Disjoint cycles with partial degree conditions in claw-free graphs

Abstract

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K1,3. Let G=(V,E) be a claw-free graph of order n and let W be a subset of V with |W|≥3k, where k is a positive integer. In this paper, we show that if the degree of each vertex of W is at least (n+1)/2, then G contains k disjoint cycles such that each of them contains at least three vertices of W. The degree condition is sharp in a sense. This generalizes Wang's result (2015) [13] in claw-free graphs and the theorem of Faudree et al. (2012) [9] when n is odd.