Monotonicity and decompositions of random regular graphs

TL;DR

This paper establishes new monotonicity and decomposition results for random regular graphs, including couplings, tools for analyzing graph models, and bounds on perfect matchings, advancing understanding of their structure.

Contribution

It introduces new methods for analyzing random regular graphs, proves a conjecture on graph couplings, and refines bounds on perfect matchings.

Findings

Existence of couplings for different degrees with high probability

New tools for contiguity and total variation analysis

Refined estimates on the number of perfect matchings

Abstract

In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters $d_1 \leq d_2$, there exists a coupling of $G(n,d_1)$ and $G(n,d_2)$ satisfying that $G(n,d_1) \subseteq G(n,d_2)$ with high probability, confirming a conjecture of Gao, Isaev and McKay in a new regime. Our contributions include new tools for analysing contiguity and total variation distance between random regular graph models, a novel procedure for generating unions of random edge-disjoint perfect matchings, and refined estimates of Gao's bounds on the number of perfect matchings in random regular graphs. In addition, we make progress towards another conjecture of Isaev, McKay, Southwell and Zhukovskii.