Exact algorithms to clique-colour graphs

Abstract

The clique-chromatic number of a graph G = (V,E) denoted by χ c (G) is the smallest integer k such that there exists a partition of the vertex set of G into k subsets with the property that no maximal clique of G is contained in any of the subsets. Such a partition is called a k-clique-colouring of G. Recently Marx proved that deciding whether a graph admits a k-clique-colouring is-complete for every fixed k ≥ 2. Our main results are an O (2 n ) time inclusion-exclusion algorithm to compute χ c (G) exactly, and a branching algorithm to decide whether a graph of bounded clique-size admits a 2-clique-colouring which runs in time O (λ n ) for some λ < 2. © 2014 Springer International Publishing Switzerland.