On hitting all maximum cliques with an independent set

Abstract

We prove that every graph $G$ for which $\omega(G) \geq 3/4(\Delta(G) + 1)$, has an independent set $I$ such that $\omega(G - I) < \omega(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $\omega(G) < 3/4(\Delta(G) + 1)$ and hence also $\chi(G) > \lceil 7/6\omega(G) \rceil$. We also prove that if for every induced subgraph $H$ of $G$ we have $\chi(H) \leq \max{\lceil 7/6\omega(H) \rceil, \lceil \frac{\omega(H) + \Delta(H) + 1}{2}\rceil}$, then we also have $\chi(G) \leq \lceil \frac{\omega(G) + \Delta(G) + 1}{2}\rceil$. This gives a generic proof of the upper bound for line graphs of multigraphs proved by King et al.