Vertex-disjoint stars in K1,r-free graphs

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 , r ≥ 3 and t ≥ 2. Fujita (2008) conjectured that if G is a K 1 , r-free graph of order at least ( k − 1 ) ( t ( r − 1 ) + 1 ) + 1 with minimum degree at least t, then G contains k disjoint copies of K 1 , t. In this paper, we show that this conjecture is true for the case r > t ≥ 3 and for the case r = t = 4, and we also show that this conjecture is true if | V ( G ) | ≥ ( k − 1 ) ( t 2 − 2 ) + 1 and t ≥ r ≥ 4.