The square of a Hamilton cycle in randomly perturbed graphs

We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given $\alpha \in (0,1)$, the union of any $n$-vertex graph with minimum degree $\alpha n$ and the binomial random graph $G(n,p)$. This is known when and we determine the exact perturbed threshold probability in all the remaining cases, that is, for each $\alpha \le 1/2$. We demonstrate that, as $\alpha$ ranges over the interval $(0,1)$, the threshold performs a countably infinite number of ‘jumps’. Our result has implications on the perturbed threshold for two-universality, where we also fully address all open cases.