SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 988-1004, January 2021.
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.

中文翻译：

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撒上一些随机边缘使功率加倍

SIAM 离散数学杂志，第 35 卷，第 2 期，第 988-1004 页，2021 年 1 月。
Komlós、Sarközy 和 Szemerédi 的一项开创性结果指出，如果具有 $n$ 个顶点的图 $G$ 的最小度数至少为 $kn/(k + 1)$，对于某些 $k \in \mathbb{N}$并且 $n$ 足够大，那么它包含哈密尔顿循环的 $k$th 次幂。考虑到这样的最小程度假设，这很容易被视为可以保证的哈密顿循环的最大幂。遵循最近研究将随机边添加到密集图的效果的趋势，该模型被称为随机扰动图，Dudek 等人。表明如果最小度数至少为 $kn/(k + 1) + \alpha n$，对于任何常数 $\alpha > 0$，则在顶部添加 $O(n)$ 随机边几乎肯定会导致包含汉密尔顿循环的 $(k + 1)$st 次幂的图。我们表明这些随机边缘的影响明显更强，即，几乎可以肯定地找到 $(2k + 1)$st 次幂。这是在这种情况下可以保证的最大功率。