any results from the text.
8. Let \( H \) be a finite abelian group, \( G \) a graph, and \( T \) a spanning tree of \( G \) . Show that every mapping from the directions of \( E\left( G\right) \smallsetminus E\left( T\right) \) to \( H \) that satisfies (F1) extends uniquely to an \( H \) -circulation on \( G \) .
Do not use the 6-flow Theorem 6.6.1 for the following three exercises.
9. Show that \( \varphi \left( G\right) < \infty \) for every bridgeless multigraph \( G \) .
10. Assume that a graph \( G \) has \( m \) spanning trees such that no edge of \( G \) lies in all of these trees. Show that \( \varphi \left( G\right) \leq {2}^{m} \) .
11. Let \( G \) be a bridgeless connected graph with \( n \) vertices and \( m \) edges. By considering a normal spanning tree of \( G \), show that \( \varphi \left( G\right) \leq m - n + 2 \) .
12. Show that every graph with a Hamilton cycle has a 4-flow. (A Hamilton cycle of \( G \) is a cycle in \( G \) that contains all the vertices of \( G \) .)
13. A family of (not necessarily distinct) cycles in a graph \( G \) is called a cycle double cover of \( G \) if every edge of \( G \) lies on exactly two of these cycles. The cycle double cover conjecture asserts that every bridgeless multigraph has a cycle double cover. Prove the conjecture for graphs with a 4-flow.
14. \( {}^{ - } \) Determine the flow number of \( {C}^{5} * {K}^{1} \), the wheel with 5 spokes.
15. Find bridgeless graphs \( G \) and \( H = G - e \) such that \( 2 < \varphi \left( G\right) < \varphi \left( H\right) \) .
16. Prove Proposition 6.4.1 without using Theorem 6.3.3.
17. The proof of the backward implication of Lemma 6.5.2 (ii) is a bit pedestrian. Use Lemmas 1.9.4 and 3.1.1, Proposition 4.6.1, and Exercise 31 of Chapter 4 for a shorter higher-level proof.
18. \( {}^{ + } \) Prove Heawood’s theorem that a plane triangulation is 3-colourable if and only if all its vertices have even degree.
19. Show that the 3-flow conjecture for planar multigraphs is equivalent to Grötzsch's Theorem 5.1.3.
20. \( {\left( \mathrm{i}\right) }^{ - } \) Show that the four colour theorem is equivalent to the non-existence of a planar snark, i.e. to the statement that every cubic bridgeless planar multigraph has a 4-flow.
(ii) Can 'bridgeless' in (i) be replaced by '3-connected'?
21. \( {}^{ + } \) Show that a graph \( G = \left( {V, E}\right) \) has a \( k \) -flow if and only if it admits an orientation \( D \) that directs, for every \( X \subseteq V \), at least \( 1/k \) of the edges in \( E\left( {X,\bar{X}}\right) \) from \( X \) towards \( \bar{X} \) .
22. \( {}^{ - } \) Generalize the 6-flow Theorem 6.6.1 to multigraphs.
## Notes
Network flow theory is an application of graph theory that has had a major and lasting impact on its development over decades. As is illustrated already by the fact that Menger's theorem can be deduced easily from the max-flow min-cut theorem (Exercise 3), the interaction between graphs and networks may go either way: while 'pure' results in areas such as connectivity, matching and random graphs have found applications in network flows, the intuitive power of the latter has boosted the development of proof techniques that have in turn brought about theoretic advances.
The classical reference for network flows is L.R. Ford & D.R. Fulkerson, Flows in Networks, Princeton University Press 1962. More recent and comprehensive accounts are given by R.K. Ahuja, T.L. Magnanti & J.B. Orlin, Network flows, Prentice-Hall 1993, by A. Frank in his chapter in the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995, and by A. Schrijver, Combinatorial optimization, Springer 2003. An introduction to graph algorithms in general is given in A. Gibbons, Algorithmic Graph Theory, Cambridge University Press 1985.
If one recasts the maximum flow problem in linear programming terms, one can derive the max-flow min-cut theorem from the linear programming duality theorem; see A. Schrijver, Theory of integer and linear programming, Wiley 1986.
The more algebraic theory of group-valued flows and \( k \) -flows has been developed largely by Tutte; he gives a thorough account in his monograph W.T. Tutte, Graph Theory, Addison-Wesley 1984. The fact that the number of \( k \) -flows of a multigraph is a polynomial in \( k \), whose values can be bounded in terms of the corresponding values of the flow polynomial, was proved by M. Kochol, Polynomials associated with nowhere-zero \( {}^{7} \) flows, J. Combin. Theory B 84 (2002), 260-269.
Tutte's flow conjectures are covered also in F. Jaeger's survey, Nowhere-zero flow problems, in (L.W. Beineke & R.J. Wilson, eds.) Selected Topics in Graph Theory 3, Academic Press 1988. For the flow conjectures, see also T.R. Jensen & B. Toft, Graph Coloring Problems, Wiley 1995. Seymour's 6- flow theorem is proved in P.D. Seymour, Nowhere-zero 6-flows, J. Combin. Theory B 30 (1981), 130-135. This paper also indicates how Tutte's 5-flow conjecture reduces to snarks. In 1998, Robertson, Sanders, Seymour and Thomas announced a proof of the 4-flow conjecture for cubic graphs.
Finally, Tutte discovered a 2-variable polynomial associated with a graph, which generalizes both its chromatic polynomial and its flow polynomial. What little is known about this Tutte polynomial can hardly be more than the tip of the iceberg: it has far-reaching, and largely unexplored, connections to areas as diverse as knot theory and statistical physics. See D.J.A. Welsh, Complexity: knots, colourings and counting (LMS Lecture Notes 186), Cambridge University Press 1993.
\( {}^{7} \) In the literature, the term ’flow’ is often used to mean what we have called ’circulation', i.e. flows are not required to be nowhere zero unless this is stated explicitly.
## Extremal Graph Theory
In this chapter we study how global parameters of a graph, such as its edge density or chromatic number, can influence its local substructures. How many edges, for instance, do we have to give a graph on \( n \) vertices to be sure that, no matter how these edges are arranged, the graph will contain a \( {K}^{r} \) subgraph for some given \( r \) ? Or at least a \( {K}^{r} \) minor? Will some sufficiently high average degree or chromatic number ensure that one of these substructures occurs?
Questions of this type are among the most natural ones in graph theory, and there is a host of deep and interesting results. Collectively, these are known as extremal graph theory.
Extremal graph problems in this sense fall neatly into two categories, as follows. If we are looking for ways to ensure by global assumptions that a graph \( G \) contains some given graph \( H \) as a minor (or topological minor), it will suffice to raise \( \parallel G\parallel \) above the value of some linear function of \( \left| G\right| \), i.e., to make \( \varepsilon \left( G\right) \) large enough. The precise value of \( \varepsilon \) needed to force a desired minor or topological minor will be our topic in Section 7.2. Graphs whose number of edges is about \( {}^{1} \) linear in their number of vertices are called sparse, so Section 7.2 is devoted to 'sparse
---
sparse
---
extremal graph theory'.
A particularly interesting way to force an \( H \) minor is to assume that \( \chi \left( G\right) \) is large. Recall that if \( \chi \left( G\right) \geq k + 1 \), say, then \( G \) has a subgraph \( {G}^{\prime } \) with \( {2\varepsilon }\left( {G}^{\prime }\right) \geq \delta \left( {G}^{\prime }\right) \geq k \) (Corollary 5.2.3). The question here is whether the effect of large \( \chi \) is limited to this indirect influence via \( \varepsilon \) , or whether an assumption of \( \chi \geq k + 1 \) can force bigger minors than
---
1 Formally, the notions of sparse and dense (below) make sense only for classes of graphs whose order tends to infinity, not for individual graphs.
---
the assumption of \( {2\varepsilon } \geq k \) can. Hadwiger’s conjecture, which we meet in Section 7.3, asserts that \( \chi \) has this quality. The conjecture can be viewed as a generalization of the four colour theorem, and is regarded by many as the most challenging open problem in graph theory.
On the othe |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 57 | interesting way to force an \( H \) minor is to assume that \( \chi \left( G\right) \) is large. Recall that if \( \chi \left( G\right) \geq k + 1 \), say, then \( G \) has a subgraph \( {G}^{\prime } \) with \( {2\varepsilon }\left( {G}^{\prime }\right) \geq \delta \left( {G}^{\prime }\right) \geq k \) (Corollary 5.2.3). The question here is whether the effect of large \( \chi \) is limited to this indirect influence via \( \varepsilon \) , or whether an assumption of \( \chi \geq k + 1 \) can force bigger minors than
---
1 Formally, the notions of sparse and dense (below) make sense only for classes of graphs whose order tends to infinity, not for individual graphs.
---
the assumption of \( {2\varepsilon } \geq k \) can. Hadwiger’s conjecture, which we meet in Section 7.3, asserts that \( \chi \) has this quality. The conjecture can be viewed as a generalization of the four colour theorem, and is regarded by many as the most challenging open problem in graph theory.
On the other hand, if we ask what global assumptions might imply the existence of some given graph \( H \) as a subgraph, it will not help to raise invariants such as \( \varepsilon \) or \( \chi \), let alone any of the other invariants discussed in Chapter 1. For as soon as \( H \) contains a cycle, there are graphs of arbitrarily large chromatic number not containing \( H \) as a subgraph (Theorem 5.2.5). In fact, unless \( H \) is bipartite, any function \( f \) such that \( f\left( n\right) \) edges on \( n \) vertices force an \( H \) subgraph must grow quadratically with \( n \) : since complete bipartite graphs can have \( \frac{1}{4}{n}^{2} \) edges, \( f\left( n\right) \) must exceed \( \frac{1}{4}{n}^{2} \) .
dense Graphs with a number of edges about quadratic in their number of edge vertices are usually called dense; the number \( \parallel G\parallel /\left( \begin{matrix} \left| G\right| \\ 2 \end{matrix}\right) \), the proportion of its potential edges that \( G \) actually has, is the edge density of \( G \) . The density question of exactly which edge density is needed to force a given subgraph is the archetypal extremal graph problem, and it is our first topic in this chapter (Section 7.1). Rather than attempting to survey the wide field of 'dense extremal graph theory', however, we shall concentrate on its two most important results: we first prove Turán's classical extremal graph theorem for \( H = {K}^{r} \) -a result that has served as a model for countless similar theorems for other graphs \( H \) - and then state the fundamental Erdős-Stone theorem, which gives precise asymptotic information for all \( H \) at once.
Although the Erdős-Stone theorem can be proved by elementary means, we shall use the opportunity of its proof to portray a powerful modern proof technique that has transformed much of extremal graph theory in recent years: Szemerédi regularity lemma. This lemma is presented and proved in Section 7.4. In Section 7.5, we outline a general method for applying it, and illustrate this in the proof of the Erdős-Stone theorem. Another application of the regularity lemma will be given in Chapter 9.2.
## 7.1 Subgraphs
Let \( H \) be a graph and \( n \geq \left| H\right| \) . How many edges will suffice to force an \( H \) subgraph in any graph on \( n \) vertices, no matter how these edges are arranged? Or, to rephrase the problem: which is the greatest possible number of edges that a graph on \( n \) vertices can have without containing a copy of \( H \) as a subgraph? What will such a graph look like? Will it be unique?
A graph \( G \nsupseteq H \) on \( n \) vertices with the largest possible number of
extremal edges is called extremal for \( n \) and \( H \) ; its number of edges is denoted by
\( \operatorname{ex}\left( {n, H}\right) \) . Clearly, any graph \( G \) that is extremal for some \( n \) and \( H \) will
\( \operatorname{ex}\left( {n, H}\right) \)
also be edge-maximal with \( H \nsubseteq G \) . Conversely, though, edge-maximality does not imply extremality: \( G \) may well be edge-maximal with \( H \nsubseteq G \) while having fewer than \( \operatorname{ex}\left( {n, H}\right) \) edges (Fig. 7.1.1).
Fig. 7.1.1. Two graphs that are edge-maximal with \( {P}^{3} \nsubseteq G \) ; is the right one extremal?
As a case in point, we consider our problem for \( H = {K}^{r} \) (with \( r > 1 \) ). A moment's thought suggests some obvious candidates for extremality here: all complete \( \left( {r - 1}\right) \) -partite graphs are edge-maximal without containing \( {K}^{r} \) . But which among these have the greatest number of edges? Clearly those whose partition sets are as equal as possible, i.e. differ in size by at most 1: if \( {V}_{1},{V}_{2} \) are two partition sets with \( \left| {V}_{1}\right| - \left| {V}_{2}\right| \geq 2 \), we may increase the number of edges in our complete \( \left( {r - 1}\right) \) -partite graph by moving a vertex from \( {V}_{1} \) to \( {V}_{2} \) .
The unique complete \( \left( {r - 1}\right) \) -partite graphs on \( n \geq r - 1 \) vertices whose partition sets differ in size by at most 1 are called Turán graphs; we denote them by \( {T}^{r - 1}\left( n\right) \) and their number of edges by \( {t}_{r - 1}\left( n\right) \) \( {T}^{r - 1}\left( n\right) \) (Fig. 7.1.2). For \( n < r - 1 \) we shall formally continue to use these
\( {t}_{r - 1}\left( n\right) \)
definitions, with the proviso that - contrary to our usual terminology-the partition sets may now be empty; then, clearly, \( {T}^{r - 1}\left( n\right) = {K}^{n} \) for all \( n \leq r - 1 \) .
Fig. 7.1.2. The Turán graph \( {T}^{3}\left( 8\right) \)
The following theorem tells us that \( {T}^{r - 1}\left( n\right) \) is indeed extremal for \( n \) and \( {K}^{r} \), and as such unique; in particular, \( \operatorname{ex}\left( {n,{K}^{r}}\right) = {t}_{r - 1}\left( n\right) \) .
Theorem 7.1.1. (Turán 1941)
\( \left\lbrack {9.2.2}\right\rbrack \)
For all integers \( r, n \) with \( r > 1 \), every graph \( G \nsupseteq {K}^{r} \) with \( n \) vertices and \( \operatorname{ex}\left( {n,{K}^{r}}\right) \) edges is a \( {T}^{r - 1}\left( n\right) \) .
We give two proofs: one using induction, the other by a very short and direct local argument.
First proof. We apply induction on \( n \) . For \( n \leq r - 1 \) we have \( G = \) \( {K}^{n} = {T}^{r - 1}\left( n\right) \) as claimed. For the induction step, let now \( n \geq r \) .
Since \( G \) is edge-maximal without a \( {K}^{r} \) subgraph, \( G \) has a subgraph \( K = {K}^{r - 1} \) . By the induction hypothesis, \( G - K \) has at most \( {t}_{r - 1}\left( {n - r + 1}\right) \) edges, and each vertex of \( G - K \) has at most \( r - 2 \) neighbours in \( K \) . Hence,
\[
\parallel G\parallel \leq {t}_{r - 1}\left( {n - r + 1}\right) + \left( {n - r + 1}\right) \left( {r - 2}\right) + \left( \begin{matrix} r - 1 \\ 2 \end{matrix}\right) = {t}_{r - 1}\left( n\right) ;
\]
(1)
the equality on the right follows by inspection of the Turán graph \( {T}^{r - 1}\left( n\right) \) (Fig. 7.1.3).
