A Note on Perfect Matchings in Uniform Hypergraphs

A Note on Perfect Matchings in Uniform Hypergraphs

We determine the exact minimum `-degree threshold for perfect matchings in kuniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than 12 ( n k−` ) . This extends our previous results that determine the minimum `-degree thresholds for perfect matchings in k-uniform hypergraphs for all ` > k/2 and provides two new (exact) thresholds: (k, `) =...

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 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.

Perfect Matchings in Random s-Uniform Hypergraphs

A Note on Perfect Matchings in Uniform Hypergraphs with Large Minimum Collective Degree

For an integer k ≥ 2 and a k-uniform hypergraph H, let δk−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. Further, let t(k, n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δk−1(H) ≥ t contains a perfect matching. The parameter t(k, n) has been completely determined for all k and large n d...