Hitting all maximum cliques with a stable set using lopsided independent transversals

TL;DR

This paper improves a known bound on the existence of a stable set intersecting all maximum cliques in a graph, extending the result to a broader class of graphs with a tighter inequality.

Contribution

It strengthens Rabern's theorem by proving the existence of such stable sets under a less restrictive condition, using new techniques involving independent transversals.

Findings

Stable set exists for graphs with omega > (2/3)(Delta+1)

The result is tight, inequality must be strict

Proof uses independent transversals in partitioned graphs

Abstract

Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a stable set meeting all maximum cliques. We strengthen this result, proving that such a stable set exists for any graph with omega > (2/3)(Delta+1). This is tight, i.e. the inequality in the statement must be strict. The proof relies on finding an independent transversal in a graph partitioned into vertex sets of unequal size.