High powers of Hamiltonian cycles in randomly augmented graphs

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 > 1 , r > 0, and l > ( r + 1 ) r, and for any alpha > k k + 1 we show that adding O ( n 2 - 2 / l ) 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 l + r )-th power of a Hamiltonian cycle. In particular, for r = 1 and l = 2 this implies that adding O ( n ) random edges to such a graph G already ensures the ( 2 k + 1 )-st power of a Hamiltonian cycle (proved independently by Nenadov and Trujic). In this instance and for several other choices of k , l, and r we can show that our result is asymptotically optimal.