High powers of Hamiltonian cycles in randomly augmented graphs

Abstract: We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers $k\geq1$, $r\geq 0$, and $\ell\geq (r+1)r$, and for any $\alpha>\frac{k}{k+1}$ we show that adding $O(n^{2-2/\ell})$ random edges to an $n$-vertex graph $G$ with minimum degree at least $\alpha n$ yields, with probability close to one, the existence of the $(k\ell+r)$-th power of a Hamiltonian cycle. In particular, for $r=1$ and $\ell=2$ this implies that adding $O(n)$ random edges to such a graph $G$ already ensures the $(2k+1)$-st power of a Hamiltonian cycle (proved independently by Nenadov and Truji\'c). In this instance and for several other choices of $k$, $\ell$, and $r$ we can show that our result is asymptotically optimal.