Rainbow Hamilton Cycles in Random Geometric Graphs

TL;DR

This paper proves that in a random geometric graph with edges colored randomly from nearly as many colors as edges, a rainbow Hamilton cycle exists with high probability when the graph's minimum degree is at least two.

Contribution

It establishes the existence of rainbow Hamilton cycles in random geometric graphs under minimal degree conditions with nearly optimal number of colors.

Findings

Rainbow Hamilton cycles exist a.a.s. under given conditions.

Minimum degree at least two suffices for rainbow Hamilton cycles.

Nearly optimal number of colors (n+o(n)) is sufficient for the result.

Abstract

Let $X_1,X_2,\ldots,X_n$ be chosen independently and uniformly at random from the unit $d$-dimensional cube $[0,1]^d$. Let $r$ be given and let $\cal X=\{X_1,X_2,\ldots,X_n\}$. The random geometric graph $G=G_{\cal X,r}$ has vertex set $\cal X$ and an edge $X_iX_j$ whenever $\|X_i-X_j\|\leq r$. 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.