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

Abstract

It was recently proved that any graph satisfying $ω> \frac 23(Δ+1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $ω\geq \frac 23(Δ+1)$ unless the graph is the strong product of $K_{ω/2}$ 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