39921005
Graph colorings
From the combination of knowledge and actions, someone can improve their skill and ability. It will lead them to live and work much better. This is why, the students, workers, or even employers should have reading habit for books. Any book will give certain knowledge to take all benefits. This is what this graph colorings tells you. It will add more knowledge of you to life and work better. Try it and prove it.
431690
Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult
AbstractWe consider the problem of coloring a planar graph with the
minimum number of colors so that each color class avoids one or
more forbidden graphs as subgraphs. We perform a detailed study of
the computational complexity of this problem. We present a complete picture for the case with a single forbidden
connected (induced or noninduced) subgraph. The 2-coloring
problem is NP-hard if the forbidden subgraph is a tree with at
least two edges, and it is polynomially solvable in all other
cases. The 3-coloring problem is NP-hard if the forbidden
subgraph is a path with at least one edge, and it is polynomially solvable in all other
cases. We also derive results for several forbidden sets of
cycles. In particular, we prove that it is NP-complete to decide if a planar
graph can be 2-colored so that no cycle of length at most 5 is
monochromatic.
11379371
Partitioning Graphs into Two Trees
17103586
The complexity of coloring graphs without long induced paths
We discuss the computational complexity of determining the chromatic number of graphs without long induced paths. We prove NP-completeness of deciding whether a Ps-free graph is 5-colorable and of deciding whether a Pi2-free graph is 4-colorable. Moreover, we give a polynomial time algorithm for deciding whether a Ps-free graph is 3-colorable.
6054148
Hamiltonian cycles in circulant digraphs with two stripes
Abstract The circulant traveling salesman problem ( CTSP ) is the problem of finding a minimum weight Hamiltonian cycle in a weighted graph with circulant distance matrix. The computational complexity of this problem is not known. In fact, even the complexity of deciding Hamiltonicity of the underlying graph is unknown. This paper provides a characterization of Hamiltonian digraphs with circulant distance matrix containing only two nonzero stripes . The corresponding conditions can be checked in polynomial time. Secondly, we show that all Hamiltonian cycles of a circulant 2-digraph are periodic. Based on these two results, a method for enumerating all Hamiltonian cycles in such digraphs is described. Moreover, two simple algorithms are derived for solving the sum and bottleneck versions of CTSP for circulant distance matrices with two nonzero stripes.
44716494
Polynomial Graph-Colorings
For directed graphs G and H, we say that G is H-colorable, if there is a graph homomorphism from G into H; that is, there is a mapping f from the vertex set of G into the vertex set of H such that whenever there is an edge (x, y) in G, then (f(x), f(y)) is an edge in H. We introduce a new technique for proving that the H-coloring problem is polynomial time decidable for some fixed graphs H. Among others, this is the case if H is a semipath (a “path” with edges directed in either direction), which has not been known before. We also show the existence of a tree T, for which the T-coloring problem is NP-complete.
22610152
Counting k-Subsets and Convex k-gons in the Plane
We consider problems of the following type: Let S be a set of n points in the plane, let p be some point in S and let K be an integer, 3≤k≤n. We show how to compute the overall number of convex k-gons with corners at points in S, for any fixed k≥4.
5145089
A faster off-line algorithm for the TCP acknowledgement problem
In a recent paper [Proceedings of STOC'98, 1998, pp. 389-398], Dooly, Goldman and Scott study a problem that is motivated by the networking problem of dynamically adjusting delays of acknowledgements in the Transmission Control Protocol (TCP). Among other results, they give an O(n2) off-line algorithm for computing the optimal way of acknowledging n packet arrivals and departures.In this brief note, we observe that there is a faster off-line algorithm for this problem with time complexity O(n).