On 1-Hamilton-connected claw-free graphs

A graph G is k-Hamilton-connected (k-hamiltonian) if G-X is Hamilton-connected (hamiltonian) for every set X subset of V(G) with vertical bar X vertical bar = k. In the paper, we prove that (i) every 5-connected claw-free graph with minimum degree at least 6 is 1-Hamilton-connected, (ii) every 4-connected claw-free hourglass-free graph is 1-Hamilton-connected. As a byproduct, we also show that every 5-connected line graph with minimum degree at least 6 is 3-hamiltonian. (C) 2013 Elsevier B.V. All rights reserved.