Collapsible graphs and Hamiltonian connectedness of line graphs

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 Ryjáček and Vrána 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.