On s-Hamiltonian-Connected Line Graphs

For an integer s >= 0, G is s-hamiltonian-connected if for any vertex subset S subset of V (G) with |S| <= s, G - S is hamiltonian-connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324]), and Kuzel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian-connected (see [Z. Ryjacek and P. Vrana, Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs, J. Graph Theory 66 (2011) 152-173]). In this paper we prove the following. (i) For s >= 3, every (s + 4)-connected line graph is s-hamiltonian-connected. (ii) For s >= 0, every (s + 4)-connected line graph of a claw-free graph is s-hamiltonian-connected.