Friendly bisections of random graphs

Abstract

Resolving a conjecture of Füredi from 1988, we prove that with high probability, the random graph G ( n , 1 / 2 ) \mathbb {G}(n,1/2) admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which n − o ( n ) n-o(n) vertices have more neighbours in their own part as across. Our proof is constructive, and in the process, we develop a new method to study stochastic processes driven by degree information in random graphs; this involves combining enumeration techniques with an abstract second moment argument.