The Distant-l Chromatic Number of Random Geometric Graphs

Details The Distant-l Chromatic Number of Random Geometric Graphs The Distant-l Chromatic Number of Random Geometric Graphs

2009-09-21

arxiv.org/abs/0909.3678v1

Mathematics

Combinatorics

7 pages 1 figure

Scientific paper

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.

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