Fig. 7.1.3. The equation from (1) for \( r = 5 \) and \( n = {14} \)
Since \( G \) is extremal for \( {K}^{r} \) (and \( {T}^{r - 1}\left( n\right) \nsupseteq {K}^{r} \) ), we have equality in (1). Thus, every vertex of \( G - K \) has exactly \( r - 2 \) neighbours in \( K - \)
\( {x}_{1},\ldots ,{x}_{r - 1} \) just like the vertices \( {x}_{1},\ldots ,{x}_{r - 1} \) of \( K \) itself. For \( i = 1,\ldots, r - 1 \) let
\( {V}_{1},\ldots ,{V}_{r - 1} \)
\[
{V}_{i} \mathrel{\text{:=}} \left\{ {v \in V\left( G\right) \mid v{x}_{i} \notin E\left( G\right) }\right\}
\]
be the set of all vertices of \( G \) whose \( r - 2 \) neighbours in \( K \) are precisely the vertices other than \( {x}_{i} \) . Since \( {K}^{r} \nsubseteq G \), each of the sets \( {V}_{i} \) is independent, and they partition \( V\left( G\right) \) . Hence, \( G \) is \( \left( {r - 1}\right) \) -partite. As \( {T}^{r - 1}\left( n\right) \) is the unique \( \left( {r - 1}\right) \) -partite graph with \( n \) vertices and the maximum number of edges, our claim that \( G = {T}^{r - 1}\left( n\right) \) follows from the assumed extremality of \( G \) .
---
vertex
duplication
---
In our second proof of Turán's theorem we shall use an operation called vertex duplication. By duplicating a vertex \( v \in G \) we mean adding to \( G \) a new vertex \( {v}^{\prime } \) and joining it to exactly the neighbours of \( v \) (but not to \( v \) itself).
Second proof. We have already seen that among the complete \( k \) -partite graphs on \( n \) vertices the Turán graphs \( {T}^{k}\left( n\right) \) have the most edges, and their degrees show that \( {T}^{r - 1}\left( n\right) \) has more edges than any \( {T}^{k}\left( n\right) \) with \( k < r - 1 \) . So it suffices to show that \( G \) is complete multipartite.
If not, then non-adjacency is not an equivalence relation on \( V\left( G\right) \) , and so there are vertices \( {y}_{1}, x,{y}_{2} \) such that \( {y}_{1}x, x{y}_{2} \notin E\left( G\right) \) but \( {y}_{1}{y}_{2} \in \) \( E\left( G\right) \) . If \( d\left( {y}_{1}\right) > d\left( x\right) \), then deleting \( x \) and duplicating \( {y}_{1} \) yields another \( {K}^{r} \) -free graph with more edges than \( G \), contradicting the choice of \( G \) . So \( d\left( {y}_{1}\right) \leq d\left( x\right) \), and similarly \( d\left( {y}_{2}\right) \leq d\left( x\right) \) . But then deleting both \( {y}_{1} \) and \( {y}_{2} \) and duplicating \( x \) twice yields a \( {K}^{r} \) -free graph with more edges than \( G \), again contradicting the choice of \( |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 58 | ght) \) have the most edges, and their degrees show that \( {T}^{r - 1}\left( n\right) \) has more edges than any \( {T}^{k}\left( n\right) \) with \( k < r - 1 \) . So it suffices to show that \( G \) is complete multipartite.
If not, then non-adjacency is not an equivalence relation on \( V\left( G\right) \) , and so there are vertices \( {y}_{1}, x,{y}_{2} \) such that \( {y}_{1}x, x{y}_{2} \notin E\left( G\right) \) but \( {y}_{1}{y}_{2} \in \) \( E\left( G\right) \) . If \( d\left( {y}_{1}\right) > d\left( x\right) \), then deleting \( x \) and duplicating \( {y}_{1} \) yields another \( {K}^{r} \) -free graph with more edges than \( G \), contradicting the choice of \( G \) . So \( d\left( {y}_{1}\right) \leq d\left( x\right) \), and similarly \( d\left( {y}_{2}\right) \leq d\left( x\right) \) . But then deleting both \( {y}_{1} \) and \( {y}_{2} \) and duplicating \( x \) twice yields a \( {K}^{r} \) -free graph with more edges than \( G \), again contradicting the choice of \( G \) .
The Turán graphs \( {T}^{r - 1}\left( n\right) \) are dense: in order of magnitude, they have about \( {n}^{2} \) edges. More exactly, for every \( n \) and \( r \) we have
\[
{t}_{r - 1}\left( n\right) \leq \frac{1}{2}{n}^{2}\frac{r - 2}{r - 1}
\]
with equality whenever \( r - 1 \) divides \( n \) (Exercise 7). It is therefore remarkable that just \( \epsilon {n}^{2} \) more edges (for any fixed \( \epsilon > 0 \) and \( n \) large) give us not only a \( {K}^{r} \) subgraph (as does Turán’s theorem) but a \( {K}_{s}^{r} \) for any given integer \( s \) -a graph itself teeming with \( {K}^{r} \) subgraphs:
Theorem 7.1.2. (Erdős & Stone 1946)
For all integers \( r \geq 2 \) and \( s \geq 1 \), and every \( \epsilon > 0 \), there exists an integer \( {n}_{0} \) such that every graph with \( n \geq {n}_{0} \) vertices and at least
\[
{t}_{r - 1}\left( n\right) + \epsilon {n}^{2}
\]
edges contains \( {K}_{s}^{r} \) as a subgraph.
A proof of the Erdős-Stone theorem will be given in Section 7.5, as an illustration of how the regularity lemma may be applied. But the theorem can also be proved directly; see the notes for references.
The Erdős-Stone theorem is interesting not only in its own right: it also has a most interesting corollary. In fact, it was this entirely unexpected corollary that established the theorem as a kind of meta-theorem for the extremal theory of dense graphs, and thus made it famous.
Given a graph \( H \) and an integer \( n \), consider the number \( {h}_{n} \mathrel{\text{:=}} \) \( \operatorname{ex}\left( {n, H}\right) /\left( \begin{matrix} n \\ 2 \end{matrix}\right) \) : the maximum edge density that an \( n \) -vertex graph can have without containing a copy of \( H \) . Could it be that this critical density is essentially just a function of \( H \), that \( {h}_{n} \) converges as \( n \rightarrow \infty \) ? Theorem 7.1.2 implies this, and more: the limit of \( {h}_{n} \) is determined by a very simple function of a natural invariant of \( H \) -its chromatic number!
Corollary 7.1.3. For every graph \( H \) with at least one edge,
\[
\mathop{\lim }\limits_{{n \rightarrow \infty }}\operatorname{ex}\left( {n, H}\right) {\left( \begin{array}{l} n \\ 2 \end{array}\right) }^{-1} = \frac{\chi \left( H\right) - 2}{\chi \left( H\right) - 1}.
\]
For the proof of Corollary 7.1.3 we need as a lemma that \( {t}_{r - 1}\left( n\right) \) never deviates much from the value it takes when \( r - 1 \) divides \( n \) (see above), and that \( {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \) converges accordingly. The proof of the lemma is left as an easy exercise with hint (Exercise 8).
\( \left\lbrack {7.1.2}\right\rbrack \)
Lemma 7.1.4.
\[
\mathop{\lim }\limits_{{n \rightarrow \infty }}{t}_{r - 1}\left( n\right) {\left( \begin{array}{l} n \\ 2 \end{array}\right) }^{-1} = \frac{r - 2}{r - 1}.
\]
Proof of Corollary 7.1.3. Let \( r \mathrel{\text{:=}} \chi \left( H\right) \) . Since \( H \) cannot be coloured with \( r - 1 \) colours, we have \( H \nsubseteq {T}^{r - 1}\left( n\right) \) for all \( n \in \mathbb{N} \), and hence
\[
{t}_{r - 1}\left( n\right) \leq \operatorname{ex}\left( {n, H}\right) .
\]
On the other hand, \( H \subseteq {K}_{s}^{r} \) for all sufficiently large \( s \), so
\[
\operatorname{ex}\left( {n, H}\right) \leq \operatorname{ex}\left( {n,{K}_{s}^{r}}\right)
\]
for all those \( s \) . Let us fix such an \( s \) . For every \( \epsilon > 0 \), Theorem 7.1.2 implies that eventually (i.e. for large enough \( n \) )
\[
\operatorname{ex}\left( {n,{K}_{s}^{r}}\right) < {t}_{r - 1}\left( n\right) + \epsilon {n}^{2}.
\]
Hence for \( n \) large,
\[
{t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \leq \operatorname{ex}\left( {n, H}\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right)
\]
\[
\leq \operatorname{ex}\left( {n,{K}_{s}^{r}}\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right)
\]
\[
< {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) + \epsilon {n}^{2}/\left( \begin{array}{l} n \\ 2 \end{array}\right)
\]
\[
= {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) + {2\epsilon }/\left( {1 - \frac{1}{n}}\right)
\]
\[
\leq {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) + {4\epsilon }\;\left( {\text{ assume }n \geq 2}\right) .
\]
Therefore, since \( {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \) converges to \( \frac{r - 2}{r - 1} \) (Lemma 7.1.4), so does \( \operatorname{ex}\left( {n, H}\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \) .
For bipartite graphs \( H \), Corollary 7.1.3 says that substantially fewer than \( \left( \begin{array}{l} n \\ 2 \end{array}\right) \) edges suffice to force an \( H \) subgraph. It turns out that
\[
{c}_{1}{n}^{2 - \frac{2}{r + 1}} \leq \operatorname{ex}\left( {n,{K}_{r, r}}\right) \leq {c}_{2}{n}^{2 - \frac{1}{r}}
\]
for suitable constants \( {c}_{1},{c}_{2} \) depending on \( r \) ; the lower bound is obtained by random graphs, \( {}^{2} \) the upper bound is calculated in Exercise 11. If \( H \) is a forest, then \( H \subseteq G \) as soon as \( \varepsilon \left( G\right) \) is large enough, so \( \operatorname{ex}\left( {n, H}\right) \) is at most linear in \( n \) (Exercise 13). Erdös and Sós conjectured in 1963 that \( \operatorname{ex}\left( {n, T}\right) \leq \frac{1}{2}\left( {k - 1}\right) n \) for all trees with \( k \geq 2 \) edges; as a general bound for all \( n \), this is best possible for every \( T \) (Exercises 14-16).
A related but rather different question is whether large values of \( \varepsilon \) or \( \chi \) can force a graph \( G \) to contain a given tree \( T \) as an induced subgraph. Of course, we need some additional assumption for this to make sense-for example, to prevent \( G \) from just being a large complete graph. The weakest sensible such assumption is that \( G \) has bounded clique number, i.e., that \( G \nsupseteq {K}^{r} \) for some fixed integer \( r \) . Then large average degree still does not force an induced copy of \( T \) -consider complete bipartite graphs - but large chromatic number might: according to a remarkable conjecture of Gyárfás (1975), there exists for every \( r \in \mathbb{N} \) and every tree \( T \) an integer \( k = k\left( {T, r}\right) \) such that every graph \( G \) with \( \chi \left( G\right) \geq k \) and \( \omega \left( G\right) < r \) contains \( T \) as an induced subgraph.
## 7.2 Minors
In this section and the next, we ask how global assumptions about a graph - on its average degree, its chromatic number, or even its girth-can force it to contain a given graph as a minor or topological minor.
For example, consider the analogue of Turán's theorem: how many edges on \( n \) vertices force a \( {K}^{r} \) minor or topological minor? We know already from Chapter 3.5 that topological \( {K}^{r} \) minors can be forced in sparse graphs, i.e., that some linear number \( {c}_{r}n \) of edges is enough. But what can we say about \( {c}_{r} \) as a function of \( r \) ? The upper bound \( h\left( r\right) \) on \( {c}_{r} \) that we found in the proof of Lemma 3.5.1 was \( {2}^{\left( \begin{matrix} r \\ 2 \end{matrix}\right) } \) ; an easy lower bound is \( \frac{1}{8}{r}^{2} \) (Exercise 25).
It was only in 1996 that this lower bound was shown to be of the right order of magnitude. With the help of Theorem 3.5.3, the proof is now just a few lines:
Theorem 7.2.1. There is a constant \( c \in \mathbb{R} \) such that, for every \( r \in \mathbb{N} \) , every graph \( G \) of average degree \( d\left( G\right) \geq c{r}^{2} \) contains \( {K}^{r} \) as a topological minor.
---
2 see Chapter 11
---
Proof. We prove the theorem with \( c = {10} \) . Let \( G \) be a graph of average degree at least \( {10}{r}^{2} \) . By Theorem 1.4.3 with \( k \mathrel{\text{:=}} {r}^{2}, G \) has an \( {r}^{2} \) -connected subgraph \( H \) with \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - {r}^{2} \geq 4{r}^{2} \) . To find a \( T{K}^{r} \) in \( H \), we start by picking \( r \) vertices as branch vertices, and \( r - 1 \) neighbours of each of these as some initial subdividing vertices. These are \( {r}^{2} \) vertices in total, so as \( \delta \left( H\right) \geq \kappa \left( H\right) \geq {r}^{2} \) they can be chosen distinct. Now all that remains is to link up the subdividing vertices in pairs, by disjoint paths in \( H \) corresponding to the edges of the \( {K}^{r} \) of which we wish to find a subdivision. Such paths exist, because \( H \) is \( \frac{1}{2}{r}^{2} \) -linked by Theorem 3.5.3.
For small \( r \), one can try to determine the exact number of edges needed to force a \( T{K}^{r} \) subgraph on \( n \) vertices. For \( r = 4 \), this number is \( {2n} - 2 \) ; see Corollary 7.3.2. For \( r = 5 \), plane triangulations yield |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 59 | 2}, G \) has an \( {r}^{2} \) -connected subgraph \( H \) with \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - {r}^{2} \geq 4{r}^{2} \) . To find a \( T{K}^{r} \) in \( H \), we start by picking \( r \) vertices as branch vertices, and \( r - 1 \) neighbours of each of these as some initial subdividing vertices. These are \( {r}^{2} \) vertices in total, so as \( \delta \left( H\right) \geq \kappa \left( H\right) \geq {r}^{2} \) they can be chosen distinct. Now all that remains is to link up the subdividing vertices in pairs, by disjoint paths in \( H \) corresponding to the edges of the \( {K}^{r} \) of which we wish to find a subdivision. Such paths exist, because \( H \) is \( \frac{1}{2}{r}^{2} \) -linked by Theorem 3.5.3.
For small \( r \), one can try to determine the exact number of edges needed to force a \( T{K}^{r} \) subgraph on \( n \) vertices. For \( r = 4 \), this number is \( {2n} - 2 \) ; see Corollary 7.3.2. For \( r = 5 \), plane triangulations yield a lower bound of \( {3n} - 5 \) (Corollary 4.2.10). The converse, that \( {3n} - 5 \) edges do force a \( T{K}^{5} \) -not just either a \( T{K}^{5} \) or a \( T{K}_{3,3} \), as they do by Corollary 4.2.10 and Kuratowski's theorem-is already a difficult theorem (Mader 1998).
Let us now turn from topological minors to general minors. The average degree needed to force a \( {K}^{r} \) minor is known almost precisely. Thomason (2001) determined, asymptotically, the smallest constant \( c \) that makes the following theorem true as \( \alpha + o\left( 1\right) \), where \( o\left( 1\right) \) stands for a function of \( r \) tending to zero as \( r \rightarrow \infty \) and \( \alpha = {0.53131}\ldots \) is an explicit constant.
