Coloring the Maximal Cliques of Graphs

Abstract

In this paper we are concerned with the so-called clique-colorations of a graph, that is, colorations of the vertices so that no maximal clique is monochromatic. On one hand, it is known to be NP-complete to decide whether a perfect graph is 2-clique-colorable, or whether a triangle-free graph is 3-clique-colorable; on the other hand, there is no example of a perfect graph where more than three colors would be necessary. We first exhibit some simple recursive methods to clique-color graphs and then relate the chromatic number, the domination number, and the maximum cardinality of a stable set to the clique-chromatic number. We show exact bounds and polynomial algorithms that find the clique-chromatic number for some classes of graphs and prove NP-completeness results for some others, trying to find the boundary between the two. For instance, while it is NP-complete to decide whether a graph of maximum degree 3 is 2-clique-colorable, K1,3 -free graphs without an odd hole turn out to be always 2-clique-colorable by a polynomial algorithm. Finally, we show that "almost" all perfect graphs are 3-clique-colorable.