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

@article{Christofides2013ANO,
title={A note on hitting maximum and maximal cliques with a stable set},
author={Demetres Christofides and Katherine L Edwards and Andrew D. King},
journal={Journal of Graph Theory},
year={2013},
volume={73},
pages={354-360}
}

It was recently proved that any graph satisfying ω > 2 3(∆ + 1) contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying ω ≥ 23(∆ + 1) unless the graph is the strong product of an odd hole and Kω/2. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.