Kim--Vu's sandwich conjecture is true for $d \gg \log^4 n$

TL;DR

The paper resolves Kim–Vu's sandwich conjecture in the sparse regime by proving that for $igl\min\{d,n-d\}\gg \,\

Abstract

Kim and Vu made the following conjecture (\textit{Advances in Mathematics}, 2004): if $d\gg \log n$, then the random $d$-regular graph $G(n,d)$ can be ``sandwiched'' between $G(n,p_*)$ and $G(n,p^*)$ where $p_*$ and $p^*$ are both asymptotically equal to $d/n$. This famous conjecture was previously proved for all $d\gg (n\log n)^{3/4}$. In this paper, we confirm the conjecture when $d \gg \log^4 n$. We also extend this result to near-regular degree sequences.

Figures (1)

• Figure 1: $\ell$-switching.

Theorems & Definitions (47)

• Conjecture 1.1: Sandwich conjecture
• Theorem 1.2
• Remark 1.3
• Definition
• Conjecture 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1