Square of Hamilton cycle in a random graph

TL;DR

This paper establishes the precise probability threshold at which a random graph almost surely contains the square of a Hamilton cycle, improving previous bounds and advancing understanding of Hamiltonian properties in random graphs.

Contribution

The paper proves that p=√(e/n) is the sharp threshold for the existence of the square of a Hamilton cycle in G_{n,p}, refining earlier results.

Findings

Identifies the sharp threshold p=√(e/n) for the square of a Hamilton cycle.

Improves previous bounds on Hamiltonian cycle properties in random graphs.

Provides rigorous proof of the threshold's sharpness.

Abstract

We show that $p=\sqrt{\frac{e}{n}}$ is a sharp threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle. This improves the previous results of K\"uhn and Osthus and also Nenadov and \v{S}kori\'c.