Matchings in 3-uniform hypergraphs of large minimum vertex degree

Abstract

Abstract We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than ( n − 1 2 ) − ( 2 n / 3 2 ) , then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we determine the minimum vertex degree threshold that ensures that H contains a matching of size d ⩽ n / 3 .