Sprinkling a few random edges doubles the power

Abstract

A seminal result by Koml\'os, Sark\"ozy, and Szemer\'edi states that if a graph $G$ with $n$ vertices has minimum degree at least $kn/(k + 1)$, for some $k \in \mathbb{N}$ and $n$ sufficiently large, then it contains the $k$-th power of a Hamilton cycle. This is easily seen to be the largest power of a Hamilton cycle one can guarantee, given such a minimum degree assumption. Following a recent trend of studying effects of adding random edges to a dense graph, the model known as the randomly perturbed graph, Dudek, Reiher, Ruci\'nski, and Schacht showed that if the minimum degree is at least $kn/(k + 1) + \alpha n$, for any constant $\alpha > 0$, then adding $O(n)$ random edges on top almost surely results in a graph which contains the $(k + 1)$-st power of a Hamilton cycle. We show that the effect of these random edges is significantly stronger, namely that one can almost surely find the $(2k + 1)$-st power. This is the largest power one can guarantee in such a setting.