The threshold for the square of a Hamilton cycle

TL;DR

This paper proves that in a random graph with edge probability 1/√n, the square of a Hamilton cycle almost surely appears, resolving a conjecture from 2012.

Contribution

It establishes the precise threshold for the appearance of the square of a Hamilton cycle in random graphs, confirming a conjecture by K"uhn and Osthus.

Findings

Threshold for the square of a Hamilton cycle is p=1/√n.

Confirms the conjecture from 2012.

Advances understanding of Hamiltonian structures in random graphs.

Abstract

Resolving a conjecture of K\"uhn and Osthus from 2012, we show that $p= 1/\sqrt{n}$ is the threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle.