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

Abstract

Let $H$ be a $k$-uniform $D$-regular simple hypergraph on $N$ vertices. Based on an analysis of the Rödl 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 $ND^{-1/(k-1)-η}$ vertices for some $η= Θ(k^{-3}) > 0$, when $N$ and $D$ are sufficiently large. Our approach consists of showing that the Rödl 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 Rödl nibble process. Based on this, we also improve the results of Kostochka and Rödl (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).

Figures (2)

• Figure 1: Improving the matching $M$ using augmenting stars
• Figure 2: An example of an instance of $X_i(x,y)$ and $Y_i(x,y)$ respectively.

Theorems & Definitions (71)

• Theorem 1.1: Alon, Kim and Spencer AKS1997
• Theorem 1.2: Kostochka and Rödl KR1998
• Theorem 1.3: Vu vu2000
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Conjecture 1.7
• Lemma 3.1: Martingale inequality AKS1997
• Lemma 3.2: Chernoff-Hoeffding inequality janson2000random
• Proposition 3.3