  We study the powers of Hamiltonian cycles in randomly augmented Dirac graphs,
that is, $n$-vertex graphs $G$ with minimum degree at least
$(1/2+\varepsilon)n$ to which some random edges are added. For any Dirac graph
and every integer $m\ge2$, 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.

Combinatorics