Perfect matchings in large uniform hypergraphs with large minimum collective degree

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.