On the chromatic number in the stochastic block model

TL;DR

This paper extends classical results on the chromatic number of random graphs to inhomogeneous models like the stochastic block model and Chung-Lu model, providing asymptotic estimates and new analytical techniques.

Contribution

It generalizes Bollobás' classical results to the stochastic block model and determines the chromatic number for the Chung-Lu model with growing blocks.

Findings

Asymptotic chromatic number estimates for stochastic block models

Extension of results to Chung-Lu model with increasing number of blocks

New bounds based on weighted independence number estimates

Abstract

We prove a generalisation of Bollob\'as' classical result on the asymptotics of the chromatic number of the binomial random graph to the stochastic block model. In addition, by allowing the number of blocks to grow, we determine the chromatic number in the Chung-Lu model. Our approach is based on the estimates for the weighted independence number, where weights are specifically designed to encapsulate inhomogeneities of the random graph.