## Theorem 7.2.2. (Kostochka 1982)
There exists a constant \( c \in \mathbb{R} \) such that, for every \( r \in \mathbb{N} \), every graph \( G \) of average degree \( d\left( G\right) \geq {cr}\sqrt{\log r} \) contains \( {K}^{r} \) as a minor. Up to the value of \( c \), this bound is best possible as a function of \( r \) .
The easier implication of the theorem, the fact that in general an average degree of \( {cr}\sqrt{\log r} \) is needed to force a \( {K}^{r} \) minor, follows from considering random graphs, to be introduced in Chapter 11. The converse implication, that this average degree suffices, is proved by methods not dissimilar to the proof of Theorem 3.5.3.
Rather than proving Theorem 7.2.2, therefore, we devote the remainder of this section to another striking aspect of forcing minors: that we can force a \( {K}^{r} \) minor in a graph simply by raising its girth (as long as we do not merely subdivide edges). At first glance, this may seem almost paradoxical. But it looks more plausible if, rather than trying to force a \( {K}^{r} \) minor directly, we instead try to force a minor just of large minimum or average degree - which suffices by Theorem 7.2.2. For if the girth \( g \) of a graph is large then the ball \( \{ v \mid d\left( {x, v}\right) < \lfloor g/2\rfloor \} \) around a vertex \( x \) induces a tree with many leaves, each of which sends all but one of its incident edges away from the tree. Contracting enough disjoint such trees we can thus hope to obtain a minor of large average degree, which in turn will have a large complete minor.
The following lemma realizes this idea.
Lemma 7.2.3. Let \( d, k \in \mathbb{N} \) with \( d \geq 3 \), and let \( G \) be a graph of minimum degree \( \delta \left( G\right) \geq d \) and girth \( g\left( G\right) \geq {8k} + 3 \) . Then \( G \) has a minor \( H \) of minimum degree \( \delta \left( H\right) \geq d{\left( d - 1\right) }^{k} \) .
Proof. Let \( X \subseteq V\left( G\right) \) be maximal with \( d\left( {x, y}\right) > {2k} \) for all \( x, y \in X \) . For each \( x \in X \) put \( {T}_{x}^{0} \mathrel{\text{:=}} \{ x\} \) . Given \( i < {2k} \), assume that we have defined disjoint trees \( {T}_{x}^{i} \subseteq G \) (one for each \( x \in X \) ) whose vertices together are precisely the vertices at distance at most \( i \) from \( X \) in \( G \) . Joining each vertex at distance \( i + 1 \) from \( X \) to a neighbour at distance \( i \), we obtain a similar set of disjoint trees \( {T}_{x}^{i + 1} \) . As every vertex of \( G \) has distance at most \( {2k} \) from \( X \) (by the maximality of \( X \) ), the trees \( {T}_{x} \mathrel{\text{:=}} {T}_{x}^{2k} \) obtained in this way partition the entire vertex set of \( G \) . Let \( H \) be the minor of \( G \) obtained by contracting every \( {T}_{x} \) .
To prove that \( \delta \left( H\right) \geq d{\left( d - 1\right) }^{k} \), note first that the \( {T}_{x} \) are induced subgraphs of \( G \), because \( \operatorname{diam}{T}_{x} \leq {4k} \) and \( g\left( G\right) > {4k} + 1 \) . Similarly, there is at most one edge in \( G \) between any two trees \( {T}_{x} \) and \( {T}_{y} \) : two such edges, together with the paths joining their ends in \( {T}_{x} \) and \( {T}_{y} \) , would form a cycle of length at most \( {8k} + 2 < g\left( G\right) \) . So all the edges leaving \( {T}_{x} \) are preserved in the contraction.
How many such edges are there? Note that, for every vertex \( u \in \) \( {T}_{x}^{k - 1} \), all its \( {d}_{G}\left( u\right) \geq d \) neighbours \( v \) also lie in \( {T}_{x} \) : since \( d\left( {v, x}\right) \leq k \) and \( d\left( {x, y}\right) > {2k} \) for every other \( y \in X \), we have \( d\left( {v, y}\right) > k \geq d\left( {v, x}\right) \) , so \( v \) was added to \( {T}_{x} \) rather than to \( {T}_{y} \) when those trees were defined. Therefore \( {T}_{x}^{k} \), and hence also \( {T}_{x} \), has at least \( d{\left( d - 1\right) }^{k - 1} \) leaves. But every leaf of \( {T}_{x} \) sends at least \( d - 1 \) edges away from \( {T}_{x} \), so \( {T}_{x} \) sends at least \( d{\left( d - 1\right) }^{k} \) edges to (distinct) other trees \( {T}_{y} \) .
Lemma 7.2.3 provides Theorem 7.2.2 with the following corollary:
Theorem 7.2.4. (Thomassen 1983)
There exists a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that every graph of minimum degree at least 3 and girth at least \( f\left( r\right) \) has a \( {K}^{r} \) minor, for all \( r \in \mathbb{N} \) .
Proof. We prove the theorem with \( f\left( r\right) \mathrel{\text{:=}} 8\log r + 4\log \log r + c \), for some constant \( c \in \mathbb{R} \) . Let \( k = k\left( r\right) \in \mathbb{N} \) be minimal with \( 3 \cdot {2}^{k} \geq {c}^{\prime }r\sqrt{\log r} \) , where \( {c}^{\prime } \in \mathbb{R} \) is the constant from Theorem 7.2.2. Then for a suitable constant \( c \in \mathbb{R} \) we have \( {8k} + 3 \leq 8\log r + 4\log \log r + c \), and the result follows by Lemma 7.2.3 and Theorem 7.2.2.
Large girth can also be used to force a topological \( {K}^{r} \) minor . We now need some vertices of degree at least \( r - 1 \) to serve as branch vertices, but if we assume a minimum degree of \( r - 1 \) to secure these, we can even get by with a girth bound that is independent of \( r \) :
\( \left\lbrack {7.3.9}\right\rbrack \) Theorem 7.2.5. (Kühn & Osthus 2002)
There exists a constant \( g \) such that \( G \supseteq T{K}^{r} \) for every graph \( G \) satisfying \( \delta \left( G\right) \geq r - 1 \) and \( g\left( G\right) \geq g \) .
## 7.3 Hadwiger's conjecture
As we saw in Section 7.2, an average degree of \( {cr}\sqrt{\log r} \) suffices to force an arbitrary graph to have a \( {K}^{r} \) minor, and an average degree of \( c{r}^{2} \) forces it to have a topological \( {K}^{r} \) minor. If we replace ’average degree’ above with ’chromatic number’ then, with almost the same constants \( c \) , the two assertions remain true: this is because every graph with chromatic number \( k \) has a subgraph of average degree at least \( k - 1 \) (Corollary 5.2.3).
Although both functions above, \( {cr}\sqrt{\log r} \) and \( c{r}^{2} \), are best possible (up to the constant \( c \) ) for the said implications with ’average degree’, the question arises whether they are still best possible with 'chromatic number' or whether some slower-growing function would do in that case. What lies hidden behind this problem about growth rates is a fundamental question about the nature of the invariant \( \chi \) : can this invariant have some direct structural effect on a graph in terms of forcing concrete substructures, or is its effect no greater than that of the 'unstructural' property of having lots of edges somewhere, which it implies trivially?
Neither for general nor for topological minors is the answer to this question known. For general minors, however, the following conjecture of Hadwiger suggests a positive answer:
## Conjecture. (Hadwiger 1943)
The following implication holds for every integer \( r > 0 \) and every graph \( G \) :
\[
\chi \left( G\right) \geq r \Rightarrow G \succcurlyeq {K}^{r}
\]
Hadwiger’s conjecture is trivial for \( r \leq 2 \), easy for \( r = 3 \) and \( r = 4 \) (exercises), and equivalent to the four colour theorem for \( r = 5 \) and \( r = 6 \) . For \( r \geq 7 \) the conjecture is open, but it is true for line graphs (Exercise 35) and for graphs of large girth (Exercise 33; see also Corollary 7.3.9). Rephrased as \( G \succcurlyeq {K}^{\chi \left( G\right) } \), it is true for almost all graphs. \( {}^{3} \) In general, the conjecture for \( r + 1 \) implies it for \( r \) (exercise).
---
3 See Chapter 11 for the notion of 'almost all'.
---
The Hadwiger conjecture for any fixed \( r \) is equivalent to the assertion that every graph without a \( {K}^{r} \) minor has an \( \left( {r - 1}\right) \) -colouring. In this reformulation, the conjecture raises the question of what the graphs without a \( {K}^{r} \) minor look like: any sufficiently detailed structural description of those graphs should enable us to decide whether or not they can be \( \left( {r - 1}\right) \) -coloured.
For \( r = 3 \), for example, the graph |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 60 | d \( r = 4 \) (exercises), and equivalent to the four colour theorem for \( r = 5 \) and \( r = 6 \) . For \( r \geq 7 \) the conjecture is open, but it is true for line graphs (Exercise 35) and for graphs of large girth (Exercise 33; see also Corollary 7.3.9). Rephrased as \( G \succcurlyeq {K}^{\chi \left( G\right) } \), it is true for almost all graphs. \( {}^{3} \) In general, the conjecture for \( r + 1 \) implies it for \( r \) (exercise).
---
3 See Chapter 11 for the notion of 'almost all'.
---
The Hadwiger conjecture for any fixed \( r \) is equivalent to the assertion that every graph without a \( {K}^{r} \) minor has an \( \left( {r - 1}\right) \) -colouring. In this reformulation, the conjecture raises the question of what the graphs without a \( {K}^{r} \) minor look like: any sufficiently detailed structural description of those graphs should enable us to decide whether or not they can be \( \left( {r - 1}\right) \) -coloured.
For \( r = 3 \), for example, the graphs without a \( {K}^{r} \) minor are precisely the forests (why?), and these are indeed 2-colourable. For \( r = 4 \), there is also a simple structural characterization of the graphs without a \( {K}^{r} \) minor:
Proposition 7.3.1. A graph with at least three vertices is edge-maximal
\( \left\lbrack {12.4.2}\right\rbrack \)
without a \( {K}^{4} \) minor if and only if it can be constructed recursively from triangles by pasting \( {}^{4} \) along \( {K}^{2} \) s.
Proof. Recall first that every \( M{K}^{4} \) contains a \( T{K}^{4} \), because \( \Delta \left( {K}^{4}\right) = 3 \)
\( \left( {1.7.2}\right) \)
(Proposition 1.7.2); the graphs without a \( {K}^{4} \) minor thus coincide with those without a topological \( {K}^{4} \) minor. The proof that any graph constructible as described is edge-maximal without a \( {K}^{4} \) minor is left as an easy exercise; in order to deduce Hadwiger’s conjecture for \( r = 4 \), we only need the converse implication anyhow. We prove this by induction on \( \left| G\right| \) .
Let \( G \) be given, edge-maximal without a \( {K}^{4} \) minor. If \( \left| G\right| = 3 \) then \( G \) is itself a triangle, so let \( \left| G\right| \geq 4 \) for the induction step. Then \( G \) is not complete; let \( S \subseteq V\left( G\right) \) be a separator of size \( \kappa \left( G\right) \), and let \( {C}_{1},{C}_{2} \) be distinct components of \( G - S \) . Since \( S \) is a minimal separator, every vertex in \( S \) has a neighbour in \( {C}_{1} \) and another in \( {C}_{2} \) . If \( \left| S\right| \geq 3 \), this implies that \( G \) contains three independent paths \( {P}_{1},{P}_{2},{P}_{3} \) between a vertex \( {v}_{1} \in {C}_{1} \) and a vertex \( {v}_{2} \in {C}_{2} \) . Since \( \kappa \left( G\right) = \left| S\right| \geq 3 \), the graph \( G - \left\{ {{v}_{1},{v}_{2}}\right\} \) is connected and contains a (shortest) path \( P \) between two different \( {P}_{i} \) . Then \( P \cup {P}_{1} \cup {P}_{2} \cup {P}_{3} = T{K}^{4} \), a contradiction.
Hence \( \kappa \left( G\right) \leq 2 \), and the assertion follows from Lemma 4.4. \( {4}^{5} \) and the induction hypothesis.
One of the interesting consequences of Proposition 7.3.1 is that all the edge-maximal graphs without a \( {K}^{4} \) minor have the same number of edges, and are thus all 'extremal':
Corollary 7.3.2. Every edge-maximal graph \( G \) without a \( {K}^{4} \) minor has \( 2\left| G\right| - 3 \) edges.
Proof. Induction on \( \left| G\right| \) .
4 This was defined formally in Chapter 5.5.
5 The proof of this lemma is elementary and can be read independently of the rest of Chapter 4.
Corollary 7.3.3. Hadwiger’s conjecture holds for \( r = 4 \) .
Proof. If \( G \) arises from \( {G}_{1} \) and \( {G}_{2} \) by pasting along a complete graph, then \( \chi \left( G\right) = \max \left\{ {\chi \left( {G}_{1}\right) ,\chi \left( {G}_{2}\right) }\right\} \) (see the proof of Proposition 5.5.2). Hence, Proposition 7.3.1 implies by induction on \( \left| G\right| \) that all edge-maximal (and hence all) graphs without a \( {K}^{4} \) minor can be 3-coloured.
It is also possible to prove Corollary 7.3.3 by a simple direct argument (Exercise 34).
By the four colour theorem, Hadwiger’s conjecture for \( r = 5 \) follows from the following structure theorem for the graphs without a \( {K}^{5} \) minor, just as it follows from Proposition 7.3.1 for \( r = 4 \) . The proof of Theorem 7.3.4 is similar to that of Proposition 7.3.1, but considerably longer. We therefore state the theorem without proof:
Theorem 7.3.4. (Wagner 1937)
Let \( G \) be an edge-maximal graph without a \( {K}^{5} \) minor. If \( \left| G\right| \geq 4 \) then \( G \) can be constructed recursively, by pasting along triangles and \( {K}^{2}s \) , from plane triangulations and copies of the graph \( W \) (Fig. 7.3.1).
Fig. 7.3.1. Three representations of the Wagner graph \( W \)
\( \left( {4.2.10}\right) \)
Using Corollary 4.2.10, one can easily compute which of the graphs constructed as in Theorem 7.3.4 have the most edges. It turns out that these extremal graphs without a \( {K}^{5} \) minor have no more edges than those that are extremal with respect to \( \left\{ {M{K}^{5}, M{K}_{3,3}}\right\} \), i.e. the maximal planar graphs:
Corollary 7.3.5. A graph with \( n \) vertices and no \( {K}^{5} \) minor has at most \( {3n} - 6 \) edges.
Since \( \chi \left( W\right) = 3 \), Theorem 7.3.4 and the four colour theorem imply Hadwiger’s conjecture for \( r = 5 \) :
Corollary 7.3.6. Hadwiger’s conjecture holds for \( r = 5 \) .
The Hadwiger conjecture for \( r = 6 \) is again substantially more difficult than the case \( r = 5 \), and again it relies on the four colour theorem. The proof shows (without using the four colour theorem) that any minimal-order counterexample arises from a planar graph by adding one vertex—so by the four colour theorem it is not a counterexample after all.
