Random regular graphs of non-constant degree: Concentration of the chromatic number

Abstract

In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model G n , d for d = o ( n 1 / 5 ) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G ( n , p ) with p = d n . Our proof is largely based on ideas of Alon and Krivelevich who proved this two-point concentration result for G ( n , p ) for p = n − δ where δ > 1 / 2 . The main tool used to derive such a result is a careful analysis of the distribution of edges in G n , d , relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model.