## Abstract

An n-vertex graph is called pancyclic if it contains a cycle of length t for all 3> t >n. In this article, we study pancyclicity of random graphs in the context of resilience, and prove that if p>n^{-1/2}, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than edges is pancyclic. This result is best possible in two ways. First, the range of p is asymptotically tight; second, the proportion of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich et al. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers).

Original language | English |
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Pages (from-to) | 142-158 |

Number of pages | 17 |

Journal | Journal of Graph Theory |

Volume | 71 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2012 |

Externally published | Yes |

## Keywords

- Hamiltonicity
- pancyclicity
- random graph
- resilience