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