Matchings in vertex-transitive bipartite graphs

TL;DR

This paper strengthens existing bounds on perfect matchings in vertex-transitive bipartite graphs, explores their asymptotic behavior, and improves bounds on matching sizes, with implications for graph sequences and combinatorial properties.

Contribution

It provides a stronger version of Gurvits's theorem for vertex-transitive bipartite graphs, establishes convergence results for perfect matchings in graph sequences, and improves bounds on matching polynomial coefficients.

Findings

Stronger lower bounds on the number of perfect matchings in vertex-transitive bipartite graphs.

Convergence of the normalized logarithm of perfect matchings in Benjamini--Schramm convergent graph sequences.

Improved bounds on the coefficients of the matching polynomial for these graphs.

Abstract

A theorem of A. Schrijver asserts that a $d$-regular bipartite graph on $2n$ vertices has at least $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n$$ perfect matchings. L. Gurvits gave an extension of Schrijver's theorem for matchings of density $p$. In this paper we give a stronger version of Gurvits's theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer $k$, there exists a positive constant $c(k)$ such that if a $d$-regular vertex-transitive bipartite graph on $2n$ vertices contains a cycle of length at most $k$, then it has at least $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}+c(k)\right)^n$$ perfect matchings. We also show that if $(G_i)$ is a Benjamini--Schramm convergent graph sequence of vertex-transitive bipartite graphs, then $$\frac{\ln pm(G_i)}{v(G_i)}$$ is convergent, where $pm(G)$ and $v(G)$ denote the number of perfect matchings and the number of vertices of $G$, respectively. We also show that if $G$ is $d$-regular vertex-transitive bipartite graph on $2n$ vertices and $m_k(G)$ denote the number of matchings of size $k$, and $$M(G,t)=1+m_1(G)t+m_2(G)t^2+\dots +m_n(G)t^n=\prod_{k=1}^n(1+\gamma_k(G)t),$$ where $\gamma_1(G)\leq \dots \leq \gamma_n(G)$, then $$\gamma_k(G)\geq \frac{d^2}{4(d-1)}\frac{k^2}{n^2},$$ and $$\frac{m_{n-1}(G)}{m_n(G)}\leq \frac{2}{d}n^2.$$ The latter result improves on a previous bound of C. Kenyon, D. Randall and A. Sinclair. There are examples of $d$-regular bipartite graphs for which these statements fail to be true without the condition of vertex-transitivity.