Clique-Colouring and biclique-colouring unichord-free graphs
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Clique-transversal sets and clique-coloring in planar graphs
European Journal of Combinatorics
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In this paper we investigate the problem of clique-coloring, which consists in coloring the vertices of a graph in such a way that no monochromatic maximal clique appears, and we focus on odd-hole-free graphs. On the one hand we do not know any odd-hole-free graph that is not 3-clique-colorable, but on the other hand it is NP-hard to decide if they are 2-clique-colorable, and we do not know if there exists any bound k0 such that they are all k0 -clique-colorable. First we will prove that (odd hole, codiamond)-free graphs are 2-clique-colorable. Then we will demonstrate that the complexity of 2-clique-coloring odd-hole-free graphs is actually Σ2 P-complete. Finally we will study the complexity of deciding whether or not a graph and all its subgraphs are 2-clique-colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 139–156, 2009