Rainbow Hamilton cycles in random geometric graphs

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.