Theorem 7.3.7. (Robertson, Seymour & Thomas 1993) Hadwiger’s conjecture holds for \( r = 6 \) .
As mentioned earlier, the challenge posed by Hadwiger's conjecture is to devise a proof technique that makes better use of the assumption of \( \chi \geq r \) than just using its consequence of \( \delta \geq r - 1 \) in a suitable subgraph, which we know cannot force a \( {K}^{r} \) minor (Theorem 7.2.2). So far, no such technique is known.
If we resign ourselves to using just \( \delta \geq r - 1 \), we can still ask what additional assumptions might help in making this force a \( {K}^{r} \) minor. Theorem 7.2.5 says that an assumption of large girth has this effect; see also Exercise 33. In fact, a much weaker assumption suffices: for any fixed \( s \in \mathbb{N} \) and all large enough \( d \) depending only on \( s \), the graphs \( G \nsupseteq {K}_{s, s} \) of average degree at least \( d \) can be shown to have \( {K}^{r} \) minors for \( r \) considerably larger than \( d \) . For Hadwiger’s conjecture, this implies the following:
Theorem 7.3.8. (Kühn & Osthus 2005)
For every integer \( s \) there is an integer \( {r}_{s} \) such that Hadwiger’s conjecture holds for all graphs \( G \nsupseteq {K}_{s, s} \) and \( r \geq {r}_{s} \) .
The strengthening of Hadwiger's conjecture that graphs of chromatic number at least \( r \) contain \( {K}^{r} \) as a topological minor has become known as Hajós's conjecture. It is false in general, but Theorem 7.2.5 implies it for graphs of large girth:
Corollary 7.3.9. There is a constant \( g \) such that all graphs \( G \) of girth at least \( g \) satisfy the implication \( \chi \left( G\right) \geq r \Rightarrow G \supseteq T{K}^{r} \) for all \( r \) .
Proof. If \( \chi \left( G\right) \geq r \) then, by Corollary 5.2.3, \( G \) has a subgraph \( H \) of minimum degree \( \delta \left( H\right) \geq r - 1 \) . As \( g\left( H\right) \geq g\left( G\right) \geq g \), Theorem 7.2.5 implies that \( G \supseteq H \supseteq T{K}^{r} \) .
## 7.4 Szemerédi's regularity lemma
Almost 30 years ago, in the course of the proof of a major result on the Ramsey properties of arithmetic progressions, Szemerédi developed a graph theoretical tool whose fundamental importance has been realized more and more in recent years: his so-called regularity or uniformity lemma. Very roughly, the lemma says that all graphs can be approximated by random graphs in the following sense: every graph can be partitioned, into a bounded number of equal parts, so that most of its edges run between different parts and the edges between any two parts are distributed fairly uniformly just as we would expect it if they had been generated at random.
In order to state the regularity lemma precisely, we need some definitions. Let \( G = \left( {V, E}\right) \) be a graph, and let \( X, Y \subseteq V \) be disjoint. Then
\( \parallel X, Y\parallel \)
we denote by \( \parallel X, Y\parallel \) the number of \( X - Y \) edges of \( G \), and call
\( d\left( {X, Y}\right) \)
\[
d\left( {X, Y}\right) \mathrel{\text{:=}} \frac{\parallel X, Y\parallel }{\left| X\right| \left| Y\right| }
\]
density the density of the pair \( \left( {X, Y}\right) \) . (This is a real number between 0 and 1.)
---
\( \epsilon \) -regular pair
---
Given some \( \epsilon > 0 \), we call a pair \( \left( {A, B}\right) \) of disjoint sets \( A, B \subseteq V \) ϵ-regular if all \( X \subseteq A \) and \( Y \subseteq B \) with
\[
\left| X\right| \geq \epsilon \left| A\right| \text{ and }\left| Y\right| \geq \epsilon \left| B\right|
\]
satisfy
\[
\left| {d\left( {X, Y}\right) - d\left( {A, B}\right) }\right| \leq \epsilon .
\]
The edges in an \( \epsilon \) -regular pair are thus distributed fairly uniformly, the more so the smaller the \( \epsilon \) we started with.
Consider a partition \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \) of \( V \) in which one |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 61 | \parallel X, Y\parallel \) the number of \( X - Y \) edges of \( G \), and call
\( d\left( {X, Y}\right) \)
\[
d\left( {X, Y}\right) \mathrel{\text{:=}} \frac{\parallel X, Y\parallel }{\left| X\right| \left| Y\right| }
\]
density the density of the pair \( \left( {X, Y}\right) \) . (This is a real number between 0 and 1.)
---
\( \epsilon \) -regular pair
---
Given some \( \epsilon > 0 \), we call a pair \( \left( {A, B}\right) \) of disjoint sets \( A, B \subseteq V \) ϵ-regular if all \( X \subseteq A \) and \( Y \subseteq B \) with
\[
\left| X\right| \geq \epsilon \left| A\right| \text{ and }\left| Y\right| \geq \epsilon \left| B\right|
\]
satisfy
\[
\left| {d\left( {X, Y}\right) - d\left( {A, B}\right) }\right| \leq \epsilon .
\]
The edges in an \( \epsilon \) -regular pair are thus distributed fairly uniformly, the more so the smaller the \( \epsilon \) we started with.
Consider a partition \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \) of \( V \) in which one set \( {V}_{0} \) has
---
exceptional set
---
been singled out as an exceptional set. (This exceptional set \( {V}_{0} \) may be empty. \( {}^{6} \) ) We call such a partition an \( \epsilon \) -regular partition of \( G \) if it satisfies the following three conditions:
---
\( \epsilon \) -regular partition
---
(i) \( \left| {V}_{0}\right| \leq \epsilon \left| V\right| \)
(ii) \( \left| {V}_{1}\right| = \ldots = \left| {V}_{k}\right| \)
(iii) all but at most \( \epsilon {k}^{2} \) of the pairs \( \left( {{V}_{i},{V}_{j}}\right) \) with \( 1 \leq i < j \leq k \) are \( \epsilon \) -regular.
The role of the exceptional set \( {V}_{0} \) is one of pure convenience: it makes it possible to require that all the other partition sets have exactly the same size. Since condition (iii) affects only the sets \( {V}_{1},\ldots ,{V}_{k} \), we may think of \( {V}_{0} \) as a kind of bin: its vertices are disregarded when the uniformity of the partition is assessed, but there are only few such vertices.
## 9.2.2] Lemma 7.4.1. (Regularity Lemma)
For every \( \epsilon > 0 \) and every integer \( m \geq 1 \) there exists an integer \( M \) such that every graph of order at least \( m \) admits an \( \epsilon \) -regular partition \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \) with \( m \leq k \leq M \) .
---
6 So \( {V}_{0} \) may be an exception also to our terminological rule that partition sets are not normally empty.
---
The regularity lemma thus says that, given any \( \epsilon > 0 \), every graph has an \( \epsilon \) -regular partition into a bounded number of sets. The upper bound \( M \) on the number of partition sets ensures that for large graphs the partition sets are large too; note that \( \epsilon \) -regularity is trivial when the partition sets are singletons, and a powerful property when they are large. The lemma also allows us to specify a lower bound \( m \) for the number of partition sets. This can be used to increase the proportion of edges running between different partition sets (i.e., of edges governed by the regularity assertion) over edges inside partition sets (about which we know nothing). See Exercise 39 for more details.
Note that the regularity lemma is designed for use with dense graphs: \( {}^{7} \) for sparse graphs it becomes trivial, because all densities of pairs - and hence their differences - tend to zero (Exercise 40).
The remainder of this section is devoted to the proof of the regularity lemma. Although the proof is not difficult, a reader meeting the regularity lemma here for the first time is likely to draw more insight from seeing how the lemma is typically applied than from studying the technicalities of its proof. Any such reader is encouraged to skip to the start of Section 7.5 now and come back to the proof at his or her leisure.
We shall need the following inequality for reals \( {\mu }_{1},\ldots ,{\mu }_{k} > 0 \) and \( {e}_{1},\ldots ,{e}_{k} \geq 0 \) :
\[
\sum \frac{{e}_{i}^{2}}{{\mu }_{i}} \geq \frac{{\left( \sum {e}_{i}\right) }^{2}}{\sum {\mu }_{i}}
\]
(1)
This follows from the Cauchy-Schwarz inequality \( \sum {a}_{i}^{2}\sum {b}_{i}^{2} \geq {\left( \sum {a}_{i}{b}_{i}\right) }^{2} \) by taking \( {a}_{i} \mathrel{\text{:=}} \sqrt{{\mu }_{i}} \) and \( {b}_{i} \mathrel{\text{:=}} {e}_{i}/\sqrt{{\mu }_{i}} \) .
Let \( G = \left( {V, E}\right) \) be a graph and \( n \mathrel{\text{:=}} \left| V\right| \) . For disjoint sets \( A, B \subseteq V\;G = \left( {V, E}\right) \) we define
\[
q\left( {A, B}\right) \mathrel{\text{:=}} \frac{\left| A\right| \left| B\right| }{{n}^{2}}{d}^{2}\left( {A, B}\right) = \frac{\parallel A, B{\parallel }^{2}}{\left| A\right| \left| B\right| {n}^{2}}.
\]
\( q\left( {A, B}\right) \)
For partitions \( \mathcal{A} \) of \( A \) and \( \mathcal{B} \) of \( B \) we set
\[
q\left( {\mathcal{A},\mathcal{B}}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{{A}^{\prime } \in \mathcal{A};{B}^{\prime } \in \mathcal{B}}}q\left( {{A}^{\prime },{B}^{\prime }}\right) ,
\]
\( q\left( {\mathcal{A},\mathcal{B}}\right) \)
and for a partition \( \mathcal{P} = \left\{ {{C}_{1},\ldots ,{C}_{k}}\right\} \) of \( V \) we let
\[
q\left( \mathcal{P}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{i < j}}q\left( {{C}_{i},{C}_{j}}\right)
\]
\( q\left( \mathcal{P}\right) \)
7 Sparse versions do exist, though; see the notes.
However, if \( \mathcal{P} = \left\{ {{C}_{0},{C}_{1},\ldots ,{C}_{k}}\right\} \) is a partition of \( V \) with exceptional set \( {C}_{0} \), we treat \( {C}_{0} \) as a set of singletons and define
\[
q\left( \mathcal{P}\right) \mathrel{\text{:=}} q\left( \widetilde{\mathcal{P}}\right)
\]
where \( \widetilde{\mathcal{P}} \mathrel{\text{:=}} \left\{ {{C}_{1},\ldots ,{C}_{k}}\right\} \cup \left\{ {\{ v\} : v \in {C}_{0}}\right\} \) .
The function \( q\left( \mathcal{P}\right) \) plays a pivotal role in the proof of the regularity lemma. On the one hand, it measures the uniformity of the partition \( \mathcal{P} \) : if \( \mathcal{P} \) has too many irregular pairs \( \left( {A, B}\right) \), we may take the pairs \( \left( {X, Y}\right) \) of subsets violating the regularity of the pairs \( \left( {A, B}\right) \) and make those sets \( X \) and \( Y \) into partition sets of their own; as we shall prove, this refines \( \mathcal{P} \) into a partition for which \( q \) is substantially greater than for \( \mathcal{P} \) . Here, ’substantial’ means that the increase of \( q\left( \mathcal{P}\right) \) is bounded below by some constant depending only on \( \epsilon \) . On the other hand,
\[
q\left( \mathcal{P}\right) = \mathop{\sum }\limits_{{i < j}}q\left( {{C}_{i},{C}_{j}}\right)
\]
\[
= \mathop{\sum }\limits_{{i < j}}\frac{\left| {C}_{i}\right| \left| {C}_{j}\right| }{{n}^{2}}{d}^{2}\left( {{C}_{i},{C}_{j}}\right)
\]
\[
\leq \frac{1}{{n}^{2}}\mathop{\sum }\limits_{{i < j}}\left| {C}_{i}\right| \left| {C}_{j}\right|
\]
\[
\leq 1\text{.}
\]
The number of times that \( q\left( \mathcal{P}\right) \) can be increased by a constant is thus also bounded by a constant - in other words, after some bounded number of refinements our partition will be \( \epsilon \) -regular! To complete the proof of the regularity lemma, all we have to do then is to note how many sets that last partition can possibly have if we start with a partition into \( m \) sets, and to choose this number as our desired bound \( M \) .
Let us make all this precise. We begin by showing that, when we refine a partition, the value of \( q \) will not decrease:
## Lemma 7.4.2.
(i) Let \( C, D \subseteq V \) be disjoint. If \( \mathcal{C} \) is a partition of \( C \) and \( \mathcal{D} \) is a partition of \( D \), then \( q\left( {\mathcal{C},\mathcal{D}}\right) \geq q\left( {C, D}\right) \) .
(ii) If \( \mathcal{P},{\mathcal{P}}^{\prime } \) are partitions of \( V \) and \( {\mathcal{P}}^{\prime } \) refines \( \mathcal{P} \), then \( q\left( {\mathcal{P}}^{\prime }\right) \geq q\left( \mathcal{P}\right) \) .
Proof. (i) Let \( \mathcal{C} = : \left\{ {{C}_{1},\ldots ,{C}_{k}}\right\} \) and \( \mathcal{D} = : \left\{ {{D}_{1},\ldots ,{D}_{\ell }}\right\} \) . Then
\[
q\left( {\mathcal{C},\mathcal{D}}\right) = \mathop{\sum }\limits_{{i, j}}q\left( {{C}_{i},{D}_{j}}\right)
\]
\[
= \frac{1}{{n}^{2}}\mathop{\sum }\limits_{{i, j}}\frac{{\begin{Vmatrix}{C}_{i},{D}_{j}\end{Vmatrix}}^{2}}{\left| {C}_{i}\right| \left| {D}_{j}\right| }
\]
\[
\underset{\left( 1\right) }{ \geq }\frac{1}{{n}^{2}}\frac{{\left( \mathop{\sum }\limits_{{i, j}}\begin{Vmatrix}{C}_{i},{D}_{j}\end{Vmatrix}\right) }^{2}}{\mathop{\sum }\limits_{{i, j}}\left| {C}_{i}\right| \left| {D}_{j}\right| }
\]
\[
= \frac{1}{{n}^{2}}\frac{\parallel C, D{\parallel }^{2}}{\left( {\mathop{\sum }\limits_{i}\left| {C}_{i}\right| }\right) \left( {\mathop{\sum }\limits_{j}\left| {D}_{j}\right| }\right) }
\]
\[
= q\left( {C, D}\right) \text{.}
\]
(ii) Let \( \mathcal{P} = : \left\{ {{C}_{1},\ldots ,{C}_{k}}\right\} \), and for \( i = 1,\ldots, k \) let \( {\mathcal{C}}_{i} \) be the partition of \( {C}_{i} \) induced by \( {\mathcal{P}}^{\prime } \) . Then
\[
q\left( \mathcal{P}\right) = \mathop{\sum }\limits_{{i < j}}q\left( {{C}_{i},{C}_{j}}\right)
\]
\[
\underset{\left( \mathrm{i}\right) }{ \leq }\mathop{\sum }\limits_{{i < j}}q\left( {{\mathcal{C}}_{i},{\mathcal{C}}_{j}}\right)
\]
\[
\leq q\left( {\mathcal{P}}^{\prime }\right)
\]
since \( q\left( {\mathcal{P}}^{\prime }\right) = \mathop{\sum }\limits_{i}q\left( {\mathcal{C}}_{i}\right) + \mathop{\sum }\limits_{{i < j}}q\left( {{\mathcal{C}}_{i},{\mathcal{C}}_{j}}\right) \) .
