Sandwiching random regular graphs between binomial random graphs

TL;DR

This paper proves a conjecture about embedding random regular graphs between binomial random graphs with high probability, providing new insights into their structure and properties.

Contribution

It establishes the Kim--Vu sandwich conjecture for all degrees significantly larger than n divided by the square root of log n, with perfect containment on both sides.

Findings

Proves the Kim--Vu sandwich conjecture for all d≫n/√log n.

Provides bounds for random graphs with near-regular degree sequences.

Analyzes phase transitions and properties like Hamiltonicity and chromatic number.

Abstract

Kim and Vu made the following conjecture (\textit{Advances in Mathematics}, 2004): if $d\gg \log n$, then the random $d$-regular graph $\mathcal G(n,d)$ can asymptotically almost surely be "sandwiched" between $\mathcal G(n,p_1)$ and $\mathcal G(n,p_2)$ where $p_1$ and $p_2$ are both $(1+o(1))d/n$. They proved this conjecture for $\log n\ll d\le n^{1/3-o(1)}$, with a defect in the sandwiching: $\mathcal G(n,d)$ contains $\mathcal G(n,p_1)$ perfectly, but is not completely contained in $\mathcal G(n,p_2)$. Recently, the embedding $\mathcal G(n,p_1) \subseteq \mathcal G(n,d)$ was improved by Dudek, Frieze, Ruci\'nski and \v{S}ileikis to $d=o(n)$. In this paper, we prove Kim--Vu's sandwich conjecture, with perfect containment on both sides, for all $d\gg n/\sqrt{\log n}$. For $d=O(n/\sqrt{\log n})$, we prove a weaker version of the sandwich conjecture with $p_2$ approximately equal to $(d/n)\log n$, without any defect. In addition to sandwiching regular graphs, our results cover graphs whose degrees are asymptotically equal. The proofs rely on estimates for the probability that a random factor of a pseudorandom graph contains a given edge, which is of independent interest. As applications, we obtain new results on the properties of random graphs with given near-regular degree sequences, including Hamiltonicity and universality in subgraph containment. We also determine several graph parameters in these random graphs, such as the chromatic number, small subgraph counts, the diameter, and the independence number. We are also able to characterise many phase transitions in edge percolation on these random graphs, such as the threshold for the appearance of a giant component.