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

A graph G is said to be K1,r-free if G does not contain an induced subgraph isomorphic to K1,r. Let k,r,t be integers with k≥2 and t≥3. In this paper, we prove that if G is a K1,r-free graph of order at least (k−1)(t(r−1)+1)+1 with δ(G)≥t and r≥2t−1, then G contains k vertex-disjoint copies of K1,t. This result shows that the conjecture in Fujita (2008) is true for r≥2t−1 and t≥3. Furthermore, we obtain a weaker version of Fujita's conjecture, that is, if G is a K1,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 K1,t.