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A polynomially searchable exponential neighbourhood for graph colouring [PDF]

Journal of the Operational Research Society, 2005
In this paper we develop a new graph colouring strategy. Our heuristic is an example of a so called "polynomially searchable exponential neighbourhood" approach. The neighbourhood is that of permutations of the colours of vertices of a subgraph.
Glass, Celia A., Prügel-Bennett, Adam
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Total Colourings of Graphs

Bulletin of the London Mathematical Society, 1989
We prove that the TCC (Total Colouring Conjecture) is true for complete r-partite graphs which extends a result of M. Rosenfeld. We also give an alternate, slightly simpler proof of an earlier result (which says that the TCC is true for graphs having maximum degree 3) obtained independently by M. Rosenfeld and N. Vijayaditya.
openaire   +1 more source

On colour critical graphs

2006
A brief survey of the theory of colour critical graphs, with an emphasis on constructive methods, is given.
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Colouring AT-Free Graphs

2012
A vertex colouring assigns to each vertex of a graph a colour such that adjacent vertices have different colours. The algorithmic complexity of the Colouring problem, asking for the smallest number of colours needed to vertex-colour a given graph, is known for a large number of graph classes.
Dieter Kratsch, Haiko Müller
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Generation of Colourings and Distinguishing Colourings of Graphs

2015
A colouring of a graph \(X\) is an assignment of colours to the vertices of \(X\). A distinguishing colouring of \(X\) is a colouring such that no non-trivial automorphism of \(X\) preserves all colours. The distinguishing number of \(X\) is the minimum number of colours in a distinguishing colouring.
William Bird, Wendy J. Myrvold
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The Number of Coloured Graphs

Canadian Journal of Mathematics, 1963
Let Fn(k) denote the total number of k-coloured graphs on n labelled nodes and let Mn(k) denote the number of graphs on n nodes that are coloured in at most kcolours ; also let fn(k) denote the number of connected k coloured graphs on n nodes. Read (3) has proved the following formulas:
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Okhuma Graphs and Coloured Chains

Order, 2004
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
M. Giraudet, John Kenneth Truss
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Colouring of distance graphs

Ars Comb., 1999
For a set \(D\) of positive integers, the distance graph \(G(D)\) has the integers as vertex set and two integers \(u,v\) are adjacent if \(|u-v|\in D\). The question of determining the chromatic number of distance graphs and first results can be found in \textit{R. B. Eggleton, P. Erdős}, and \textit{D. K. Skilton} [J. Comb. Theory, Ser. B 39, 86-100 (
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Colourings and orderings in a graph

Sci. Ann. Cuza Univ., 1992
Throughout this paper \(G= (V, E)\) denotes a finite, simple graph with vertex set \(V\) the set \(\{1, 2, \dots, n\}\). A stable set in \(G\) is a set of pairwise nonadjacent vertices in \(G\). The family of all stable sets of \(G\) is denoted by \({\mathcal S}_G\).
Cornelius Croitoru, Costel Radu
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Coloured Graph Decompositions

Let G be a graph in which each vertex has been coloured using one of k colours, say C1, c2, …, ck. If a graph H in G has ni vertices coloured ci i = 1, 2, ... , k, and |ni –nj| < 1 for any i , j Є {1, 2, . . . , k}, then H is said to be equitably k-coloured.
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