Sharp thresholds for spanning regular subgraphs

TL;DR

This paper establishes the precise probability threshold for the emergence of the square of a Hamilton cycle in random graphs and determines the exact asymptotics for the appearance of spanning regular subgraphs, confirming longstanding conjectures.

Contribution

It proves the sharp threshold for the square of a Hamilton cycle in G(n,p) and finds exact asymptotics for spanning regular subgraphs, advancing understanding of phase transitions in random graphs.

Findings

Sharp threshold at (1+o(1))√(e/n) for the square of a Hamilton cycle

Exact asymptotics for spanning regular subgraphs in G(n,p)

Confirmation of conjectures by Kahn, Narayanan, and Park

Abstract

We prove that $(1+o(1))\sqrt{e/n}$ is the sharp threshold for the appearance of the square of a Hamilton cycle in $G(n,p)$, confirming the conjecture of Kahn, Narayanan, and Park. We also find the exact asymptotics of the threshold for the emergence of a spanning subgraph isomorphic to a fixed graph $F$ for a wide family of $d$-regular graphs $F$. This family includes almost all $d$-regular graphs.