Powers of Hamiltonian cycles in randomly augmented Dirac graphs—The complete collection

We study the powers of Hamiltonian cycles in randomly augmented Dirac graphs, that is, $$n$$-vertex graphs $$G$$ with minimum degree at least $$(1\unicode{x02215}2+\varepsilon )n$$ to which some random edges are added. For any Dirac graph and every integer $$m\ge 2$$, we accurately estimate the threshold probability $$p=p(n)$$ for the event that the random augmentation $$G\cup G(n,p)$$ contains the $$m$$-th power of a Hamiltonian cycle.