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arXiv · 2003.02998

Rainbow Hamilton Cycles in Random Geometric Graphs

Alan Frieze·Xavier Pérez-Giménez
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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.

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Alan Frieze, Xavier Pérez-Giménez. 2023-09-12. Rainbow Hamilton Cycles in Random Geometric Graphs. https://arxiv.org/abs/2003.02998

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