Packing perfect matchings in random hypergraphs

TL;DR

This paper introduces online sprinkling, a new method for generating random hypergraphs, and proves that such hypergraphs contain nearly optimal numbers of edge-disjoint perfect matchings under certain probabilistic conditions.

Contribution

The paper presents a novel online sprinkling technique for random hypergraph generation and establishes near-optimal bounds on the number of edge-disjoint perfect matchings.

Findings

Hypergraphs contain asymptotically optimal numbers of perfect matchings.

The online sprinkling method effectively generates random hypergraphs.

Results are optimal up to polylogarithmic factors in n.

Abstract

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial $k$-uniform random hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matchings, provided $p\geq \frac{\log^{C}n}{n^{k-1}}$, where $C:=C(k)$ is an integer depending only on $k$. Our result for $N$ is asymptotically best optimal and for $p$ is optimal up to the $polylog(n)$ factor.