We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given α∈ (0, 1 ), the union of any n-vertex graph with minimum degree αn and the binomial random graph G(n, p). This is known when α> 1 / 2, and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each α≤ 1 / 2. Our result has implications on the perturbed threshold for 2-universality, where we also fully address all open cases
