## Abstract

A graph G is quadrangularly connected if for every pair of edges e_{1} and e_{2} in E (G), G has a sequence of l-cycles (3 ≤ l ≤ 4)C_{1}, C_{2}, ..., C_{r} such that e_{1} ∈ E (C_{1}) and e_{2} ∈ E (C_{r}) and E (C_{i}) ∩ E (C_{i + 1}) ≠ ∅ for i = 1, 2, ..., r - 1. In this paper, we show that every quadrangularly connected claw-free graph without vertices of degree 1, which does not contain an induced subgraph H isomorphic to either G_{1} or G_{2} such that N_{1} (x, G) of every vertex x of degree 4 in H is disconnected is hamiltonian, which implies a result by Z. Ryjáček [Hamiltonian circuits in N_{2}-locally connected K_{1, 3}-free graphs, J. Graph Theory 14 (1990) 321-331] and other known results.

Original language | English |
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Pages (from-to) | 1205-1211 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 307 |

Issue number | 9-10 |

DOIs | |

Publication status | Published - 6 May 2007 |

## Keywords

- Claw-free graph
- Cycle
- Quadrangularly connected