Square of a Hamilton cycle in a random graph

Abstract

We show that the threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle is $p=\frac{1}{\sqrt{n}}$. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić. In addition we consider how many random edges need to be added to a graph of order $n$ with minimum degree $\alpha n$ in order that it contains the square of a Hamilton cycle w.h.p.

There is an error in our proof