Decomposing and Clique-Coloring (Diamond, Odd-Hole)-Free Graphs

A diamond is a graph on four vertices with exactly one pair of nonadjacent vertices, and an odd hole is an induced cycle of odd length greater than 3. If G and H are graphs, G is H-free if no induced subgraph of G is isomorphic to H. A clique-coloring of G is an assignment of colors to the vertices of G such that no inclusion-wise maximal clique of size at least 2 is monochromatic. We show that every (diamond, odd-hole)-free graph is 3-clique-colorable, answering a question of Bacsó et al. (SIAM J Discrete Math 17(3) (2004), 361–376).