A note on hitting maximum and maximal cliques with a stable set

Abstract

It was recently proved that any graph satisfying $\omega >\frac{2}{3}(\Delta +1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $\omega \ge \frac{2}{3}(\Delta +1)$ unless the graph is the strong product of $K_{\omega /2}$ and an odd hole. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.

7 pages, two figures, accepted to J. Graph Theory