On Pósa's Conjecture for Random Graphs

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 Komlos, Sarkozy, and Szemeredi. Here we prove that if $p \ge n^{-1/2+\varepsilon}$, 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 $p\le n^{-1/2}$.