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Let $\alpha(G)$ and $\beta(G)$, denote the size of a largest independent setand the clique cover number of an undirected graph $G$. Let $H$ be an intervalgraph with $V(G)=V(H)$ and $E(G)\subseteq E(H)$, and let $\phi(G,H)$ denote themaximum of ${\beta(G[W])\over \alpha(G[W])}$ overall induced subgraphs $G[W]$of $G$ that are cliques in $H$. The main result of this paper is to prove thatfor any graph $G$ $${\beta(G)}\le 2 \alpha(H)\phi(G,H)(\log \alpha(H)+1),$$where, $\alpha(H)$ is the size of a largest independent set in $H$. We furtherprovide a generalization that significantly unifies or improves some pastalgorithmic and structural results concerning the clique cover number for somewell known intersection graphs.

A new upper bound for the clique cover number with applications