Edge clique covers in graphs with independence number two

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