Sandwiching between random regular graphs and Erd\H{o}s-R\'enyi graphs: configuration model and unions of perfect matchings

TL;DR

This paper develops new couplings among various random graph models, proving the Kim-Vusandwich conjecture for large degrees and a weakened version for smaller degrees, advancing understanding of random regular graphs.

Contribution

It introduces a coupling framework connecting random regular graphs with Erdős-Rényi graphs, resolving the Kim-Vusandwich conjecture for large degrees.

Findings

Verified the Kim-Vusandwich conjecture for all large degrees.

Proved a weakened version of the conjecture for smaller degrees.

Established new couplings among random graph models.

Abstract

We establish new couplings among several random graph and multigraph models related to the random regular graph $G(n,d)$, including the configuration model and unions of random perfect matchings. As a main result, we verify the Kim-Vusandwich conjecture for all large degrees $d=n-O(\log^4 n)$ and prove a weakened version for $d=O(\log^4 n)$, which are the only remaining open cases. Our approach introduces a coupling framework that links $G(n,d)$ and $G(n,p)$ through a chain of intermediate models.