### Every 3-connected, essentially 11-connected line graph is hamiltonian

*Hong-Jian Lai, Yehong Shao, Ju Zhou, Hehui Wu*

#### Abstract

Thomassen conjectured that every 4-connected line graph is hamiltonian. A vertex cut X of G is essential if G-X has at least two nontrivial components. We prove that every 3-connected, essentially 11-connected line graph is hamiltonian. Using Ryjáček's line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is hamiltonian.

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