Next, we show that refining a partition by subpartitioning an irregular pair of partition sets increases the value of \( q \) a little; since we are dealing here with a single pair only, the amount of this increase will still be less than any constant.
Lemma 7.4.3. Let \( \epsilon > 0 \), and let \( C, D \subseteq V \) be disjoint. If \( \left( {C, D}\right) |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 62 | ight\} \), and for \( i = 1,\ldots, k \) let \( {\mathcal{C}}_{i} \) be the partition of \( {C}_{i} \) induced by \( {\mathcal{P}}^{\prime } \) . Then
\[
q\left( \mathcal{P}\right) = \mathop{\sum }\limits_{{i < j}}q\left( {{C}_{i},{C}_{j}}\right)
\]
\[
\underset{\left( \mathrm{i}\right) }{ \leq }\mathop{\sum }\limits_{{i < j}}q\left( {{\mathcal{C}}_{i},{\mathcal{C}}_{j}}\right)
\]
\[
\leq q\left( {\mathcal{P}}^{\prime }\right)
\]
since \( q\left( {\mathcal{P}}^{\prime }\right) = \mathop{\sum }\limits_{i}q\left( {\mathcal{C}}_{i}\right) + \mathop{\sum }\limits_{{i < j}}q\left( {{\mathcal{C}}_{i},{\mathcal{C}}_{j}}\right) \) .
Next, we show that refining a partition by subpartitioning an irregular pair of partition sets increases the value of \( q \) a little; since we are dealing here with a single pair only, the amount of this increase will still be less than any constant.
Lemma 7.4.3. Let \( \epsilon > 0 \), and let \( C, D \subseteq V \) be disjoint. If \( \left( {C, D}\right) \) is not \( \epsilon \) - regular, then there are partitions \( \mathcal{C} = \left\{ {{C}_{1},{C}_{2}}\right\} \) of \( C \) and \( \mathcal{D} = \left\{ {{D}_{1},{D}_{2}}\right\} \) of \( D \) such that
\[
q\left( {\mathcal{C},\mathcal{D}}\right) \geq q\left( {C, D}\right) + {\epsilon }^{4}\frac{\left| C\right| \left| D\right| }{{n}^{2}}.
\]
Proof. Suppose \( \left( {C, D}\right) \) is not \( \epsilon \) -regular. Then there are sets \( {C}_{1} \subseteq C \) and \( {D}_{1} \subseteq D \) with \( \left| {C}_{1}\right| > \epsilon \left| C\right| \) and \( \left| {D}_{1}\right| > \epsilon \left| D\right| \) such that
\[
\left| \eta \right| > \epsilon
\]
(2)
for \( \eta \mathrel{\text{:=}} d\left( {{C}_{1},{D}_{1}}\right) - d\left( {C, D}\right) \) . Let \( \mathcal{C} \mathrel{\text{:=}} \left\{ {{C}_{1},{C}_{2}}\right\} \) and \( \mathcal{D} \mathrel{\text{:=}} \left\{ {{D}_{1},{D}_{2}}\right\} \) , where \( {C}_{2} \mathrel{\text{:=}} C \smallsetminus {C}_{1} \) and \( {D}_{2} \mathrel{\text{:=}} D \smallsetminus {D}_{1} \) .
Let us show that \( \mathcal{C} \) and \( \mathcal{D} \) satisfy the conclusion of the lemma. We
shall write \( {c}_{i} \mathrel{\text{:=}} \left| {C}_{i}\right| ,{d}_{i} \mathrel{\text{:=}} \left| {D}_{i}\right| ,{e}_{ij} \mathrel{\text{:=}} \begin{Vmatrix}{{C}_{i},{D}_{j}}\end{Vmatrix}, c \mathrel{\text{:=}} \left| C\right|, d \mathrel{\text{:=}} \left| D\right| \) \( {c}_{i},{d}_{i},{e}_{ij} \) and \( e \mathrel{\text{:=}} \parallel C, D\parallel \) . As in the proof of Lemma 7.4.2, \( c, d, e \)
\[
q\left( {\mathcal{C},\mathcal{D}}\right) = \frac{1}{{n}^{2}}\mathop{\sum }\limits_{{i, j}}\frac{{e}_{ij}^{2}}{{c}_{i}{d}_{j}}
\]
\[
= \frac{1}{{n}^{2}}\left( {\frac{{e}_{11}^{2}}{{c}_{1}{d}_{1}} + \mathop{\sum }\limits_{{i + j > 2}}\frac{{e}_{ij}^{2}}{{c}_{i}{d}_{j}}}\right)
\]
\[
\underset{\left( 1\right) }{ \geq }\frac{1}{{n}^{2}}\left( {\frac{{e}_{11}^{2}}{{c}_{1}{d}_{1}} + \frac{{\left( e - {e}_{11}\right) }^{2}}{{cd} - {c}_{1}{d}_{1}}}\right) .
\]
By definition of \( \eta \), we have \( {e}_{11} = {c}_{1}{d}_{1}e/{cd} + \eta {c}_{1}{d}_{1} \), so
\[
{n}^{2}q\left( {\mathcal{C},\mathcal{D}}\right) \geq \frac{1}{{c}_{1}{d}_{1}}{\left( \frac{{c}_{1}{d}_{1}e}{cd} + \eta {c}_{1}{d}_{1}\right) }^{2}
\]
\[
+ \frac{1}{{cd} - {c}_{1}{d}_{1}}{\left( \frac{{cd} - {c}_{1}{d}_{1}}{cd}e - \eta {c}_{1}{d}_{1}\right) }^{2}
\]
\[
= \frac{{c}_{1}{d}_{1}{e}^{2}}{{c}^{2}{d}^{2}} + \frac{{2e\eta }{c}_{1}{d}_{1}}{cd} + {\eta }^{2}{c}_{1}{d}_{1}
\]
\[
+ \frac{{cd} - {c}_{1}{d}_{1}}{{c}^{2}{d}^{2}}{e}^{2} - \frac{{2e\eta }{c}_{1}{d}_{1}}{cd} + \frac{{\eta }^{2}{c}_{1}^{2}{d}_{1}^{2}}{{cd} - {c}_{1}{d}_{1}}
\]
\[
\geq \frac{{e}^{2}}{cd} + {\eta }^{2}{c}_{1}{d}_{1}
\]
\[
\mathop{\sum }\limits_{\left( 2\right) }\frac{{e}^{2}}{cd} + {\epsilon }^{4}{cd}
\]
since \( {c}_{1} \geq {\epsilon c} \) and \( {d}_{1} \geq {\epsilon d} \) by the choice of \( {C}_{1} \) and \( {D}_{1} \) .
Finally, we show that if a partition has enough irregular pairs of partition sets to fall short of the definition of an \( \epsilon \) -regular partition, then subpartitioning all those pairs at once results in an increase of \( q \) by a constant:
Lemma 7.4.4. Let \( 0 < \epsilon \leq 1/4 \), and let \( \mathcal{P} = \left\{ {{C}_{0},{C}_{1},\ldots ,{C}_{k}}\right\} \) be a partition of \( V \), with exceptional set \( {C}_{0} \) of size \( \left| {C}_{0}\right| \leq {\epsilon n} \) and \( \left| {C}_{1}\right| = \ldots = \left| {C}_{k}\right| = : c \) . If \( \mathcal{P} \) is not \( \epsilon \) -regular, then there is a partition \( {\mathcal{P}}^{\prime } = \left\{ {{C}_{0}^{\prime },{C}_{1}^{\prime },\ldots ,{C}_{\ell }^{\prime }}\right\} \) of \( V \) with exceptional set \( {C}_{0}^{\prime } \), where \( k \leq \ell \leq k{4}^{k} \) , such that \( \left| {C}_{0}^{\prime }\right| \leq \left| {C}_{0}\right| + n/{2}^{k} \), all other sets \( {C}_{i}^{\prime } \) have equal size, and
\[
q\left( {\mathcal{P}}^{\prime }\right) \geq q\left( \mathcal{P}\right) + {\epsilon }^{5}/2
\]
\( {\mathcal{C}}_{ij} \) Proof. For all \( 1 \leq i < j \leq k \), let us define a partition \( {\mathcal{C}}_{ij} \) of \( {C}_{i} \) and a partition \( {\mathcal{C}}_{ji} \) of \( {C}_{j} \), as follows. If the pair \( \left( {{C}_{i},{C}_{j}}\right) \) is \( \epsilon \) -regular, we let \( {\mathcal{C}}_{ij} \mathrel{\text{:=}} \left\{ {C}_{i}\right\} \) and \( {\mathcal{C}}_{ji} \mathrel{\text{:=}} \left\{ {C}_{j}\right\} \) . If not, then by Lemma 7.4.3 there are partitions \( {\mathcal{C}}_{ij} \) of \( {C}_{i} \) and \( {\mathcal{C}}_{ji} \) of \( {C}_{j} \) with \( \left| {\mathcal{C}}_{ij}\right| = \left| {\mathcal{C}}_{ji}\right| = 2 \) and
\[
q\left( {{\mathcal{C}}_{ij},{\mathcal{C}}_{ji}}\right) \geq q\left( {{C}_{i},{C}_{j}}\right) + {\epsilon }^{4}\frac{\left| {C}_{i}\right| \left| {C}_{j}\right| }{{n}^{2}} = q\left( {{C}_{i},{C}_{j}}\right) + \frac{{\epsilon }^{4}{c}^{2}}{{n}^{2}}.
\]
(3)
For each \( i = 1,\ldots, k \), let \( {\mathcal{C}}_{i} \) be the unique minimal partition of \( {C}_{i} \) that refines every partition \( {\mathcal{C}}_{ij} \) with \( j \neq i \) . (In other words, if we consider two
elements of \( {C}_{i} \) as equivalent whenever they lie in the same partition set of \( {\mathcal{C}}_{ij} \) for every \( j \neq i \), then \( {\mathcal{C}}_{i} \) is the set of equivalence classes.) Thus, \( \left| {\mathcal{C}}_{i}\right| \leq {2}^{k - 1} \) . Now consider the partition
\[
\mathcal{C} \mathrel{\text{:=}} \left\{ {C}_{0}\right\} \cup \mathop{\bigcup }\limits_{{i = 1}}^{k}{\mathcal{C}}_{i}
\]
of \( V \), with \( {C}_{0} \) as exceptional set. Then \( \mathcal{C} \) refines \( \mathcal{P} \), and
\[
k \leq \left| \mathcal{C}\right| \leq k{2}^{k}
\]
(4)
Let \( {\mathcal{C}}_{0} \mathrel{\text{:=}} \left\{ {\{ v\} : v \in {C}_{0}}\right\} \) . Now if \( \mathcal{P} \) is not \( \epsilon \) -regular, then for more than \( \epsilon {k}^{2} \) of the pairs \( \left( {{C}_{i},{C}_{j}}\right) \) with \( 1 \leq i < j \leq k \) the partition \( {\mathcal{C}}_{ij} \) is non-trivial. Hence, by our definition of \( q \) for partitions with exceptional set, and Lemma 7.4.2 (i),
\[
q\left( \mathcal{C}\right) = \mathop{\sum }\limits_{{1 \leq i < j}}q\left( {{\mathcal{C}}_{i},{\mathcal{C}}_{j}}\right) + \mathop{\sum }\limits_{{1 \leq i}}q\left( {{\mathcal{C}}_{0},{\mathcal{C}}_{i}}\right) + \mathop{\sum }\limits_{{0 \leq i}}q\left( {\mathcal{C}}_{i}\right)
\]
\[
\geq \mathop{\sum }\limits_{{1 \leq i < j}}q\left( {{\mathcal{C}}_{ij},{\mathcal{C}}_{ji}}\right) + \mathop{\sum }\limits_{{1 \leq i}}q\left( {{\mathcal{C}}_{0},\left\{ {C}_{i}\right\} }\right) + q\left( {\mathcal{C}}_{0}\right)
\]
\[
\underset{\left( 3\right) }{ \geq }\mathop{\sum }\limits_{{1 \leq i < j}}q\left( {{C}_{i},{C}_{j}}\right) + \epsilon {k}^{2}\frac{{\epsilon }^{4}{c}^{2}}{{n}^{2}} + \mathop{\sum }\limits_{{1 \leq i}}q\left( {{\mathcal{C}}_{0},\left\{ {C}_{i}\right\} }\right) + q\left( {\mathcal{C}}_{0}\right)
\]
\[
= q\left( \mathcal{P}\right) + {\epsilon }^{5}{\left( \frac{kc}{n}\right) }^{2}
\]
\[
\geq q\left( \mathcal{P}\right) + {\epsilon }^{5}/2
\]
(For the last inequality, recall that \( \left| {C}_{0}\right| \leq {\epsilon n} \leq \frac{1}{4}n \), so \( {kc} \geq \frac{3}{4}n \) .)
In order to turn \( \mathcal{C} \) into our desired partition \( {\mathcal{P}}^{\prime } \), all that remains to do is to cut its sets up into pieces of some common size, small enough that all remaining vertices can be collected into the exceptional set without making this too large. Let \( {C}_{1}^{\prime },\ldots ,{C}_{\ell }^{\prime } \) be a maximal collection of disjoint sets of size \( d \mathrel{\text{:=}} \left\lfloor {c/{4}^{k}}\right\rfloor \) such that each \( {C}_{i}^{\prime } \) is contained in some \( C \in \mathcal{C} \smallsetminus \left\{ {C}_{0}\right\} \), and put \( {C}_{0}^{\prime } \mathrel{\text{:=}} V \smallsetminus \bigcup {C}_{i}^{\prime } \) . Then \( {\mathcal{P}}^{\prime } = \left\{ {{C}_{0}^{\prime },{C}_{1}^{\prime },\ldots ,{C}_{\ell }^{\prime }}\right\} \) is indeed a partition of \( V \) . Moreover, \( {\widetilde{\mathcal{P}}}^{\prime } \) refines \( \widetilde{\mathcal{C}} \), so
\[
q\left( {\mathcal{P}}^{\prime }\right) \geq q\left( \mathcal{C}\right) \geq q\left( \mathcal{P}\right) + {\epsilon }^{5}/2
\]
by Lemma 7.4.2 (ii). Since each set \( {C}_{i}^{\prime } \neq {C}_{0}^{\prime } \) is also contained in one of the sets \( {C}_{1},\ldots ,{C}_{k} \), but no more than \( {4}^{k} \) sets \( {C}_{i}^{\prime } \) can lie inside the same \( {C}_{j} \) (by the choice of \( d \) ), we also have \( k \leq \ell \leq k{4}^{k} \) as required. Finally, the sets \( {C}_{1}^{\prime },\ldots ,{C}_{\ell }^{\prime } \) use all but at most \( d \) vertices from each set \( C \neq {C}_{0} \) of \( \mathcal{C} \) . Hence,
\[
\left| {C}_{0}^{\prime }\right| \leq \left| {C}_{0}\right| + d\left| \mathcal{C}\right|
\]
\[
{ \leq }_{\left( 4\right |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 63 | cal{P}}^{\prime } = \left\{ {{C}_{0}^{\prime },{C}_{1}^{\prime },\ldots ,{C}_{\ell }^{\prime }}\right\} \) is indeed a partition of \( V \) . Moreover, \( {\widetilde{\mathcal{P}}}^{\prime } \) refines \( \widetilde{\mathcal{C}} \), so
\[
q\left( {\mathcal{P}}^{\prime }\right) \geq q\left( \mathcal{C}\right) \geq q\left( \mathcal{P}\right) + {\epsilon }^{5}/2
\]
by Lemma 7.4.2 (ii). Since each set \( {C}_{i}^{\prime } \neq {C}_{0}^{\prime } \) is also contained in one of the sets \( {C}_{1},\ldots ,{C}_{k} \), but no more than \( {4}^{k} \) sets \( {C}_{i}^{\prime } \) can lie inside the same \( {C}_{j} \) (by the choice of \( d \) ), we also have \( k \leq \ell \leq k{4}^{k} \) as required. Finally, the sets \( {C}_{1}^{\prime },\ldots ,{C}_{\ell }^{\prime } \) use all but at most \( d \) vertices from each set \( C \neq {C}_{0} \) of \( \mathcal{C} \) . Hence,
\[
\left| {C}_{0}^{\prime }\right| \leq \left| {C}_{0}\right| + d\left| \mathcal{C}\right|
\]
\[
{ \leq }_{\left( 4\right) }\left| {C}_{0}\right| + \frac{c}{{4}^{k}}k{2}^{k}
\]
\[
= \left| {C}_{0}\right| + {ck}/{2}^{k}
\]
\[
\leq \left| {C}_{0}\right| + n/{2}^{k}\text{.}
\]
The proof of the regularity lemma now follows easily by repeated application of Lemma 7.4.4:
Proof of Lemma 7.4.1. Let \( \epsilon > 0 \) and \( m \geq 1 \) be given; without loss of generality, \( \epsilon \leq 1/4 \) . Let \( s \mathrel{\text{:=}} 2/{\epsilon }^{5} \) . This number \( s \) is an upper bound on the number of iterations of Lemma 7.4.4 that can be applied to a partition of a graph before it becomes \( \epsilon \) -regular; recall that \( q\left( \mathcal{P}\right) \leq 1 \) for all partitions \( \mathcal{P} \) .
