Abstract
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)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.
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