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We study 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. It follows from the theorems of Dirac and of Komlós, Sarközy, and Szemerédi that for every  k  ≥ 1 and sufficiently large  n  already the minimum degree   δ  (  G  )  ≥    k    k  +  1    n   for an  n ‐vertex graph  G  alone suffices to ensure the existence of a  k th power of a Hamiltonian cycle. Here we show that under essentially the same degree assumption the addition of just  O ( n ) random edges ensures the presence of the ( k  + 1)st power of a Hamiltonian cycle with probability close to one.

Powers of Hamiltonian cycles in randomly augmented graphs