## Abstract

For a graph G, δ denotes the minimum degree of G. In 1971, Bondy proved that, if G is a 2-connected graph of order n and d(x)+d(y)<n for each pair of non-adjacent vertices x, y in G, then G is pancyclic or G=Kn _{2,n2}. In 2006, Wu et al. proved that, if G is a 2-connected graph of order n<6 and |N(x)∪N(y)|+δ<n for each pair of non-adjacent vertices x, y of d(x,y)=2 in G, then G is pancyclic or G=Kn_{2,n2}. In this paper, we introduce a new condition which generalizes two conditions of degree sum and neighborhood union and prove that, if G is a 2-connected graph of order n<6 and |N(x)∪N(y)|+d(w)<n for any three vertices x, y, w of d(x,y)=2 and wx or wy∉E(G) in G, then G is pancyclic or G=Kn _{2,n2}. This result also generalizes the above two results.

Original language | English (US) |
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Pages (from-to) | 218-223 |

Number of pages | 6 |

Journal | Discrete Applied Mathematics |

Volume | 160 |

Issue number | 3 |

DOIs | |

State | Published - Feb 2012 |

Externally published | Yes |

## Keywords

- Degree sum
- Neighborhood union
- Pancyclic graphs