Clique-chromatic numbers of claw-free graphs (Q2923093)

A proper \(k\)-clique coloring of a graph \(G\) is a \(k\)-coloring of \(G\) such that an no maximal clique of size at least 2 is monochromatic. The minimum \(k\) for which \(G\) is \(k\)-clique colorable is the clique-chromatic number of \(G\). Several authors have proved that \(F\)-free graphs have bounded clique-chromatic number when \(F=(P_5, C_3)\) or (diamond, odd hole) and so on. Claw-free graphs do not have a bounded clique-chromatic number. In this paper, the authors have proved that (claw, paw)-free graphs and (claw, diamond)-free graphs have clique-chromatic number 2 or 3 and hence these families of graphs have a bounded clique-chromatic number.