122331567
A new upper bound for the harmonious chromatic number
A harmonious coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colors in such a coloring. We obtain a new upper bound for the harmonious chromatic number of general graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 257–261, 1998
2588408
On monochromatic component size for improper colourings
This paper concerns improper @l-colourings of graphs and focuses on the sizes of the monochromatic components (i.e., components of the subgraphs induced by the colour classes). Consider the following three simple operations, which should, heuristically, help reduce monochromatic component size: (a) assign to a vertex the colour that is least popular among its neighbours; (b) change the colours of any two adjacent differently coloured vertices, if doing so reduces the number of monochromatic edges; and (c) change the colour of a vertex, if by so doing you can reduce the size of the largest monochromatic component containing it without increasing the number of monochromatic edges. If a colouring cannot be further improved by these operations, then we regard it as locally optimal. We show that, for such a locally optimal 2-colouring of a graph of maximum degree 4, the maximum monochromatic component size is O(2^(^2^l^o^g^"^2^n^)^^^1^^^/^^^2). The operation set (a)-(c) appears to be one of the simplest that achieves a o(n) bound on monochromatic component size. Recent work by Alon, Ding, Oporowski and Vertigan, and then Haxell, Szabo and Tardos, has shown that some algorithms can do much better, achieving a constant bound on monochromatic component size. However, the simplicity of our operation set, and of the associated local search algorithm, make the algorithm, and our locally optimal colourings, of interest in their own right.
17705689
The Harmonious Chromatic Number of Almost All Trees
General rights Copyright and moral rights for the publications made accessible in Discovery Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from Discovery Research Portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain. • You may freely distribute the URL identifying the publication in the public portal. Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. For any positive integer m, let Q(m) be the least positive integer k such that (*) S: m. We show that for almost all unlabelled, unrooted trees T, h(T) = Q(m), where m is the number of edges of T.
20062312
The harmonious chromatic number of complete r-ary trees
A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. We define Q(m) to be the least positive integer k such that k2⩾m. Then h(G)⩾Q(m) for any graph G with m edges. We consider the complete r-ary tree of height H, denoted Tr,H. We show that for any r⩾2,H⩾3, if m is the number of edges of Tr,H, then h(Tr,H)=Q(m), except that h(T2,3)=7.
6923895
Edge decomposition of complete tripartite graphs
A graph H decomposes a graph G if and only if the edges of G can be partitioned into disjoint subsets each of which induces a graph isomorphic to H. Wilson (in: C.St.J.A. Nash-Williams, J. Sheehan (Eds.), Proceedings of the Fifth British Combinatorial Conference, Aberdeen, 1975) showed, among other things, that for any graph H, there is an integer n such that H decomposes K"n, andHaggkvist (in: J. Siemons (Ed.), Surveys in Combinatories 1989, Invited papers for 12th British Combinatorial Conference, LMS Lecture Notes 141, Cambridge University Press, Cambridge, 1989, pp. 115-147) showed that for any bipartite H, there is an n such that H decomposes K"n","n. In this paper, we extend this result to tripartite graphs, by showing that for any tripartite graph H, there is an integer n such that H decomposes K"n","n","n.
16759367
The Complexity of Harmonious Colouring for Trees
A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. It was shown by Hopcroft and Krishnamoorthy (1983) that the problem of determining the harmonious chromatic number of a graph is NP-hard. We show here that the problem remains hard even when restricted to trees.
109748790
Jini Example By Example
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60662595
Real-Time Structured Methods: Systems Analysis
Modelling Complex Systems Behaviour Modelling Tools The Strategic Model The Project Modelling Procedure Essential Environment Modelling Essential Behaviour Modelling Validation of the Essential Model Appendices.