Abstract
Let C be a clique covering for E(G) and let v be a vertex of G. The valency of vertex v (with respect to C), denoted by valC(v), is the number of cliques in C containing v. The local clique cover number of G, denoted by lcc(G), is defined as the smallest integer k, for which there exists a clique covering for E(G) such that valC(v) is at most k, for every vertex v∈V(G). In this paper, among other results, we prove that if G is a claw-free graph, then lcc(G)+χ(G)≤n+1.
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