SIAM Journal on Discrete Mathematics, Volume 35, Issue 3, Page 1569-1577, January 2021.
Given an $n$-vertex graph $H$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $H \cup \mathbb{G}(n,p)$ over the supergraphs of $H$ is referred to as a (random) perturbation of $H$. We consider the distribution of edge-colored graphs arising from assigning each edge of the random perturbation $H \cup \mathbb{G}(n,p)$ a color, chosen independently and uniformly at random from a set of colors of size $r := r(n)$. We prove that edge-colored graphs which are generated in this manner asymptotically almost surely admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) \geq C/n$ for some fixed $C > 0$ and $r = (1 + o(1))n$. The number of colors used is clearly asymptotically best possible. In particular, this improves on a recent result of Anastos and Frieze [J. Graph Theory, 92 (2019), pp. 405--414] in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-colored sparse pseudorandom graphs asymptotically almost surely admit an almost spanning rainbow path.

中文翻译：

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随机着色随机扰动密集图中的彩虹汉密尔顿循环

SIAM 离散数学杂志，第 35 卷，第 3 期，第 1569-1577 页，2021 年 1 月。
给定一个 $n$-顶点图 $H$，对于某些固定的 $d > 0$，最小度数至少为 $dn$，$H \cup \mathbb{G}(n,p)$ 在 $ 的超图上的分布H$被称为$H$的（随机）扰动。我们考虑通过为随机扰动 $H \cup \mathbb{G}(n,p)$ 的每条边分配一种颜色而产生的边彩色图的分布，该颜色从一组大小为 $ 的颜色中独立且均匀地随机选择r := r(n)$。我们证明，只要随机扰动的边密度满足 $p := p(n) \geq C/n$ 对于某些固定的 $C > 0$ 和 $r = (1 + o(1))n$。使用的颜色数量显然是渐近最好的。特别是，这改进了 Anastos 和 Frieze [J. Graph Theory, 92 (2019), pp. 405--414] 在这方面。作为可能具有独立兴趣的中间结果，我们证明随机边着色的稀疏伪随机图渐近地几乎可以肯定地承认几乎跨越的彩虹路径。