A note on hitting maximum and maximal cliques with a stable set
  It was recently proved that any graph satisfying $\omega > \frac 23(\Delta+1)$ contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying $\omega \geq \frac 23(\Delta+1)$ 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. 
Demetres Christofides, Katherine Edwards, and Andrew D. King
cs.DM math.CO