Sharp thresholds for spanning regular graphs

TL;DR

This paper establishes precise conditions under which random graphs almost surely contain a given regular graph as a subgraph, confirming a conjecture on sharp thresholds for certain graph structures.

Contribution

It provides a tight bound on the edge expansion of regular graphs that guarantees a sharp threshold for their appearance in random graphs, resolving a conjecture by Kahn, Narayanan, and Park.

Findings

Identifies a tight bound on edge expansion for sharp thresholds.

Proves the conjecture on sharp thresholds for squares of Hamilton cycles.

Shows that for most regular graphs, the threshold is asymptotically (e/n)^{2/d}.

Abstract

Let $d\geq 3$ be a constant and let $F$ be a $d$-regular graph on $[n]$ with not too many symmetries. By the union bound, the probability threshold for the existence of a spanning subgraph in $G(n,p)$ isomorphic to $F$ is at least $p^*(n)=(1+o(1))(e/n)^{2/d}$. We give a tight bound on the edge expansion of $F$ guaranteeing that the probability threshold for the appearance of a copy of $F$ has the same order of magnitude as $p^*$. We also prove that, within a slight strengthening of this bound, the probability threshold is asymptotically equal to $p^*$. In particular, it proves the conjecture of Kahn, Narayanan and Park on a sharp threshold for the containment of a square of a Hamilton cycle. It also implies that, for $d\geq 4$ and (asymptotically) almost all $d$-regular graphs $F$ on $[n]$, $p(n)=(e/n)^{2/d}$ is a sharp threshold for $F$-containment.