### Abstract

A vertex v of a graph G is locally connected if the set of neighbors N(v) of v induces a connected subgraph in G. Let B(G) denote the set of vertices of G that are not locally connected. Then G is almost locally connected if B(G) is an independent set and for any vertex x in B(G), there is a vertex y in V(G) - x such that N(x) ∪ y induces a connected subgraph of G. The main result of this paper is that an almost locally connected claw-free graph on at least 4 vertices is Hamilton-connected if and only if it is 3-connected. This generalizes the result by Asratian that a locally connected claw-free graph on at least 4 vertices is Hamilton-connected if and only if it is 3-connected [Journal of Graph Theory 23 (1996) 191–201].

Original language | Undefined |
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Pages (from-to) | 95-109 |

Number of pages | 15 |

Journal | Ars combinatoria |

Volume | 124 |

Publication status | Published - Jan 2016 |

### Keywords

- EWI-27151
- MSC-05C
- IR-101070
- METIS-318496
- Hamilton-connected
- Hamiltonian
- almost locally connected
- Claw-free graph

## Cite this

Chen, X., Li, M., Liao, W., & Broersma, H. J. (2016). Hamiltonian properties of almost locally connected claw-free graphs.

*Ars combinatoria*,*124*, 95-109.