The Distant-l Chromatic Number of Random Geometric Graphs

TL;DR

This paper studies the asymptotic behavior of the distant-l chromatic number in random geometric graphs, revealing how it compares to the standard chromatic number as the number of vertices grows.

Contribution

It provides a comprehensive analysis of the ratios of the distant-l chromatic number to the chromatic number in high-dimensional random geometric graphs, with almost sure convergence results.

Findings

Ratios of _l(G_n) to (G_n) are characterized asymptotically.

Complete description of the asymptotic ratios for different parameters.

Results hold with probability approaching 1 as n .

Abstract

A random geometric graph $G_n$ is given by picking $n$ vertices in $\mathbb{R}^d$ independently under a common bounded probability distribution, with two vertices adjacent if and only if their $l^p$-distance is at most $r_n$. We investigate the distant-$l$ chromatic number $\chi_l(G_n)$ of $G_n$ for $l\ge1$. Complete picture of the ratios of $\chi_l(G_n)$ to the chromatic number $\chi(G_n)$ are given in the sense of almost sure convergence.