There is one formal requirement which a partition \( \left\{ {{C}_{0},{C}_{1},\ldots ,{C}_{k}}\right\} \) with \( \left| {C}_{1}\right| = \ldots = \left| {C}_{k}\right| \) has to satisfy before Lemma 7.4.4 can be (re-) applied: the size \( \left| {C}_{0}\right| \) of its exceptional set must not exceed \( {\epsilon n} \) . With each iteration of the lemma, however, the size of the exceptional set can grow by up to \( n/{2}^{k} \) . (More precisely, by up to \( n/{2}^{\ell } \), where \( \ell \) is the number of other sets in the current partition; but \( \ell \geq k \) by the lemma, so \( n/{2}^{k} \) is certainly an upper bound for the increase.) We thus want to choose \( k \) large enough that even \( s \) increments of \( n/{2}^{k} \) add up to at most \( \frac{1}{2}{\epsilon n} \), and \( n \) large enough that, for any initial value of \( \left| {C}_{0}\right| < k \), we have \( \left| {C}_{0}\right| \leq \frac{1}{2}{\epsilon n} \) . (If we give our starting partition \( k \) non-exceptional sets \( {C}_{1},\ldots ,{C}_{k} \), we should allow an initial size of up to \( k \) for \( {C}_{0} \), to be able to achieve \( \left. {\left| {C}_{1}\right| = \ldots = \left| {C}_{k}\right| \text{.}}\right) \)
So let \( k \geq m \) be large enough that \( {2}^{k - 1} \geq s/\epsilon \) . Then \( s/{2}^{k} \leq \epsilon /2 \) , and hence
\[
k + \frac{s}{{2}^{k}}n \leq {\epsilon n}
\]
(5)
whenever \( k/n \leq \epsilon /2 \), i.e. for all \( n \geq {2k}/\epsilon \) .
Let us now choose \( M \) . This should be an upper bound on the number of (non-exceptional) sets in our partition after up to \( s \) iterations of Lemma 7.4.4, where in each iteration this number may grow from its current value \( r \) to at most \( r{4}^{r} \) . So let \( f \) be the function \( x \mapsto x{4}^{x} \), and take \( M \mathrel{\text{:=}} \max \left\{ {{f}^{s}\left( k\right) ,{2k}/\epsilon }\right\} \) ; the second term in the maximum ensures that any \( n \geq M \) is large enough to satisfy (5).
We finally have to show that every graph \( G = \left( {V, E}\right) \) of order at least \( m \) has an \( \epsilon \) -regular partition \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \) with \( m \leq k \leq M \) . So let \( G \) be given, and let \( n \mathrel{\text{:=}} \left| G\right| \) . If \( n \leq M \), we partition \( G \) into \( k \mathrel{\text{:=}} n \) singletons, choosing \( {V}_{0} \mathrel{\text{:=}} \varnothing \) and \( \left| {V}_{1}\right| = \ldots = \left| {V}_{k}\right| = 1 \) . This partition of \( G \) is clearly \( \epsilon \) -regular. Suppose now that \( n > M \) . Let \( {C}_{0} \subseteq V \) be minimal such that \( k \) divides \( \left| {V \smallsetminus {C}_{0}}\right| \), and let \( \left\{ {{C}_{1},\ldots ,{C}_{k}}\right\} \) be any partition of \( V \smallsetminus {C}_{0} \) into sets of equal size. Then \( \left| {C}_{0}\right| < k \), and hence \( \left| {C}_{0}\right| \leq {\epsilon n} \) by (5). Starting with \( \left\{ {{C}_{0},{C}_{1},\ldots ,{C}_{k}}\right\} \) we apply Lemma 7.4.4 again and again, until the partition of \( G \) obtained is \( \epsilon \) -regular; this will happen after at most \( s \) iterations, since by (5) the size of the exceptional set in the partitions stays below \( {\epsilon n} \), so the lemma could indeed be reapplied up to the theoretical maximum of \( s \) times.
## 7.5 Applying the regularity lemma
The purpose of this section is to illustrate how the regularity lemma is typically applied in the context of (dense) extremal graph theory. Suppose we are trying to prove that a certain edge density of a graph \( G \) suffices to force the occurrence of some given subgraph \( H \), and that we have an \( \epsilon \) -regular partition of \( G \) . For most of the pairs \( \left( {{V}_{i},{V}_{j}}\right) \) of partition sets, the edges between \( {V}_{i} \) and \( {V}_{j} \) are distributed fairly uniformly; their density, however, may depend on the pair. But since \( G \) has many edges, this density cannot be zero for all the pairs: some sizeable proportion of the pairs will have positive density. Now if \( G \) is large, then so are the pairs: recall that the number of partition sets is bounded, and they have equal size. But any large enough bipartite graph with equal partition sets, fixed positive edge density (however small) and a uniform distribution of edges will contain any given bipartite subgraph \( {}^{8} \) -this will be made precise below. Thus if enough pairs in our partition of \( G \) have positive density that \( H \) can be written as the union of bipartite graphs each arising in one of those pairs, we may hope that \( H \subseteq G \) as desired.
These ideas will be formalized by Lemma 7.5.2 below. We shall then use this and the regularity lemma to prove the Erdős-Stone theorem from Section 7.1; another application will be given later, in the proof of Theorem 9.2.2. We wind up the section with an informal review of the application of the regularity lemma that we have seen, summarizing what it can teach us for similar applications. In particular, we look at how the various parameters involved depend on each other, and in which order they have to be chosen to make the lemma work.
Let us begin by noting a simple consequence of the \( \epsilon \) -regularity of a pair \( \left( {A, B}\right) \) . For any subset \( Y \subseteq B \) that is not too small, most vertices of \( A \) have about the expected number of neighbours in \( Y \) :
---
8 Readers already acquainted with random graphs may find it instructive to compare this statement with Proposition 11.3.1.
---
Lemma 7.5.1. Let \( \left( {A, B}\right) \) be an \( \epsilon \) -regular pair, of density \( d \) say, and let \( Y \subseteq B \) have size \( \left| Y\right| \geq \epsilon \left| B\right| \) . Then all but fewer than \( \epsilon \left| A\right| \) of the vertices in \( A \) have (each) at least \( \left( {d - \epsilon }\right) \left| Y\right| \) neighbours in \( Y \) .
Proof. Let \( X \subseteq A \) be the set of vertices with fewer than \( \left( {d - \epsilon }\right) \left| Y\right| \) neighbours in \( Y \) . Then \( \parallel X, Y\parallel < \left| X\right| \left( {d - \epsilon }\right) \left| Y\right| \), so
\[
d\left( {X, Y}\right) = \frac{\parallel X, Y\parallel }{\left| X\right| \left| Y\right| } < d - \epsilon = d\left( {A, B}\right) - \epsilon .
\]
As \( \left( {A, B}\right) \) is \( \epsilon \) -regular and \( \left| Y\right| \geq \epsilon \left| B\right| \), this implies that \( \left| X\right| < \epsilon \left| A\right| \) .
Let \( G \) be a graph with an \( \epsilon \) -regular partition \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \), with \( R \) exceptional set \( {V}_{0} \) and \( \left| {V}_{1}\right| = \ldots = \left| {V}_{k}\right| = : \ell \) . Given \( d \in \left\lbrack {0,1}\right\rbrack \), let \( R \) be the graph on \( \left\{ {{V}_{1},\ldots ,{V}_{k}}\right\} \) in which two vertices \( {V}_{i},{V}_{j} \) are adjacent if and only if they form an \( \epsilon \) -regular pair in \( G \) of density \( \geq d \) . We shall call
regularity \( R \) a regularity graph of \( G \) with parameters \( \epsilon ,\ell \) and \( d \) . Given \( s \in \mathbb{N} \), let graph \( {V}_{i}^{s} \) us now replace every vertex \( {V}_{i} \) of \( R \) by a set \( {V}_{i}^{s} \) of \( s \) vertices, and every edge by a complete bipartite graph between the corresponding \( s \) -sets. \( {R}_{s} \) The resulting graph will be denoted by \( {R}_{s} \) . (For \( R = {K}^{r} \), for example, we have \( {R}_{s} = {K}_{s}^{r} \) .)
The following lemma says that subgraphs of \( {R}_{s} \) can also be found in \( G \), provided that \( d > 0 \), that \( \epsilon \) is small enough, and that the \( {V}_{i} \) are large enough. In fact, the values of \( \epsilon \) and \( \ell \) required depend only on ( \( d \) and) the maximum degree of the subgraph:
\( \left\lbrack {9.2.2}\right\rbrack \)
Lemma 7.5.2. For all \( d \in (0,1\rbrack \) and \( \Delta \geq 1 \) there exists an \( {\epsilon }_{0} > 0 \) with the following property: if \( G \) is any graph, \( H \) is a graph with \( \Delta \left( H\right) \leq \Delta \) , \( s \in \mathbb{N} \), and \( R \) is any regularity graph of \( G \) with parameters |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 64 | ry vertex \( {V}_{i} \) of \( R \) by a set \( {V}_{i}^{s} \) of \( s \) vertices, and every edge by a complete bipartite graph between the corresponding \( s \) -sets. \( {R}_{s} \) The resulting graph will be denoted by \( {R}_{s} \) . (For \( R = {K}^{r} \), for example, we have \( {R}_{s} = {K}_{s}^{r} \) .)
The following lemma says that subgraphs of \( {R}_{s} \) can also be found in \( G \), provided that \( d > 0 \), that \( \epsilon \) is small enough, and that the \( {V}_{i} \) are large enough. In fact, the values of \( \epsilon \) and \( \ell \) required depend only on ( \( d \) and) the maximum degree of the subgraph:
\( \left\lbrack {9.2.2}\right\rbrack \)
Lemma 7.5.2. For all \( d \in (0,1\rbrack \) and \( \Delta \geq 1 \) there exists an \( {\epsilon }_{0} > 0 \) with the following property: if \( G \) is any graph, \( H \) is a graph with \( \Delta \left( H\right) \leq \Delta \) , \( s \in \mathbb{N} \), and \( R \) is any regularity graph of \( G \) with parameters \( \epsilon \leq {\epsilon }_{0} \) , \( \ell \geq {2s}/{d}^{\Delta } \) and \( d \), then
\[
H \subseteq {R}_{s} \Rightarrow H \subseteq G.
\]
\( d,\Delta ,{\epsilon }_{0} \) Proof. Given \( d \) and \( \Delta \), choose \( {\epsilon }_{0} > 0 \) small enough that \( {\epsilon }_{0} < d \) and
\[
{\left( d - {\epsilon }_{0}\right) }^{\Delta } - \Delta {\epsilon }_{0} \geq \frac{1}{2}{d}^{\Delta }
\]
(1)
\( G, H, R,{R}_{s} \)
such a choice is possible, since \( {\left( d - \epsilon \right) }^{\Delta } - {\Delta \epsilon } \rightarrow {d}^{\Delta } \) as \( \epsilon \rightarrow 0 \) . Now let \( {V}_{i} \) \( G, H, s \) and \( R \) be given as stated. Let \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \) be the \( \epsilon \) -regular \( \epsilon, k,\ell \) partition of \( G \) that gave rise to \( R \) ; thus, \( \epsilon \leq {\epsilon }_{0}, V\left( R\right) = \left\{ {{V}_{1},\ldots ,{V}_{k}}\right\} \) and \( \left| {V}_{1}\right| = \ldots = \left| {V}_{k}\right| = \ell \geq {2s}/{d}^{\Delta } \) . Let us assume that \( H \) is actually \( {u}_{i}, h \) a subgraph of \( {R}_{s} \) (not just isomorphic to one), with vertices \( {u}_{1},\ldots ,{u}_{h} \) say. Each vertex \( {u}_{i} \) lies in one of the \( s \) -sets \( {V}_{j}^{s} \) of \( {R}_{s} \), which defines a \( \sigma \) map \( \sigma : i \mapsto j \) . Our aim is to define an embedding \( {u}_{i} \mapsto {v}_{i} \in {V}_{\sigma \left( i\right) } \) of \( H \) \( {v}_{i} \) in \( G \) as a subgraph; thus, \( {v}_{1},\ldots ,{v}_{h} \) will be distinct, and \( {v}_{i}{v}_{j} \) will be an edge of \( G \) whenever \( {u}_{i}{u}_{j} \) is an edge of \( H \) .
Our plan is to choose the vertices \( {v}_{1},\ldots ,{v}_{h} \) inductively. Throughout the induction, we shall have a ’target set’ \( {Y}_{i} \subseteq {V}_{\sigma \left( i\right) } \) assigned to each \( {u}_{i} \) ; this contains the vertices that are still candidates for the choice of \( {v}_{i} \) . Initially, \( {Y}_{i} \) is the entire set \( {V}_{\sigma \left( i\right) } \) . As the embedding proceeds, \( {Y}_{i} \) will get smaller and smaller (until it collapses to \( \left\{ {v}_{i}\right\} \) when \( {v}_{i} \) is chosen): whenever we choose a vertex \( {v}_{j} \) with \( j < i \) and \( {u}_{j}{u}_{i} \in E\left( H\right) \), we delete all those vertices from \( {Y}_{i} \) that are not adjacent to \( {v}_{j} \) . The set \( {Y}_{i} \) thus evolves as
\[
{V}_{\sigma \left( i\right) } = {Y}_{i}^{0} \supseteq \ldots \supseteq {Y}_{i}^{i} = \left\{ {v}_{i}\right\}
\]
where \( {Y}_{i}^{j} \) denotes the version of \( {Y}_{i} \) current after the definition of \( {v}_{j} \) and the resulting deletion of vertices from \( {Y}_{i}^{j - 1} \) .
