We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of ⌊ n / k ⌋ disjoint edges. Let δk - 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 δk - 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 δk - 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.
- Minimum degree
- Perfect matching
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics