Edge clique covers in graphs with independence number two

The edge clique cover number ecc(G) of a graph G is size of the smallest collection of complete sub-graphs whose union covers all edges of G. Chen, Jacobson, Kezdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw-free, then ecc(G) is bounded above by its order (denoted n). Recently, Javadi and Hajebi verified this conjecture for claw-free graphs with independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw-free. We give the first known proof of a linear bound in n for ecc(G) for such graphs, improving upon the bound of O (n 4/3 log 1/3 n) due to Javadi, Maleki and Omoomi. More precisely we prove that ecc(G) is at most the minimum of n + δ (G) and 2n − Ω (n log n), where δ (G) is the minimum degree of G. In the fractional version of the problem, we improve these upper bounds to 3 2 n. We also verify the conjecture for some specific subfamilies, for example when the edge packing number with respect to cliques (a lower bound for ecc(G)) equals n, and when G contains no induced subgraph isomorphic to H where H is any fixed graph of order 4.