Near Perfect Matchings in $k$-uniform Hypergraphs II

TL;DR

This paper investigates the conditions under which near perfect matchings exist in $k$-uniform hypergraphs, introducing a divisibility barrier construction and proposing a conjecture on degree thresholds, verified in specific cases.

Contribution

It generalizes the divisibility barrier concept for perfect matchings to near perfect matchings and proposes a new conjecture on degree thresholds, using lattice-based absorbing methods.

Findings

Divisibility barrier construction prevents near perfect matchings.

Conjecture on minimum degree thresholds for near perfect matchings.

Verification of the conjecture in various cases.

Abstract

Suppose $k\nmid n$ and $H$ is an $n$-vertex $k$-uniform hypergraph. A near perfect matching in $H$ is a matching of size $\lfloor n/k\rfloor$. We give a divisibility barrier construction that prevents the existence of near perfect matchings in $H$. This generalizes the divisibility barrier for perfect matchings. We give a conjecture on the minimum $d$-degree threshold forcing a (near) perfect matching in $H$ which generalizes a well-known conjecture on perfect matchings. We also verify our conjecture in various cases. Our proof makes use of the lattice-based absorbing method that the author used recently to solve two other problems on matching and tilings for hypergraphs.