One-point concentration of the clique and chromatic numbers of the random Cayley graph on F_2^n

TL;DR

This paper determines the precise asymptotic concentration of clique and chromatic numbers in random Cayley graphs on _2^n, showing they are tightly concentrated on a single value for large n.

Contribution

It establishes the exact constants for the clique number bounds and proves one-point concentration for both clique and chromatic numbers in these graphs.

Findings

Clique number is concentrated on a single value for almost all large n.

Exact asymptotic constants for the clique number are identified.

Chromatic number also exhibits one-point concentration, confirming conjectures.

Abstract

Green showed that there exist constants $C_1,C_2>0$ such that the clique number $\omega$ of the random Cayley graph on $\mathbb{F}_2^n$ satisfies $\lim_{n\to\infty}\mathbb{P}(C_1n\log n < \omega < C_2n\log n)=1$. In this paper we find the best possible $C_1$ and $C_2$. Moreover, we prove that for $n$ in a set of density $1$, clique number is actually concentrated on a single value. As a simple consequence of these results, we also prove the one-point concentration result for the chromatic number, thus proving the $\mathbb{F}_2^n$ analogue of the famous conjecture by Bollob\'{a}s and giving almost the complete answer to the question by Green.