A proof of the Kim-Vu sandwich conjecture

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 $d\gg n/\sqrt{\log n}$.