### Abstract

A subset C of edges in a k-uniform hypergraph H is a loose Hamilton cycle if C covers all the vertices of H and there exists a cyclic ordering of these vertices such that the edges in C are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random k-uniform hypergraph H ^{k} _{n,p} has vertex set [n] and an edge set E obtained by adding each k-tuple e ∈ (^{[n]} _{k}) to E with probability p, independently at random. Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but o(|E|) edges, referred to as the packing problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle in H ^{k} _{n,p} is {equation presented} the best known bounds for the packing problem are around p = polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: for {equation presented}, a random k-uniform hypergraph H ^{k} _{n,p} with high probability contains {equation presented} edge-disjoint loose Hamilton cycles. Our proof utilizes and modifies the idea of 'online sprinkling' recently introduced by Vu and the first author.

Original language | English (US) |
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Pages (from-to) | 839-849 |

Number of pages | 11 |

Journal | Combinatorics Probability and Computing |

Volume | 26 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2017 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

### Keywords

- 2010 Mathematics subject classification:Primary 05C80 Secondary 05C65

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## Cite this

*Combinatorics Probability and Computing*,

*26*(6), 839-849. https://doi.org/10.1017/S0963548317000402