On Posa's conjecture for random graphs

TL;DR

This paper establishes that for random graphs with edge probability above n^{-1/2+ ext{epsilon}}, the graph almost surely contains the square of a Hamilton cycle, advancing understanding of Posa's conjecture in probabilistic settings.

Contribution

It proves an approximate threshold for the appearance of the square of a Hamilton cycle in random graphs, extending Posa's conjecture to probabilistic models.

Findings

Threshold p > n^{-1/2+ε} ensures the square of a Hamilton cycle appears almost surely.

The result delineates the boundary where the property transitions from unlikely to likely.

The paper confirms the conjecture's probabilistic analogue for large n.

Abstract

The famous Posa conjecture states that every graph of minimum degree at least 2n/3 contains the square of a Hamilton cycle. This has been proved for large n by Koml\'os, Sark\"ozy and Szemer\'edi. Here we prove that if p > n^{-1/2+\eps}, then asymptotically almost surely, the binomial random graph G_{n,p} contains the square of a Hamilton cycle. This provides an `approximate threshold' for the property in the sense that the result fails to hold if p< n^{-1/2}.