New bounds on the size of Nearly Perfect Matchings in almost regular hypergraphs

TL;DR

This paper improves bounds on the size of nearly perfect matchings in almost regular hypergraphs by analyzing the R"odl nibble process and introducing well-distributed augmenting stars, leading to broader combinatorial and design applications.

Contribution

It provides a tighter bound on the size of matchings in almost regular hypergraphs for all k > 3, using an enhanced analysis of the R"odl nibble process and augmenting structures.

Findings

Improved bounds on matchings in hypergraphs with large D and N.

Enhanced analysis of the R"odl nibble process with augmenting stars.

Broader implications for combinatorial designs and hypergraph coloring.

Abstract

Let $H$ be a $k$-uniform $D$-regular simple hypergraph on $N$ vertices. Based on an analysis of the R\"odl nibble, Alon, Kim and Spencer (1997) proved that if $k \ge 3$, then $H$ contains a matching covering all but at most $ND^{-1/(k-1)+o(1)}$ vertices, and asked whether this bound is tight. In this paper we improve their bound by showing that for all $k > 3$, $H$ contains a matching covering all but at most $ND^{-1/(k-1)-\eta}$ vertices for some $\eta = \Theta(k^{-3}) > 0$, when $N$ and $D$ are sufficiently large. Our approach consists of showing that the R\"odl nibble process not only constructs a large matching but it also produces many well-distributed `augmenting stars' which can then be used to significantly improve the matching constructed by the R\"odl nibble process. Based on this, we also improve the results of Kostochka and R\"odl (1998) and Vu (2000) on the size of matchings in almost regular hypergraphs with small codegree. As a consequence, we improve the best known bounds on the size of large matchings in combinatorial designs with general parameters. Finally, we improve the bounds of Molloy and Reed (2000) on the chromatic index of hypergraphs with small codegree (which can be applied to improve the best known bounds on the chromatic index of Steiner triple systems and more general designs).