Scaling Laws for Maximum Coloring of Random Geometric Graphs

TL;DR

This paper investigates the maximum vertex coloring of random geometric graphs across dimensions, establishing convergence laws using subadditivity, with exact results in one dimension and bounds in higher dimensions.

Contribution

It introduces a novel approach based on subadditivity to analyze maximum coloring in random geometric graphs, overcoming the challenge of non-scale-invariance.

Findings

Convergence laws for maximum coloring established

Exact values obtained for one-dimensional case

Bounds provided for higher dimensions

Abstract

We examine maximum vertex coloring of random geometric graphs, in an arbitrary but fixed dimension, with a constant number of colors. Since this problem is neither scale-invariant nor smooth, the usual methodology to obtain limit laws cannot be applied. We therefore leverage different concepts based on subadditivity to establish convergence laws for the maximum number of vertices that can be colored. For the constants that appear in these results, we provide the exact value in dimension one, and upper and lower bounds in higher dimensions.