Geodesic
On $s$-hamiltonian-connected line graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 297-315

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For an integer s≥ 0, 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 [Reflections on graph theory, J. Graph Theory 10 (1986) 309–324]), and Kužel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian-connected (see [Z. Ryjáček and P. Vrána, 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.
Keywords: line graph, claw-free graph, $s$-hamiltonian-connected, collapsible graphs, reductions 
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Ma, Xiaoling; Lai, Hong-Jian; Zhan, Mingquan; Zhang, Taoye; Zhou, Ju. On $s$-hamiltonian-connected line graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 1, pp. 297-315. http://geodesic.mathdoc.fr/item/DMGT_2024_44_1_a14/