Matchings in Benjamini-Schramm convergent graph sequences

TL;DR

This paper studies the asymptotic behavior of matchings in large sparse graphs, introducing the matching measure, and proves estimability of the normalized logarithm of matchings in certain graph sequences, with applications to random regular bipartite graphs.

Contribution

It introduces the matching measure for finite graphs, analyzes its behavior in graph sequences, and establishes estimability of the normalized logarithm of matchings, including perfect matchings in specific cases.

Findings

Normalized log of matchings is estimable in Benjamini-Schramm convergent sequences.

Normalized log of perfect matchings converges for d-regular bipartite graphs converging to the d-regular tree.

Limit matches Schrijver's lower bound for perfect matchings in random d-regular bipartite graphs.

Abstract

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini-Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d-regular bipartite graphs for any fixed d at least 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge. However, for any sequence of d-regular bipartite graphs converging to the d-regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d-regular bipartite graphs. We show that the limit equals to the exponent in Schrijver's lower bound on the number of perfect matchings. Our analytic approach also yields a short proof for the Nguyen-Onak (also Elek--Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs.