On s-hamiltonian line graphs of claw-free graphs

For an integer s≥0, a graph G is s-hamiltonian if for any vertex subset S⊆V(G) with |S|≤s, G−S is hamiltonian, and G is s-hamiltonian connected if for any vertex subset S⊆V(G) with |S|≤s, G−S is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Kučzel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjáček and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of s-hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for s≥2, a line graph L(G) is s-hamiltonian if and only if L(G) is (s+2)-connected. In this paper we prove the following. (i) For an integer s≥2, the line graph L(G) of a claw-free graph G is s-hamiltonian if and only if L(G) is (s+2)-connected. (ii) The line graph L(G) of a claw-free graph G is 1-hamiltonian connected if and only if L(G) is 4-connected.