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 ⊂ V ( G ) with | X | = k . In the paper, we prove that (i) 5-connected claw-free graph with minimum degree at least 6 is 1-Hamilton-connected, 4-connected claw-free hourglass-free graph is 1-Hamilton-connected. byproduct, we also show that every 5-connected line graph with minimum degree at least 6 is 3-hamiltonian.