Disjoint cycles in star-free graphs

A graph is claw-free if it does not contain K_1.3 as an induced subgraph. It is K_1.r-free if it does not contain K_1.r as an induced subgraph. We show that if a graph is K_1.r-free (r ≥ 4), only p + 2r − 1 edges are needed to insure that G has two disjoint cycles. As an easy consequence we get a well-known result of Pósa's. © 1996 John Wiley & Sons, Inc.