The Chromatic Number of Random Graphs for Most Average Degrees

Abstract

For a fixed number d> 0 and n large, let G(n,d/n) be the random graph on n vertices in which any two vertices are connected with probability d/n independently. The problem of determining the chromatic number of G(n,d/n) goes back to the famous 1960 article of Erdős and Renyi that started the theory of random graphs [Magayar Tud. Akad. Mat. Kutato Int. Kozl. 5 (1960) 17–61]. Progress culminated in the landmark paper of Achlioptas and Naor [Ann. Math. 162 (2005) 1333–1349], in which they calculate the chromatic number precisely for all d in a set S⊂ (0,∞) of asymptotic density limz→∞ z ∫z 0 1S = 2 , and up to an additive error of one for the remaining d. Here we obtain a near-complete answer by determining the chromatic number of G(n,d/n) for all d in a set of asymptotic density 1.