On the chromatic number of random d-regular graphs

TL;DR

This paper determines the asymptotic chromatic number of random d-regular graphs for fixed d, showing it is either k-1 or k, and exactly k under certain conditions, improving previous bounds.

Contribution

It establishes the precise asymptotic chromatic number of random d-regular graphs, refining earlier bounds and eliminating the possibility of the chromatic number being k+1.

Findings

Chromatic number is asymptotically almost surely k-1 or k.

Exact chromatic number determined when d>(2k-3)log(k-1).

Improves previous results by Achlioptas and Moore.

Abstract

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k-1 or k. If moreover d>(2k-3)log(k-1), then the value k-1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small subgraph conditioning method to the number of balanced k-colourings, where a colouring is balanced if the number of vertices of each colour is equal.