On Hamiltonicity of 3-Connected Claw-Free Graphs

Lai, Shao and Zhan (J Graph Theory 48:142-146, 2005) showed that every 3-connected N₂-locally connected claw-free graph is Hamiltonian. In this paper, we generalize this result and show that every 3-connected claw-free graph G such that every locally disconnected vertex lies on some induced cycle of length at least 4 with at most 4 edges contained in some triangle of G is Hamiltonian. It is best possible in some sense.