The number of perfect matchings, and the nesting properties, of random regular graphs

TL;DR

This paper establishes the asymptotic normality of the number of perfect matchings in certain random regular graphs and explores their nesting properties, providing new distributional results and coupling techniques for spanning subgraphs.

Contribution

It proves the first distributional result for spanning subgraphs of random regular graphs as degree grows, and introduces coupling methods for subgraph inclusion.

Findings

Number of perfect matchings is asymptotically normal under specified conditions.

Coupling of ${ m G}(n,d-1)$ and ${ m G}(n,d)$ with high probability.

Almost sure subgraph inclusion relations for ${ m G}(n,d)$ and ${ m G}(n,d')$.

Abstract

We prove that the number of perfect matchings in ${\mathcal G}(n,d)$ is asymptotically normal when $n$ is even, $d\to\infty$ as $n\to\infty$, and $d=O(n^{1/7}/\log^2 n)$. This is the first distributional result of spanning subgraphs of ${\mathcal G}(n,d)$ when $d\to\infty$. Moreover, we prove that ${\mathcal G}(n,d-1)$ and ${\mathcal G}(n,d)$ can be coupled so that ${\mathcal G}(n,d-1)$ is a subgraph of ${\mathcal G}(n,d)$ with high probability when $d\to\infty$ and $d=o(n^{1/3})$. Further, if $d=\Omega(\log^7 n)$, $d=O(n^{1/7}/\log^2n)$, and $d\le d'\le n-1$ then ${\mathcal G}(n,d)$ and ${\mathcal G}(n,d')$ can be coupled so that asymptotically almost surely ${\mathcal G}(n,d)$ is a subgraph of ${\mathcal G}(n,d')$.