### Abstract

Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m, m + 1,..., n}, if G has a cycle of length l, then G is called [m, n]-pancyclic, and if G has a cycle of length l which contains u, then G is called [m, n]-vertex pancyclic. Let δ(G) be a minimum degree of G and let N_{G}(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81-91] Liu, Lou and Zhao proved that if |N_{G}(u) ∪ N_{G}(v)| + δ(G) ≥ n + 1 for any nonadjacent vertices u, v of G, then G is [3, n]-vertex pancyclic. In this paper, we prove if n ≥ 6 and |N_{G}(u)∪N_{G}(v)|+d_{G}(w) ≥ n for every triple independent vertices u, v, w of G, then (i) G is [3,n]-pancyclic or isomorphic to the complete bipartite graph K_{n/2,n/2}, and (ii) G is [5, n]-vertex pancyclic or isomorphic to the complete bipartite graph K_{n/2,n/2}.

Original language | English |
---|---|

Pages (from-to) | 15-25 |

Number of pages | 11 |

Journal | Australasian Journal of Combinatorics |

Volume | 40 |

Publication status | Published - 2008 |

Externally published | Yes |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*40*, 15-25.

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

Research output: Contribution to journal › Article

*Australasian Journal of Combinatorics*, vol. 40, pp. 15-25.

}

TY - JOUR

T1 - A neighborhood and degree condition for pancyclicity and vertex pancyclicity

AU - Matsubara, Ryota

AU - Tsugaki, Masao

AU - Yamashita, Tomoki

PY - 2008

Y1 - 2008

N2 - Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m, m + 1,..., n}, if G has a cycle of length l, then G is called [m, n]-pancyclic, and if G has a cycle of length l which contains u, then G is called [m, n]-vertex pancyclic. Let δ(G) be a minimum degree of G and let NG(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81-91] Liu, Lou and Zhao proved that if |NG(u) ∪ NG(v)| + δ(G) ≥ n + 1 for any nonadjacent vertices u, v of G, then G is [3, n]-vertex pancyclic. In this paper, we prove if n ≥ 6 and |NG(u)∪NG(v)|+dG(w) ≥ n for every triple independent vertices u, v, w of G, then (i) G is [3,n]-pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2, and (ii) G is [5, n]-vertex pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2.

AB - Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m, m + 1,..., n}, if G has a cycle of length l, then G is called [m, n]-pancyclic, and if G has a cycle of length l which contains u, then G is called [m, n]-vertex pancyclic. Let δ(G) be a minimum degree of G and let NG(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81-91] Liu, Lou and Zhao proved that if |NG(u) ∪ NG(v)| + δ(G) ≥ n + 1 for any nonadjacent vertices u, v of G, then G is [3, n]-vertex pancyclic. In this paper, we prove if n ≥ 6 and |NG(u)∪NG(v)|+dG(w) ≥ n for every triple independent vertices u, v, w of G, then (i) G is [3,n]-pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2, and (ii) G is [5, n]-vertex pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2.

UR - http://www.scopus.com/inward/record.url?scp=84885718082&partnerID=8YFLogxK

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M3 - Article

VL - 40

SP - 15

EP - 25

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

ER -