On 1-Hamilton-connected claw-free graphs

A graph GG is kk-Hamilton-connected (kk-hamiltonian) if G−XG−X is Hamilton-connected (hamiltonian) for every set X⊂V(G)X⊂V(G) with |X|=k|X|=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.