The two possible values of the chromatic number of a random graph

TL;DR

This paper establishes that for a random graph with edge probability d/n, the chromatic number almost surely takes one of two specific values, k_d or k_d+1, based on a precise threshold involving d and k.

Contribution

The paper precisely determines the almost sure possible values of the chromatic number for sparse random graphs, refining previous bounds and understanding.

Findings

Chromatic number of G(n,d/n) is almost surely either k_d or k_d+1

Defines k_d as the smallest integer satisfying d < 2k log k

Provides a threshold-based characterization of chromatic number values

Abstract

Given d \in (0,infty) let k_d be the smallest integer k such that d < 2k\log k. We prove that the chromatic number of a random graph G(n,d/n) is either k_d or k_d+1 almost surely.