Clique minors in claw-free graphs

Hadwiger’s 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, Hadwiger’s conjecture implies that G has a clique minor of size n α . 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 n α .