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 K\"uhn and Osthus and also Nenadov and \v{S}kori\'c. 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.