In order to make this approach work, we have to ensure that the target sets \( {Y}_{i} \) do not get too small. When we come to embed a vertex \( {u}_{j} \) , we consider all the indices \( i > j \) with \( {u}_{j}{u}_{i} \in E\left( H\right) \) ; there are at most \( \Delta \) such \( i \) . For each of these \( i \), we wish to select \( {v}_{j} \) so that
\[
{Y}_{i}^{j} = N\left( {v}_{j}\right) \cap {Y}_{i}^{j - 1}
\]
(2)
is still relatively large: smaller than \( {Y}_{i}^{j - 1} \) by no more than a constant factor such as \( \left( {d - \epsilon }\right) \) . Now this can be done by Lemma 7.5.1 (with \( A = {V}_{\sigma \left( j\right) }, B = {V}_{\sigma \left( i\right) } \) and \( \left. {Y = {Y}_{i}^{j - 1}}\right) \) : provided that \( {Y}_{i}^{j - 1} \) still has size at least \( \epsilon \ell \) (which induction will ensure), all but at most \( \epsilon \ell \) choices of \( {v}_{j} \) will be such that the new set \( {Y}_{i}^{j} \) as in (2) satisfies
\[
\left| {Y}_{i}^{j}\right| \geq \left( {d - \epsilon }\right) \left| {Y}_{i}^{j - 1}\right| .
\]
(3)
Excluding the bad choices for \( {v}_{j} \) for all the relevant values of \( i \) simultaneously, we find that all but at most \( {\Delta \epsilon }\ell \) choices of \( {v}_{j} \) from \( {V}_{\sigma \left( j\right) } \), and in particular from \( {Y}_{j}^{j - 1} \subseteq {V}_{\sigma \left( j\right) } \), satisfy (3) for all \( i \) .
It remains to show that the sets \( {Y}_{i}^{j - 1} \) considered above as \( Y \) for Lemma 7.5.1 never fall below the size of \( \epsilon \ell \), and that when we come to select \( {v}_{j} \in {Y}_{j}^{j - 1} \) we have a choice of at least \( s \) suitable candidates: since before \( {u}_{j} \) at most \( s - 1 \) vertices \( u \) were given an image in \( {V}_{\sigma \left( j\right) } \), we can then choose \( {v}_{j} \) distinct from these.
But all this follows from our choice of \( {\epsilon }_{0} \) . Indeed, the initial target sets \( {Y}_{i}^{0} \) have size \( \ell \), and each \( {Y}_{i} \) shrinks at most \( \Delta \) times by a factor of \( \left( {d - \epsilon }\right) \) when some \( {v}_{j} \) with \( j < i \) and \( {u}_{j}{u}_{i} \in E\left( H\right) \) is defined. Thus,
\[
\left| {Y}_{i}^{j - 1}\right| - {\Delta \epsilon }\ell \underset{\left( 3\right) }{ \geq }{\left( d - \epsilon \right) }^{\Delta }\ell - {\Delta \epsilon }\ell \geq {\left( d - {\epsilon }_{0}\right) }^{\Delta }\ell - \Delta {\epsilon }_{0}\ell \underset{\left( 1\right) }{ \geq }\frac{1}{2}{d}^{\Delta }\ell \geq s
\]
for all \( j \leq i \) ; in particular, we have \( \left| {Y}_{i}^{j - 1}\right| \geq \epsilon \ell \) and \( \left| {Y}_{j}^{j - 1}\right| - {\Delta \epsilon }\ell \geq s \) as desired.
We are now ready to prove the Erdős-Stone theorem.
(7.1.1) Proof of Theorem 7.1.2. Let \( r \geq 2 \) and \( s \geq 1 \) be given as in the (7.1.4) statement of the theorem. For \( s = 1 \) the assertion follows from Turán’s \( r, s \) theorem, so we assume that \( s \geq 2 \) . Let \( \gamma > 0 \) be given; this \( \gamma \) will play \( \gamma \) the role of the \( \epsilon \) of the theorem. If any graph \( G \) with \( \left| G\right| = : n \) has
\( \parallel G\parallel \)
\[
\parallel G\parallel \geq {t}_{r - 1}\left( n\right) + \gamma {n}^{2}
\]
edges, then \( \gamma < 1 \) . We want to show that \( {K}_{s}^{r} \subseteq G \) if \( n \) is large enough.
Our plan is to use the regularity lemma to show that \( G \) has a regularity graph \( R \) dense enough to contain a \( {K}^{r} \) by Turán’s theorem. Then \( {R}_{s} \) contains a \( {K}_{s}^{r} \), so we may hope to use Lemma 7.5.2 to deduce that \( {K}_{s}^{r} \subseteq G. \)
---
\( d,\Delta ,{\epsilon }_{0} \)
\( m,\epsilon \)
---
On input \( d \mathrel{\text{:=}} \gamma \) and \( \Delta \mathrel{\text{:=}} \Delta \left( {K}_{s}^{r}\right) \) Lemma 7.5.2 returns an \( {\epsilon }_{0} > 0 \) . To apply the regularity lemma, let \( m > 1/\gamma \) and choose \( \epsilon > 0 \) small enough that \( \epsilon \leq {\epsilon }_{0} \) ,
\[
\epsilon < \gamma /2 < 1
\]
(1)
and
\[
\delta \mathrel{\text{:=}} {2\gamma } - {\epsilon }^{2} - {4\epsilon } - d - \frac{1}{m} > 0;
\]
this is possible, since \( {2\gamma } - d - \frac{1}{m} > 0 \) . On input \( \epsilon \) and \( m \), the regularity \( M \) lemma returns an integer \( M \) . Let us assume that
\[
n \geq \frac{2Ms}{{d}^{\Delta }\left( {1 - \epsilon }\right) }.
\]
Since this number is at least \( m \), the regularity lemma provides us with an \( \epsilon \) -regular partition \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \) of \( G \), where \( m \leq k \leq M \) ; let \( \left| {V}_{1}\right| = \ldots = \left| {V}_{k}\right| = : \ell \) . Then
\[
n \geq {kl}
\]
(2)
and
\[
\ell = \frac{n - \left| {V}_{0}\right| }{k} \geq \frac{n - {\epsilon n}}{M} = n\frac{1 - \epsilon }{M} \geq \frac{2s}{{d}^{\Delta }}
\]
by the choice of \( n \) . Let \( R \) be the regularity graph of \( G \) with parameters \( \epsilon ,\ell, d \) corresponding to the above partition. Then Lemma 7.5.2 will imply \( {K}_{s}^{r} \subseteq G \) as desired if \( {K}^{r} \subseteq R \) (and hence \( {K}_{s}^{r} \subseteq {R}_{s} \) ).
Our plan was to show \( {K}^{r} \subseteq R \) by Turán’s theorem. We thus have to check that \( R \) has enough edges, i.e. that enough \( \epsilon \) -regular pairs \( \left( {{V}_{i},{V}_{j}}\right) \) have density at least \( d \) . This should follow from our assumption that \( G \) has at least \( {t}_{r - 1}\left( n\right) + \gamma {n}^{2} \) edges, i.e. an edge density of about \( \frac{r - 2}{r - 1} + {2\gamma } \) : this lies substantially above the approximate density of \( \frac{r - 2}{r - 1} \) of the Turán graph \( {T}^{r - 1}\left( k\right) \), and hence substantially above any density that \( G \) could derive from \( {t}_{r - 1}\left( k\right) \) dense pairs alone, even if all these had density 1 .
Let us then estimate \( \parallel R\parallel \) more precisely. How many edges of \( G \) lie outside \( \epsilon \) -regular pairs? At most \( \left( \begin{matrix} \left| {V}_{0}\right| \\ 2 \end{matrix}\right) \) edges lie inside \( {V}_{0} \), and by condition (i) in the definition of \( \epsilon \) -regularity these ar |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 65 | urán’s theorem. We thus have to check that \( R \) has enough edges, i.e. that enough \( \epsilon \) -regular pairs \( \left( {{V}_{i},{V}_{j}}\right) \) have density at least \( d \) . This should follow from our assumption that \( G \) has at least \( {t}_{r - 1}\left( n\right) + \gamma {n}^{2} \) edges, i.e. an edge density of about \( \frac{r - 2}{r - 1} + {2\gamma } \) : this lies substantially above the approximate density of \( \frac{r - 2}{r - 1} \) of the Turán graph \( {T}^{r - 1}\left( k\right) \), and hence substantially above any density that \( G \) could derive from \( {t}_{r - 1}\left( k\right) \) dense pairs alone, even if all these had density 1 .
Let us then estimate \( \parallel R\parallel \) more precisely. How many edges of \( G \) lie outside \( \epsilon \) -regular pairs? At most \( \left( \begin{matrix} \left| {V}_{0}\right| \\ 2 \end{matrix}\right) \) edges lie inside \( {V}_{0} \), and by condition (i) in the definition of \( \epsilon \) -regularity these are at most \( \frac{1}{2}{\left( \epsilon n\right) }^{2} \) edges. At most \( \left| {V}_{0}\right| k\ell \leq {\epsilon nk}\ell \) edges join \( {V}_{0} \) to other partition sets. The at most \( \epsilon {k}^{2} \) other pairs \( \left( {{V}_{i},{V}_{j}}\right) \) that are not \( \epsilon \) -regular contain at most \( {\ell }^{2} \) edges each, together at most \( \epsilon {k}^{2}{\ell }^{2} \) . The \( \epsilon \) -regular pairs of insufficient density \( \left( { < d}\right) \) each contain no more than \( d{\ell }^{2} \) edges, altogether at most \( \frac{1}{2}{k}^{2}d{\ell }^{2} \) edges. Finally, there are at most \( \left( \begin{array}{l} \ell \\ 2 \end{array}\right) \) edges inside each of the partition sets \( {V}_{1},\ldots ,{V}_{k} \), together at most \( \frac{1}{2}{\ell }^{2}k \) edges. All other edges of \( G \) lie in \( \epsilon \) -regular pairs of density at least \( d \), and thus contribute to edges of \( R \) . Since each edge of \( R \) corresponds to at most \( {\ell }^{2} \) edges of \( G \) , we thus have in total
\[
\parallel G\parallel \leq \frac{1}{2}{\epsilon }^{2}{n}^{2} + {\epsilon nk}\ell + \epsilon {k}^{2}{\ell }^{2} + \frac{1}{2}{k}^{2}d{\ell }^{2} + \frac{1}{2}{\ell }^{2}k + \parallel R\parallel {\ell }^{2}.
\]
Hence, for all sufficiently large \( n \) ,
\[
\parallel R\parallel \geq \frac{1}{2}{k}^{2}\frac{\parallel G\parallel - \frac{1}{2}{\epsilon }^{2}{n}^{2} - {\epsilon nk}\ell - \epsilon {k}^{2}{\ell }^{2} - \frac{1}{2}d{k}^{2}{\ell }^{2} - \frac{1}{2}k{\ell }^{2}}{\frac{1}{2}{k}^{2}{\ell }^{2}}
\]
\[
\underset{\left( 1,2\right) }{ \geq }\frac{1}{2}{k}^{2}\left( {\frac{{t}_{r - 1}\left( n\right) + \gamma {n}^{2} - \frac{1}{2}{\epsilon }^{2}{n}^{2} - {\epsilon nk}\ell }{{n}^{2}/2} - {2\epsilon } - d - \frac{1}{k}}\right)
\]
\[
\underset{\left( 2\right) }{ \geq }\frac{1}{2}{k}^{2}\left( {\frac{{t}_{r - 1}\left( n\right) }{{n}^{2}/2} + {2\gamma } - {\epsilon }^{2} - {4\epsilon } - d - \frac{1}{m}}\right)
\]
\[
= \frac{1}{2}{k}^{2}\left( {{t}_{r - 1}\left( n\right) {\left( \begin{array}{l} n \\ 2 \end{array}\right) }^{-1}\left( {1 - \frac{1}{n}}\right) + \delta }\right)
\]
\[
> \frac{1}{2}{k}^{2}\frac{r - 2}{r - 1}
\]
\[
\geq {t}_{r - 1}\left( k\right) \text{.}
\]
(The strict inequality follows from Lemma 7.1.4.) Therefore \( {K}^{r} \subseteq R \) by Theorem 7.1.1, as desired.
Having seen a typical application of the regularity lemma in full detail, let us now step back and try to separate the wheat from the chaff: what were the main ideas, how do the various parameters depend on each other, and in which order were they chosen?
The task was to show that \( \gamma {n}^{2} \) more edges than can be accommodated on \( n \) vertices without creating a \( {K}^{r} \) force a \( {K}_{s}^{r} \) subgraph, provided that \( G \) is large enough. The plan was to do this using Lemma 7.5.2, which asks for the input of two parameters: \( d \) and \( \Delta \) . As we wish to find a copy of \( H = {K}_{s}^{r} \) in \( G \), it is clear that we must choose \( \Delta \mathrel{\text{:=}} \Delta \left( {K}_{s}^{r}\right) \) . We shall return to the question of how to choose \( d \) in a moment.
Given \( d \) and \( \Delta \), Lemma 7.5.2 tells us how small we must choose \( \epsilon \) to make the regularity lemma provide us with a suitable partition. The regularity lemma also requires the input of a lower bound \( m \) for the number of partition classes; we shall discuss this below, together with \( d \) .
All that remains now is to choose \( G \) large enough that the partition classes have size at least \( {2s}/{d}^{\Delta } \), as required by Lemma 7.5.2. (The \( s \) here depends on the graph \( H \) we wish to embed, and \( s \mathrel{\text{:=}} \left| H\right| \) would certainly be big enough. In our case, we can use the \( s \) from our \( H = {K}_{s}^{r} \) .) How large is ’large enough’ for \( \left| G\right| \) follows straight from the upper bound \( M \) on the number of partition classes returned by the regularity lemma: roughly, i.e. disregarding \( {V}_{0} \), an assumption of \( \left| G\right| \geq {2Ms}/{d}^{\Delta } \) suffices.
So far, everything was entirely straightforward, and standard for any application of the regularity lemma of this kind. But now comes the interesting bit, the part specific to this proof: the observation that, if only \( d \) is small enough, our \( \gamma {n}^{2} \) ’additional edges’ force an ’additional dense \( \epsilon \) -regular pair’ of partition sets, giving us more than \( {t}_{r - 1}\left( k\right) \) dense \( \epsilon \) -regular pairs in total (where ’dense’ means ’of density at least \( d \) ’), thus forcing \( R \) to contain a \( {K}^{r} \) and hence \( {R}_{s} \) to contain a \( {K}_{s}^{r} \) .
