On the chromatic number of random Cayley graphs

TL;DR

This paper investigates the chromatic number of random Cayley graphs on abelian groups, establishing an asymptotically sharp upper bound that depends on the group's size and structure, especially for prime cyclic groups.

Contribution

It provides the first asymptotically sharp bounds for the chromatic number of random Cayley graphs on abelian groups, extending previous results to a broader class of groups.

Findings

Chromatic number is at most (1 + o(1))N/2 log_2 N for large N

Asymptotic sharpness established for cyclic groups of prime order

Results hold for abelian groups with (N,6)=1

Abstract

Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic number chi(Gamma_A) is at most (1 + o(1))N/2 log_2 N. This is asymptotically sharp when G = Z/NZ, N prime. Presented at the conference in honour of Bela Bollobas on his 70th birthday, Cambridge August 2013.