Clique-Coloring Claw-Free Graphs

Abstract

A clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no clique of G is monochromatic. Bacso et al. (SIAM J Discrete Math 17:361---376, 2004) noted that the clique-coloring number is unbounded even for the line graphs of complete graphs. In this paper, we prove that a claw-free graph with maximum degree at most 7, except an odd cycle longer than 3, has a 2-clique-coloring by using a decomposition theorem of Chudnovsky and Seymour (J Combin Theory Ser B 98:839---938, 2008) and the limitation of the degree 7 is necessary since the line graph of $$K_{6}$$K6 is a graph with maximum degree 8 but its clique-coloring number is 3 by the Ramsey number $$R(3,3)=6$$R(3,3)=6. In addition, we point out that, if an arbitrary line graph of maximum degree at most d is m-clique-colorable ($$m\ge 3$$m?3), then an arbitrary claw-free graph of maximum degree at most d is also m-clique-colorable.