Perfect Matchings in Random r-regular, s-uniform Hypergraphs

Perfect Matchings in Random r-regular, s-uniform Hypergraphs

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Perfect Matchings in Random s-Uniform Hypergraphs

Perfect matchings in random uniform hypergraphs

In the random k-uniform hypergraph Hk(n, p) on a vertex set V of size n, each subset of size k of V independently belongs to it with probability p. Motivated by a theorem of Erdős and Rényi [6] regarding when a random graph G(n, p) = H2(n, p) has a perfect matching, Schmidt and Shamir [14] essentially conjectured the following. Conjecture Let k|n for fixed k ≥ 3, and the expected degree d(n, p)...

Nearly perfect matchings in regular simple hypergraphs

For every fixed k ≥ 3 there exists a constant ck with the following property. Let H be a kuniform, D-regular hypergraph on N vertices, in which no two edges contain more than one common vertex. If k > 3 then H contains a matching covering all vertices but at most ckND. If k = 3, then H contains a matching covering all vertices but at most c3ND lnD. This improves previous estimates and implies, ...

Perfect matchings in 4-uniform hypergraphs

A perfect matching in a 4-uniform hypergraph is a subset of b4 c disjoint edges. We prove that if H is a sufficiently large 4-uniform hypergraph on n = 4k vertices such that every vertex belongs to more than ( n−1 3 ) − ( 3n/4 3 ) edges then H contains a perfect matching. This bound is tight and settles a conjecture of Hán, Person and Schacht.