Vertex-disjoint copies of K1, t in K1, r-free graphs

Abstract A graph G is said to be K 1 , r -free if G does not contain an induced subgraph isomorphic to K 1 , r . Let k , r , t be integers with k ≥ 2 and t ≥ 3 . In this paper, we prove that if G is a K 1 , r -free graph of order at least ( k − 1 ) ( t ( r − 1 ) + 1 ) + 1 with δ ( G ) ≥ t and r ≥ 2 t − 1 , then G contains k vertex-disjoint copies of K 1 , t . This result shows that the conjecture in Fujita (2008) is true for r ≥ 2 t − 1 and t ≥ 3 . Furthermore, we obtain a weaker version of Fujita’s conjecture, that is, if G is a K 1 , r -free graph of order at least ( k − 1 ) ( t ( r − 1 ) + 1 + ( t − 1 ) ( t − 2 ) ) + 1 with δ ( G ) ≥ t and r ≥ 6 , then G contains k vertex-disjoint copies of K 1 , t .