Vertex-Disjoint Stars In K-1,K-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, r >= 3 and t >= 2. Fujita (2008) conjectured that if G is a K-1,K-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,K-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 vertical bar V(G)vertical bar >= (k - 1)(t(2) - 2) + 1 and t >= r >= 4. (C) 2021 Elsevier B.V. All rights reserved.