Vertex-disjoint stars 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,r≥3 and t≥2. Fujita (2008) conjectured that if G is a K1,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 K1,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)(t2−2)+1 and t≥r≥4.