### Abstract

We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of ⌊ n / k ⌋ disjoint edges. Let δ_{k - 1} (H) be the largest integer d such that every (k - 1)-element set of vertices of H belongs to at least d edges of H. In this paper we study the relation between δ_{k - 1} (H) and the presence of a perfect matching in H for k ≥ 3. Let t (k, n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δ_{k - 1} (H) ≥ t contains a perfect matching. For large n divisible by k, we completely determine the values of t (k, n), which turn out to be very close to n / 2 - k. For example, if k is odd and n is large and even, then t (k, n) = n / 2 - k + 2. In contrast, for n not divisible by k, we show that t (k, n) ∼ n / k. In the proofs we employ a newly developed "absorbing" technique, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.

Original language | English |
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Pages (from-to) | 613-636 |

Number of pages | 24 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 116 |

Issue number | 3 |

DOIs | |

Publication status | Published - Apr 1 2009 |

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### Keywords

- Hypergraph
- Minimum degree
- Perfect matching

### ASJC Scopus subject areas

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

### Cite this

*Journal of Combinatorial Theory. Series A*,

*116*(3), 613-636. https://doi.org/10.1016/j.jcta.2008.10.002