Let $X_1,X_2,\dots ,X_n$ be chosen independently and uniformly at random from the unit $d$-dimensional cube ${[0,1]}^d$. Let $r$ be given and let $𝒳=\left\{X_1,X_2,\dots ,X_n\right\}$. The random geometric graph $G=G_{𝒳,r}$ has vertex set $𝒳$ and an edge $X_i{X}_j$ whenever $\Vert X_i-X_j\Vert \le r$. We show that if each edge of $G$ is colored independently from one of $n+o(n)$ colors and $r$ has the smallest value such that $G$ has minimum degree at least two, then $G$ contains a rainbow Hamilton cycle asymptotically almost surely.

中文翻译：

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随机几何图中的彩虹汉密尔顿循环

让$X_1,X_2,\mathrm{\dots \dots },X_n$从单位中独立、统一、随机选择$d$维立方体${[0,1]}^d$。让$r$被给予并让$𝒳=\left\{X_1,X_2,\mathrm{\dots \dots },X_n\right\}$。随机几何图$G=G_{𝒳,r}$已设置顶点$𝒳$和一个边缘$X_{我}{X}_j$每当$\Vert X_{我}-X_j\Vert \le r$。我们证明如果每条边$G$颜色独立于其中之一$n+哦（n）$颜色和$r$具有最小值使得$G$最小度数至少为二，那么$G$几乎肯定渐近地包含彩虹汉密尔顿循环。