Abstract
Abstract Let G=(V,E) be a graph with n vertices. A clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. A k-clique-colouring is a clique-colouring that uses k colours. The clique-chromatic number of a graph G is the minimum k such that G has a k-clique-colouring. In this paper we will use the primeval decomposition technique to find the clique-chromatic number and the clique-colouring of well known classes of graphs that in some local sense contain few P 4’s. In particular we shall consider the classes of extended P 4-laden graphs, p-trees (graphs which contain exactly n−3 P 4’s) and (q,q−3)-graphs, q≥7, such that no set of at most q vertices induces more that q−3 distincts P 4’s. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cographs, P 4-reducible graphs, P 4-sparse graphs, extended P 4-reducible graphs, extended P 4-sparse graphs, P 4-extendible graphs, P 4-lite graphs, P 4-tidy graphs and P 4-laden graphs that are included in the class of extended P 4-laden graphs.
Highlights
In this paper we are concerned with the so called cliquecolouring of a graph
A hypergraph is a pair H = (V, E), where V is the set of vertices of H and E is a family of non-empty subsets of V called edges
In other words: a clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured
Summary
In this paper we are concerned with the so called cliquecolouring of a graph. To introduce this concept we need the following definitions. In other words: a clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. In particular we shall consider the classes of extended P4-laden [16], p-trees [3] (graphs which contain exactly n − 3 P4’s) and (q, q − 3)-graphs, q ≥ 7, [2] such that no set of at most q vertices induces more that q − 3 distincts P4’s. 5 we find the clique-chromatic number and the clique-colouring of the remaining classes of graphs mentioned above. These results lead to polynomial time algorithms for finding the clique-colouring and the cliquechromatic number of the above classes
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