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

A graph G is said to be K-1,K-r-free if G does not contain an induced subgraph isomorphic to K-1,K-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,K-r-free graph of order at least (k - 1)(t(r - 1)+1) + 1 with delta(G) >= t and r >= 2t 1, then G contains k vertex-disjoint copies of K-1,K- 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 K-1,K-r-free graph of order at least (k - 1)(t(r - 1) + 1 (t - 1)(t - 2)) + 1 with delta(G) >= t and r >= 6, then G contains k vertex-disjoint copies of K-1,K-t. (C) 2016 Elsevier B.V. All rights reserved.