Matchings in 3-uniform hypergraphs

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−12)−(2n/32), then H contains a perfect matching. This bound is tight and answers a question of Hàn, Person and Schacht. More generally, we show that H contains a matching of size d⩽n/3 if its minimum vertex degree is greater than (n−12)−(n−d2), which is also best possible. This extends a result of Bollobás, Daykin and Erdős.