## Abstract

A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on n vertices with minimum out-degree and in-degree at least n/2 contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph D(n,p), that is, a directed graph in which every ordered pair (u, v) becomes an arc with probability p independently of all other pairs. Motivated by the study of resilience of properties of random graphs, we prove that if p ≫ log n/√n, then a.a.s. every subdigraph of D(n,p) with minimum out-degree and in-degree at least (1/2+o(1))np contains a directed Hamilton cycle. The constant 1/2 is asymptotically best possible. Our result also strengthens classical results about the existence of directed Hamilton cycles in random directed graphs.

Original language | English |
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Pages (from-to) | 345-362 |

Number of pages | 18 |

Journal | Random Structures and Algorithms |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - 1 Sep 2016 |

Externally published | Yes |

## Keywords

- Hamilton cycle
- random digraph
- resilience