Hamiltonian properties of almost locally connected claw-free graphs

A vertex v of a graph G is locally connected if the set of neighbors N(v) of v induces a connected subgraph in G. Let B(G) denote the set of vertices of G that are not locally connected. Then G is almost locally connected if B(G) is an independent set and for any x ∈ B(G), there is a vertex y in V (G) \ {x} such that N(x) ∪ {y} induces a connected subgraph of G. The main result of this paper is that an almost locally connected claw-free graph on at least 4 vertices is Hamilton-connected if and only if it is 3-connected. This generalizes the result by Asratian that a locally connected claw-free graph on at least 4 vertices is Hamilton-connected if and only if it is 3-connected [Journal of Graph Theory 23 (1996) 191–201].