On facets of stable set polytopes of claw-free graphs with stability number three

Abstract

Abstract Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem since almost twenty years. Eisenbrandt et al. recently achieved a breakthrough for the subclass of quasi-line graphs. As a consequence, every non-trivial facet of their stable set polytope is of the form k ∑ v ∈ V 1 x v + ( k + 1 ) ∑ v ∈ V 2 x v ⩽ b for some positive integers k and b, and non-empty sets of vertices V 1 and V 2 . Roughly speaking, this states that the facets of the stable set polytope of quasi-line graphs have at most two left coefficients. For stable set polytopes of claw-free graphs with maximum stable set size at least four, Stauffer conjectured in 2005 that this still holds. It is already known that some stable set polytopes of claw-free graphs with maximum stable set size three may have facets with up to 5 left coefficients. We prove that the situation is even worse: for every positive integer b, there is a claw-free graph with stability number three whose stable set polytope has a facet with b left coefficients.