Edge clique cover of claw-free graphs

The smallest number of cliques, covering all edges of a graph G, is called the (edge) clique cover number of G and is denoted by cc(G). It is an easy observation that if G is a line graph on n vertices, then cc(G) <= n. G. Chen et al. [Discrete Math. 219 (2000), no. 1-3, 17-26; MR1761707] extended this observation to all quasi-line graphs and questioned if the same assertion holds for all claw-free graphs. In this paper, using the celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour, we give an affirmative answer to this question for all claw-free graphs with independence number at least three. In particular, we prove that if G is a connected claw-free graph on n vertices with three pairwise nonadjacent vertices, then cc(G) <= n and the equality holds if and only if G is either the graph of icosahedron, or the complement of a graph on 10 vertices called "twister" or the pth power of the cycle C-n, for some positive integer p <= left perpendicular (n - 1)/3 right perpendicular.