Hamiltonian claw-free graphs with locally disconnected vertices

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: For each locally disconnected vertex v of G with degree at least 3, 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-3 locally connected vertices; for each locally disconnected vertex v of G with degree 2, 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-2 locally connected vertices and such that the subgraph induced by those vertices of C that have degree two in G is a path or a cycle, then G is Hamiltonian, and it is best possible in some sense. Our result is a common extension of two known results in Bielak (2000) and in Li (2002); hence also of the results in Oberly and Sumner (1979) and in Ryjáček (1990).