## Abstract

Let H_{r}(n,p) denote the maximum number of Hamiltonian cycles in an n-vertex r-graph with density p∈(0,1). The expected number of Hamiltonian cycles in the random r-graph model G_{r}(n,p) is E(n,p)=p^{n}(n−1)!/2 and in the random graph model G_{r}(n,m) with m=p(nr) it is, in fact, slightly smaller than E(n,p). For graphs, H_{2}(n,p) is proved to be only larger than E(n,p) by a polynomial factor and it is an open problem whether a quasi-random graph with density p can be larger than E(n,p) by a polynomial factor. For hypergraphs (i.e. r≥3) the situation is drastically different. For all r≥3 it is proved that H_{r}(n,p) is larger than E(n,p) by an exponential factor and, moreover, there are quasi-random r-graphs with density p whose number of Hamiltonian cycles is larger than E(n,p) by an exponential factor.

Original language | English |
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Pages (from-to) | 195-222 |

Number of pages | 28 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 153 |

DOIs | |

State | Published - Mar 2022 |

### Bibliographical note

Funding Information:This research was supported by the Israel Science Foundation (grant No. 1082/16 ).

Publisher Copyright:

© 2021 Elsevier Inc.

## Keywords

- Hamiltonian cycle
- Quasi-random hypergraph
- r-graph

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics