Vertex Disjoint Copies Of K-1,K-4 In Claw-Free Graphs

A complete bipartite graph with partite sets X and Y, where vertical bar X vertical bar = 1 and vertical bar Y vertical bar = r, is denoted by K-1,K-r. A graph G is said to be claw-free if G does not contain K-1,K-3 as an induced subgraph. There are several well-known and important families of graphs that are claw-free such as line graphs and complements of triangle-free graphs. Claw-free graphs have numerous interesting properties and applications. This paper considers vertex disjoint K(1,4)s in claw-free graphs. Let k be an integer with k >= 2 and let G be a claw-free graph with vertical bar V(G)vertical bar >= 10k - 9. We prove that if the minimum degree of G is at least 4, then it contains k vertex disjoint K(1,4)s. This result answers the question in [Jiang, Chiba, Fujita, Yan, Discrete Math. 340 (2017) 649-654]. (C) 2020 Elsevier Inc. All rights reserved.