2-Factors in claw-free graphs with locally disconnected vertices

Abstract

An edge of G is singular if it does not lie on any triangle of G; otherwise, it is non-singular. A vertex u of a graph G is called locally connected if the induced subgraph G[N(u)] by its neighborhood is connected; otherwise, it is called locally disconnected.In this paper, we prove that if a connected claw-free graph G of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex v of degree at least 3 in G, there is a nonnegative integer s such that v lies on an induced cycle of length at least 4 with at most s non-singular edges and with at least s-5 locally connected vertices; (ii) for each locally disconnected vertex v of degree 2 in G, there is a nonnegative integer s such that v lies on an induced cycle C with at most s non-singular edges and with at least s-3 locally connected vertices and such that G[V (C)∩V 2(G)] is a path or a cycle, then G has a 2-factor, and it is the best possible in some sense. This result generalizes two known results in Faudree, Faudree and Ryjáček (2008) and in Ryjáček, Xiong and Yoshimoto (2010).