Minimum degree and disjoint cycles in generalized claw-free graphs

For s ≥ 3 a graph is K 1 , s -free if it does not contain an induced subgraph isomorphic to K 1 , s . Cycles in K 1 , 3 -free graphs, called claw-free graphs, have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to K 1 , s -free graphs, normally called generalized claw-free graphs. In particular, we prove that if G is K 1 , s -free of sufficiently large order n = 3 k with δ ( G ) ≥ n / 2 + c for some constant c = c ( s ) , then G contains k disjoint triangles. Analogous results with the complete graph K 3 replaced by a complete graph K m for m ≥ 3 will be proved. Also, the existence of 2 -factors for K 1 , s -free graphs with minimum degree conditions will be shown.