A proof of the Kim-Vu sandwich conjecture

TL;DR

This paper proves the long-standing Kim-Vu sandwich conjecture, establishing a coupling between random regular graphs and binomial graphs for all degrees d=( ext{log} n), enabling transfer of properties between these models.

Contribution

The paper provides a complete proof of the Kim-Vu sandwich conjecture, extending previous results to all degrees d=( ext{log} n).

Findings

Confirmed the existence of the coupling for all d=( ext{log} n).

Extended the analysis of the coupling procedure to broader degree ranges.

Abstract

In 2004, Kim and Vu conjectured that, when $d=\omega(\log n)$, the random $d$-regular graph $G_d(n)$ can be sandwiched with high probability between two random binomial graphs $G(n,p)$ with edge probabilities asymptotically equal to $\frac{d}{n}$. That is, there should exist $p_*=(1-o(1))\frac{d}{n}$, $p^*=(1+o(1))\frac{d}{n}$ and a coupling $(G_*,G,G^*)$ such that $G_*\sim G(n,p_*)$, $G\sim G_d(n)$, $G^*\sim G(n,p^*)$, and $\mathbb{P}(G_*\subset G\subset G^*)=1-o(1)$. Known as the sandwich conjecture, such a coupling is desirable as it would allow properties of the random regular graph to be inferred from those of the more easily studied binomial random graph. The conjecture was recently shown to be true when $d\gg\log^4n$ by Gao, Isaev and McKay. In this paper, we prove the sandwich conjecture in full. We do so by analysing a natural coupling procedure introduced in earlier work by Gao, Isaev and McKay, which had only previously been done when $d\gg n/\sqrt{\log n}$.