Let us examine why this is so. Suppose we have at most \( {t}_{r - 1}\left( k\right) \) dense \( \epsilon \) -regular pairs . Inside these, \( G \) has at most
\[
\frac{1}{2}{k}^{2}\frac{r - 2}{r - 1}{\ell }^{2} \leq \frac{1}{2}{n}^{2}\frac{r - 2}{r - 1}
\]
edges, even if we use those pairs to their full capacity of \( {\ell }^{2} \) edges each (where \( \ell \) is again the common size of the partition sets other than \( {V}_{0} \), so that \( k\ell \) is nearly \( n \) ). Thus, we have almost exactly our \( \gamma {n}^{2} \) additional edges left to accommodate elsewhere in the graph: either in \( \epsilon \) -regular pairs of density less than \( d \), or in some exceptional way, i.e. in irregular pairs, inside a partition set, or with an end in \( {V}_{0} \) . Now the number of edges in low-density \( \epsilon \) -regular pairs is less than
\[
\frac{1}{2}{k}^{2}d{\ell }^{2} \leq \frac{1}{2}d{n}^{2}
\]
and hence less than half of our extra edges if \( d \leq \gamma \) . The other half, the remaining \( \frac{1}{2}\gamma {n}^{2} \) edges, are more than can be accommodated in exceptional ways, provided we choose \( m \) large enough and \( \epsilon \) small enough (giving an additional upper bound for \( \epsilon \) ). It is now a routine matter to compute the values of \( m \) and \( \epsilon \) that will work.
## Exercises
1. \( {}^{ - } \) Show that \( {K}_{1,3} \) is extremal without a \( {P}^{3} \) .
2. \( {}^{ - } \) Given \( k > 0 \), determine the extremal graphs of chromatic number at most \( k \) .
3. Determine the value of \( \operatorname{ex}\left( {n,{K}_{1, r}}\right) \) for all \( r, n \in \mathbb{N} \) .
4. Is there a graph that is edge-maximal without a \( {K}^{3} \) minor but not extremal?
5. \( {}^{ + } \) Given \( k > 0 \), determine the extremal graphs without a matching of size \( k \) .
(Hint. Theorem 2.2.3 and Ex. 15, Ch. 2.)
6. Without using Turán's theorem, show that the maximum number of edges in a triangle-free graph of order \( n > 1 \) is \( \left\lfloor {{n}^{2}/4}\right\rfloor \) .
7. Show that
\[
{t}_{r - 1}\left( n\right) \leq \frac{1}{2}{n}^{2}\frac{r - 2}{r - 1}
\]
with equality whenever \( r - 1 \) divides \( n \) .
8. Show that \( {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \) converges to \( \left( {r - 2}\right) /\left( {r - 1}\right) \) as \( n \rightarrow \infty \) .
(Hint. \( {t}_{r - 1}\left( {\left( {r - 1}\right) \left\lfloor \frac{n}{r - 1}\right\rfloor }\right) \leq {t}_{r - 1}\left( n\right) \leq {t}_{r - 1}\left( {\left( {r - 1}\right) \left\lceil \frac{n}{r - 1}\right\rceil }\right) \) .)
9. Show that deleting at most \( \left( {m - s}\right) \left( {n - t}\right) /s \) edges from a \( {K}_{m, n} \) will never destroy all its \( {K}_{s, t} \) subgraphs.
10. For \( 0 < s \leq t \leq n \) let \( z\left( {n, s, t}\right) \) denote the maximum number of edges in a bipartite graph whose partition sets both have size \( n \), and which does not contain a \( {K}_{s, t} \) . Show that \( 2\operatorname{ex}\left( {n,{K}_{s, t}}\right) \leq z\left( {n, s, t}\right) \leq \operatorname{ex}\left( {{2n},{K}_{s, t}}\right) \) .
11. \( {}^{ + } \) Let \( 1 \leq r \leq n \) be integers. Let \( G \) be a bipartite graph with bipartition \( \{ A, B\} \), where \( \left| A\right| = \left| B\right| = n \), and assume that \( {K}_{r, r} \nsubseteq G \) . Show that
\[
\mathop{\sum }\limits_{{x \in A}}\left( \begin{matrix} d\left( x\right) \\ r \end{matrix}\right) \leq \left( {r - 1}\right) \left( \begin{array}{l} n \\ r \end{array}\right)
\]
Using the previous exercise, deduce that \( \operatorname{ex}\left( {n,{K}_{r, r}}\right) \leq c{n}^{2 - 1/r} \) for some constant \( c \) depending only on \( r \) .
12. The upper density of an infinite graph \( G \) is the infimum of all reals \( \alpha \) such that the finite graphs \( H \subseteq G \) with \( \parallel H\parallel {\left( \begin{matrix} \left| H\right| \\ 2 \end{matrix}\right) }^{-1} > \alpha \) have bounded order. Use the Erdős-Stone theorem to show that this number always takes one of the countably many values \( 0,1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\ldots \) .
13. Given a tree \( T \), find an |
1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论） | 66 | be integers. Let \( G \) be a bipartite graph with bipartition \( \{ A, B\} \), where \( \left| A\right| = \left| B\right| = n \), and assume that \( {K}_{r, r} \nsubseteq G \) . Show that
\[
\mathop{\sum }\limits_{{x \in A}}\left( \begin{matrix} d\left( x\right) \\ r \end{matrix}\right) \leq \left( {r - 1}\right) \left( \begin{array}{l} n \\ r \end{array}\right)
\]
Using the previous exercise, deduce that \( \operatorname{ex}\left( {n,{K}_{r, r}}\right) \leq c{n}^{2 - 1/r} \) for some constant \( c \) depending only on \( r \) .
12. The upper density of an infinite graph \( G \) is the infimum of all reals \( \alpha \) such that the finite graphs \( H \subseteq G \) with \( \parallel H\parallel {\left( \begin{matrix} \left| H\right| \\ 2 \end{matrix}\right) }^{-1} > \alpha \) have bounded order. Use the Erdős-Stone theorem to show that this number always takes one of the countably many values \( 0,1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\ldots \) .
13. Given a tree \( T \), find an upper bound for \( \operatorname{ex}\left( {n, T}\right) \) that is linear in \( n \) and independent of the structure of \( T \), i.e. depends only on \( \left| T\right| \) .
14. Show that, as a general bound for arbitrary \( n \), the bound on \( \operatorname{ex}\left( {n, T}\right) \) claimed by the Erdős-Sós conjecture is best possible for every tree \( T \) . Is it best possible even for every \( n \) and every \( T \) ?
15. \( {}^{ - } \) Prove the Erdős-Sós conjecture for the case when the tree considered is a star.
16. Prove the Erdős-Sós conjecture for the case when the tree considered is a path.
(Hint. Use Exercise 7 of Chapter 1.)
17. \( {}^{ + } \) For which trees \( T \) is there a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) tending to infinity, such that every graph \( G \) with \( \chi \left( G\right) < f\left( {d\left( G\right) }\right) \) contains an induced copy of \( T \) ? (In other words: can we force the chromatic number up by raising the average degree, as long as \( T \) does not occur as an induced subgraph? Or, as in Gyárfás's conjecture: will a large average degree force an induced copy of \( T \) if the chromatic number is kept small?)
18. Given two graph invariants \( {i}_{1} \) and \( {i}_{2} \), write \( {i}_{1} \leq {i}_{2} \) if we can force \( {i}_{2} \) arbitrarily high on a subgraph of \( G \) by making \( {i}_{1}\left( G\right) \) large enough. (Formally: write \( {i}_{1} \leq {i}_{2} \) if there exists a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that, given any \( k \in \mathbb{N} \), every graph \( G \) with \( {i}_{1}\left( G\right) \geq f\left( k\right) \) has a subgraph \( H \) with \( \left. {{i}_{2}\left( H\right) \geq k\text{.}}\right) \) If \( {i}_{1} \leq {i}_{2} \) as well as \( {i}_{1} \geq {i}_{2} \), write \( {i}_{1} \sim {i}_{2} \) . Show that this is an equivalence relation for graph invariants, and sort the following invariants into equivalence classes ordered by \( < \) : minimum degree; average degree; connectivity; arboricity; chromatic number; colouring number; choice number; \( \max \left\{ {r \mid {K}^{r} \subseteq G}\right\} \) ; \( \max \left\{ {r \mid T{K}^{r} \subseteq G}\right\} \) ; \( \max \left\{ {r \mid {K}^{r} \preccurlyeq G}\right\} \) ; min max \( {d}^{ + }\left( v\right) \), where the maximum is taken over all vertices \( v \) of the graph, and the minimum over all its orientations.
19. \( {}^{ + } \) Prove, from first principles, the theorem of Wagner (1964) that every graph of chromatic number at least \( {2}^{r} \) contains \( {K}^{r} \) as a minor.
(Hint. Use induction on \( r \) . For the induction step, contract a connected subgraph chosen so that the remaining graph still needs at least half as many colours as the given graph.)
20. Let \( G \) be a graph of average degree at least \( {2}^{r - 2} \) . By considering the neighbourhood of a vertex in a minimal minor \( H \preccurlyeq G \) with \( \varepsilon \left( H\right) \geq \varepsilon \left( G\right) \) , prove Mader’s (1967) theorem that \( G \succcurlyeq {K}^{r} \) .
21. \( {}^{ - } \) Derive Wagner’s theorem (Ex. 19) from Mader’s theorem (Ex. 20).
22. \( {}^{ + } \) Given a graph \( G \) with \( \varepsilon \left( G\right) \geq k \in \mathbb{N} \), find a minor \( H \preccurlyeq G \) such that both \( \delta \left( H\right) \geq k \) and \( \delta \left( H\right) \geq \left| H\right| /2 \) .
23. \( {}^{ + } \) Find a constant \( c \) such that every graph with \( n \) vertices and at least \( n + {2k}\left( {\log k + \log \log k + c}\right) \) edges contains \( k \) edge-disjoint cycles (for all \( k \in \mathbb{N} \) ). Deduce an edge-analogue of the Erdős-Pósa theorem (2.3.2).
(Hint. Assuming \( \delta \geq 3 \), delete the edges of a short cycle and apply induction. The calculations are similar to the proof of Lemma 2.3.1.)
24. \( {}^{ - } \) Use Exercise 22 of Chapter 3 to reduce the constant \( c \) in Theorem 7.2.1 from 10 to 5 .
25. \( {}^{ + } \) Show that any function \( h \) as in Lemma 3.5.1 satisfies the inequality \( h\left( r\right) > \frac{1}{8}{r}^{2} \) for all even \( r \), and hence that Theorem 7.2.1 is best possible up to the value of the constant \( c \) .
26. Characterize the graphs with \( n \) vertices and more than \( {3n} - 6 \) edges that contain no \( T{K}_{3,3} \) . In particular, determine \( \operatorname{ex}\left( {n, T{K}_{3,3}}\right) \) .
(Hint. By a theorem of Wagner, every edge-maximal graph without a \( {K}_{3,3} \) minor can be constructed recursively from maximal planar graphs and copies of \( {K}^{5} \) by pasting along \( {K}^{2} \) s.)
27. \( {}^{ - } \) Derive the four colour theorem from Hadwiger’s conjecture for \( r = 5 \) .
28. \( {}^{ - } \) Show that Hadwiger’s conjecture for \( r + 1 \) implies the conjecture for \( r \) .
29. \( {}^{ - } \) Prove the following weakening of Hadwiger’s conjecture: given any \( \epsilon > 0 \), every graph of chromatic number at least \( {r}^{1 + \epsilon } \) has a \( {K}^{r} \) minor, provided that \( r \) is large enough.
30. Show that any graph constructed as in Proposition 7.3.1 is edge-maximal without a \( {K}^{4} \) minor.
31. Prove the implication \( \delta \left( G\right) \geq 3 \Rightarrow G \supseteq T{K}^{4} \) .
(Hint. Proposition 7.3.1.)
32. A multigraph is called series-parallel if it can be constructed recursively from a \( {K}^{2} \) by the operations of subdividing and of doubling edges. Show that a 2-connected multigraph is series-parallel if and only if it has no (topological) \( {K}^{4} \) minor.
33. Without using Theorem 7.3.8, prove Hadwiger's conjecture for all graphs of girth at least 11 and \( r \) large enough. Without using Corollary 7.3.9, show that there is a constant \( g \in \mathbb{N} \) such that all graphs of girth at least \( g \) satisfy Hadwiger’s conjecture, irrespective of \( r \) .
34. \( {}^{ + } \) Prove Hadwiger’s conjecture for \( r = 4 \) from first principles.
35. \( {}^{ + } \) Prove Hadwiger’s conjecture for line graphs.
36. Prove Corollary 7.3.5.
37. \( {}^{ - } \) In the definition of an \( \epsilon \) -regular pair, what is the purpose of the requirement that \( \left| X\right| > \epsilon \left| A\right| \) and \( \left| Y\right| > \epsilon \left| B\right| \) ?
38. \( {}^{ - } \) Show that any \( \epsilon \) -regular pair in \( G \) is also \( \epsilon \) -regular in \( \bar{G} \) .
39. Consider a partition \( \left\{ {{V}_{1}\ldots {V}_{k}}\right\} \) of a finite set \( V \) . Show that the complete graph on \( V \) has about \( k - 1 \) as many edges between different partition sets as edges inside partition sets. Explain how this leads to the choice of \( m \mathrel{\text{:=}} 1/\gamma \) in the proof of the Erdős-Stone theorem.
40. (i) Deduce the regularity lemma from the assumption that it holds, given \( \epsilon > 0 \) and \( m \geq 1 \), for all graphs of order at least some \( n = n\left( {\epsilon, m}\right) \) .
(ii) Prove the regularity lemma for sparse graphs, that is, for every sequence \( {\left( {G}_{n}\right) }_{n \in \mathbb{N}} \) of graphs \( {G}_{n} \) of order \( n \) such that \( \begin{Vmatrix}{G}_{n}\end{Vmatrix}/{n}^{2} \rightarrow 0 \) as \( n \rightarrow \infty \) .
## Notes
The standard reference work for results and open problems in extremal graph theory (in a very broad sense) is still B. Bollobás, Extremal Graph Theory, Academic Press 1978. A kind of update on the book is given by its author in his chapter of the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995. An instructive survey of extremal graph theory in the narrower sense of Section 7.1 is given by M. Simonovits in (L.W. Beineke & R.J. Wilson, eds.) Selected Topics in Graph Theory 2, Academic Press 1983. This paper focuses among other things on the particular role played by the Turán graphs. A more recent survey by the same author can be found in (R.L. Graham & J. Nešetřil, eds.) The Mathematics of Paul Erdős, Vol. 2, Springer 1996.
Turán's theorem is not merely one extremal result among others: it is the result that sparked off the entire line of research. Our first proof of Turán's theorem is essentially the original one; the second is a version of a proof of Zykov due to Brandt.
Our version of the Erdős-Stone theorem is a slight simplification of the original. A direct proof, not using the regularity lemma, is given in L. Lovász, Combinatorial Problems and Exercises (2nd edn.), North-Holland 1993. Its most fundamental application, Corollary 7.1.3, was only found 20 years after the theorem, by Erdős and Simonovits (1966).
Of our two bounds on \( \operatorname{ex}\left( {n,{K}_{r, r}}\right) \) the upper one is thought to give the correct order of magnitude. For vastly off-diagonal complete bipartite graphs this was verified by J. Kollár, L. Rónyai & T. Szabó, Norm-graphs and bipartite Turán numbers, Combinatorica 16 (1996), 399-406, who proved that \( \operatorname |