### Abstract

An hourglass is the only graph with degree sequence 4, 2, 2, 2, 2 (i.e. two triangles meeting in exactly one vertex). There are infinitely many claw-free graphs G such that G is not hamiltonian connected while its Ryjác̆ek closure cl(G) is hamiltonian connected. This raises such a problem what conditions can guarantee that a claw-free graph G is hamiltonian connected if and only if cl(G) is hamiltonian connected. In this paper, we will do exploration toward the direction, and show that a 3-connected $claw, (P_6)^2, hourglass$-free graph G with minimum degree at least 4 is hamiltonian connected if and only if cl(G) is hamiltonian connected, where $(P_6)^2$ is the square of a path $P_6$ on 6 vertices. Using the result, we prove that every 4-connected $claw, (P_6)^2, hourglass$-free graph is hamiltonian connected, hereby generalizing the result that every 4-connected hourglass-free line graph is hamiltonian connected by Kriesell [J Combinatorial Theory (B) 82 (2001), 306–315].

Original language | Undefined |
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Pages (from-to) | 285-298 |

Number of pages | 14 |

Journal | Journal of graph theory |

Volume | 68 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2011 |

### Keywords

- EWI-21342
- MSC-05C
- Claw-free graph
- hourglass-free graph
- IR-79452
- Hamiltonian connectedness

## Cite this

Li, M., Chen, X., & Broersma, H. J. (2011). Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs.

*Journal of graph theory*,*68*(4), 285-298. https://doi.org/10.1002/jgt.20558