On computing a conditional edge-connectivity of a graph
Information Processing Letters
On Hamiltonian line graphs and connectivity
Discrete Mathematics
Graphs without spanning closed trails
Discrete Mathematics
On a closure concept in claw-free graphs
Journal of Combinatorial Theory Series B
Graph Theory With Applications
Graph Theory With Applications
Every 3-connected, essentially 11-connected line graph is Hamiltonian
Journal of Combinatorial Theory Series B
Hamilton connectivity of line graphs and claw-free graphs
Journal of Graph Theory
Hamiltonian connectedness in 3-connected line graphs
Discrete Applied Mathematics
Note: Hamiltonicity of 6-connected line graphs
Discrete Applied Mathematics
Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs
Journal of Graph Theory
Hamilton cycles in 5-connected line graphs
European Journal of Combinatorics
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Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai [Z.-H. Chen, H.-J. Lai, Reduction techniques for super-Eulerian graphs and related topics-an update, in: Ku Tung-Hsin (Ed.), Combinatorics and Graph Theory, vol. 95, World Scientific, Singapore/London, 1995, pp. 53-69, Conjecture 8.6] conjectured that every 3-edge connected, essentially 6-edge connected graph is collapsible. In this paper, we prove the following results. (1) Every 3-edge connected, essentially 6-edge connected graph with edge-degree at least 7 is collapsible. (2) Every 3-edge connected, essentially 5-edge connected graph with edge-degree at least 6 and at most 24 vertices of degree 3 is collapsible which implies that 5-connected line graph with minimum degree at least 6 of a graph with at most 24 vertices of degree 3 is Hamiltonian. (3) Every 3-connected, essentially 11-connected line graph is Hamilton-connected which strengthens the result in [H.-J. Lai, Y. Shao, H. Wu, J. Zhou, Every 3-connected, essentially 11-connected line graph is Hamiltonian, J. Combin. Theory, Ser. B 96 (2006) 571-576] by Lai et al. (4) Every 7-connected line graph is Hamiltonian connected which is proved by a method different from Zhan's. By using the multigraph closure introduced by Ryjacek and Vrana which turns a claw-free graph into the line graph of a multigraph while preserving its Hamilton-connectedness, the results (3) and (4) can be extended to claw-free graphs.