Clique minors in claw-free graphs

Hadwigerʼs conjecture states that every graph with chromatic number χ has a clique minor of size χ. Let G be a graph on n vertices with chromatic number χ and stability number α. Then since χα⩾n, Hadwigerʼs conjecture implies that G has a clique minor of size . In this paper we prove that this is true for connected claw-free graphs with α⩾3. We also show that this result is tight by providing an infinite family of claw-free graphs with α⩾3 that do not have a clique minor of size larger than .