Clique Colourings of Geometric Graphs

Colin Mcdiarmid,Dieter Mitsche,Pawel Prałat

doi:10.37236/7159

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 $\mathbf{x}_1, \ldots,\mathbf{x}_n$ 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 $\mathbf{x}_i$ and $\mathbf{x}_j$ 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 $\mathcal{G}$ 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.

Highlights

Introduction and main results we introduce clique colourings and geometric graphs; and we present our main results, on clique colourings of deterministic and random geometric graphs.Recall that a proper colouring of a graph is a labeling of its vertices with colours such that no two vertices sharing the same edge have the same colour; and the smallest number of colours in a proper colouring of a graph G = (V, E) is its chromatic number, denoted by χ(G).We are concerned here with another notion of vertex colouring
A clique colouring of a graph G is a colouring of the vertices such that no maximal clique is monochromatic, ignoring isolated vertices
The two results summarise what we know about the clique chromatic number χc of a random geometric graph G in the plane; but first here is an overview

Summary

Introduction and main results

We introduce clique colourings and geometric graphs; and we present our main results, on clique colourings of deterministic and random geometric graphs. Our first theorem shows that the clique chromatic number is uniformly bounded for geometric graphs in the plane. Let χmc ax(Rd) denote the maximum value of χc(G) over geometric graphs G in Rd. Clearly χmc ax(R2) is at least 3 (consider C5) so we have 3 χmc ax(R2) 9: it would be interesting to improve these bounds. (We could work with the unit square [0, 1]2.) Note that, with probability 1, no point in Sn is chosen more than once, so we may identify each vertex v ∈ V with its corresponding geometric position v = (vx, vy) ∈ Sn. The (usual) chromatic number of G(n, r) was studied in [21, 23], see [28]. The two results summarise what we know about the clique chromatic number χc of a random geometric graph G in the plane; but first here is an overview. See [28] for more general models of random geometric graphs, and see [9] in particular for models in high dimensions

Deterministic results

Random results

Concluding Remarks