Almost perfect matchings in random 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)...

Perfect Matchings in Random s-Uniform Hypergraphs

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

The copyright law of the United States (title 17, U.S. Code) governs the making of photocopies or other reproductions of copyrighted material. Any copying of this document without permission of its author may be prohibited by law. Perfect matchings in random i—regular, 5—uniform hypergraphs.

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.

Packing perfect matchings in random hypergraphs

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer k ≥ 3, the binomial k-uniform random hypergraph H n,p contains N := (1 − o(1)) ( n−1 k−1 ) p edge-disjoint perfect matchings, provided p ≥ log C n nk−1 , where C := C(k) is an integer dependin...