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 $\cal X={X_1,X_2,\ldots,X_n}$\mathrm{.}Therandomgeometricgraph$G=G_{\cal X,r}$ has vertex set $\mathcal{X}$ and an edge $X_i{X}_j$ whenever $\mathrm{\parallel }{X}_i-X_j\mathrm{\parallel }\le r$. 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 a.a.s.