Title: The Square of a Hamilton Cycle in Randomly Perturbed Graphs
Abstract: 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 $$\alpha > 1/2$$ , and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each $$\alpha \le 1/2$$ . Our result has implications on the perturbed threshold for 2-universality, where we also fully address all open cases.