On the chromatic number of graphons

TL;DR

This paper generalizes classical results on the chromatic number of random graphs to graphons, showing that for certain classes of graphons, the chromatic number can be effectively approximated using a finite number of color class types.

Contribution

It extends Bollobas' result to exchangeable random graph models defined by graphons, providing bounds and strategies for coloring based on block graphon approximations.

Findings

Asymptotically optimal coloring strategies use finitely many color class types.

For block graphons with k×k blocks, k color class types suffice.

Coloring strategies for block-increasing or block-Lipschitz graphons approximate the chromatic number within an O(1/k) error.

Abstract

We extend Bollobas' classical result on the chromatic number of a binomial random graph to the exchangeable random graph model $\mathcal{G}(n,W)$ defined by a graphon $W:[0,1]^2 \rightarrow [0,1]$, which is a symmetric measurable function. In the case when $W$ can be approximated by block graphons in $\mathcal{L}^{\infty}$-norm, we show that asymptotically optimal value of the number of colours required for $\mathcal{G}(n,W)$ is determined by colouring strategies that use a finite number of different types of colour classes. Furthermore, if $W$ is a block graphon with $k\times k$ blocks then $k$ types of colour classes are sufficient. We also show that if $W$ is block-increasing or block-Lipschitz then such colouring strategies that use $k$ types determine the chromatic number up to a multiplicative error of order $O(k^{-1})$.