A seminal result by Komlós, Sarközy, and Szemerédi 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 et al. 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.

Consider the following simple coloring algorithm for a graph on n vertices. Each vertex chooses a color from {1, ..., Δ(G) + 1} uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC'16] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected O(n log Δ) steps, which is optimal and proves a conjecture of Chakrabarty and de Supinski [SOSA'20].