### Abstract

Original language | Undefined |
---|---|

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

*Journal of graph theory*,

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

}

*Journal of graph theory*, vol. 68, no. 4, pp. 285-298. https://doi.org/10.1002/jgt.20558

**Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs.** / Li, MingChu; Chen, Xiaodong; Broersma, Haitze J.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Hamiltonian connectedness in 4-connected hourglass-free claw-free graphs

AU - Li, MingChu

AU - Chen, Xiaodong

AU - Broersma, Haitze J.

PY - 2011/12

Y1 - 2011/12

N2 - 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].

AB - 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].

KW - EWI-21342

KW - MSC-05C

KW - Claw-free graph

KW - hourglass-free graph

KW - IR-79452

KW - Hamiltonian connectedness

U2 - 10.1002/jgt.20558

DO - 10.1002/jgt.20558

M3 - Article

VL - 68

SP - 285

EP - 298

JO - Journal of graph theory

JF - Journal of graph theory

SN - 0364-9024

IS - 4

ER -