# Clique colourings of geometric graphs

## TL;DR

This paper studies the clique chromatic number of geometric graphs, establishing an upper bound of 9 in the plane and analyzing its asymptotic behavior in random geometric graphs as the distance threshold varies.

## Contribution

It provides the first universal upper bound for the clique chromatic number of planar geometric graphs and explores its probabilistic behavior in random settings.

## Key findings

- Clique chromatic number is at most 9 in the plane.
- Asymptotic behavior of the clique chromatic number in random geometric graphs is characterized.
- The clique chromatic number transitions from 1 to 2 to at least 3 and back to 2 as the radius increases.

## Abstract

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given $n$ points $x_1, \ldots,x_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $x_i$ and $x_j$ is at most $r$. We investigate the clique chromatic number of such graphs. We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $RG$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.