Asymptotics of the Upper Matching Conjecture

TL;DR

This paper establishes near-optimal upper bounds on the number of matchings of a given size in bipartite and regular graphs, advancing the understanding of the Upper Matching Conjecture.

Contribution

It provides the best known upper bounds for matchings in regular graphs, approaching the conjectured limits up to a small error term.

Findings

Bounds are tight up to an exponential error factor diminishing with degree d.

Progress represents the most significant advancement on the Upper Matching Conjecture to date.

Results apply to both bipartite and general graphs with specified degrees.

Abstract

We give upper bounds for the number $\Phi_\ell(G)$ of matchings of size $\ell$ in (i) bipartite graphs $G=(X\cup Y, E)$ with specified degrees $d_x$ ($x\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular, for $d$-regular, $N$-vertex graphs, our bound is best possible up to an error factor of the form $\exp[o_d(1)N]$, where $o_d(1) \rightarrow 0$ as $d \rightarrow \infty$. This represents the best progress to date on the "Upper Matching Conjecture" of Friedland, Krop, Lundow and Markstr\"om. Some further possibilities are also suggested.