Geodesic
Every $3$-connected, essentially $11$-connected line graph is hamiltonian
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005).

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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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     author = {Lai, Hong-Jian and Shao, Yehong and Zhou, Ju and Wu, Hehui},
     title = {Every $3$-connected, essentially $11$-connected line graph is hamiltonian},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)},
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     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3452/}
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Lai, Hong-Jian; Shao, Yehong; Zhou, Ju; Wu, Hehui. Every $3$-connected, essentially $11$-connected line graph is hamiltonian. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005). doi : 10.46298/dmtcs.3452. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3452/

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