### 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 |
---|---|

Pages (from-to) | 3-10 |

Number of pages | 8 |

Journal | Australasian Journal of Combinatorics |

Volume | 47 |

Publication status | Published - 2010 |

Externally published | Yes |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*47*, 3-10.

**A neighborhood and degree condition for panconnectivity.** / Matsubara, Ryota; Tsugaki, Masao; Yamashita, Tomoki.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - A neighborhood and degree condition for panconnectivity

AU - Matsubara, Ryota

AU - Tsugaki, Masao

AU - Yamashita, Tomoki

PY - 2010

Y1 - 2010

N2 - 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].

AB - 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].

UR - http://www.scopus.com/inward/record.url?scp=77953158515&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953158515&partnerID=8YFLogxK

M3 - Article

VL - 47

SP - 3

EP - 10

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

ER -