Clique Coverings and Claw-free Graphs

Let $\cal C$ be a clique covering for $E(G)$ and let $v$ be a vertex of $G$. The valency of vertex $v$ (with respect to $\cal C$), denoted by $val_{\cal C}(v)$, is the number of cliques in $\cal C$ 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 $val_{\cal C}(v)$ is at most $k$, for every vertex $v\in V(G)$. In this paper, among other results, we prove that if $G$ is a claw-free graph, then $lcc(G)+\chi(G)\leq n+1$.