### Abstract

Let G be a 2-connected graph of order n with x,y ∈ V(G). For u,v ∈ V(G), let P _{i}[u, v] denote the path with i vertices which connects u and v. In this paper, we prove that if n ≥ 5 and |N _{G}(u)∪N _{G}(v)| +d _{G}(w) ≥ n+1 for every triple of independent vertices u,v,w of G, then there exists a P _{i}[x,y] in G for 5 ≤ i ≤ n, or G belongs to one of three exceptional classes. This implies a positive answer to a conjecture by Wei and Zhu [Graphs Combin. 14 (1998), 263-274].

Original language | English |
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Pages (from-to) | 3-10 |

Number of pages | 8 |

Journal | Australasian Journal of Combinatorics |

Volume | 47 |

Publication status | Published - 2010 Jun 10 |

Externally published | Yes |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

Matsubara, R., Tsugaki, M., & Yamashita, T. (2010). A neighborhood and degree condition for panconnectivity.

*Australasian Journal of Combinatorics*,*47*, 3-10.