Friendly bisections of random graphs

TL;DR

This paper proves that with high probability, the random graph G(n,1/2) can be partitioned into two nearly equal parts where most vertices have at least as many neighbors in their own part as across, resolving a long-standing conjecture.

Contribution

The paper introduces a novel method combining enumeration and second moment techniques to analyze degree-driven stochastic processes in random graphs, proving a conjecture from 1988.

Findings

Random graphs G(n,1/2) admit friendly bisections with high probability.

Most vertices in such bisections have at least as many neighbors in their own part as across.

The new method effectively studies degree-based stochastic processes in random graphs.

Abstract

Resolving a conjecture of F\"uredi from 1988, we prove that with high probability, the random graph $G(n,1/2)$ 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 $n-o(n)$ vertices have at least as many neighbours in their own part as across. The engine of our proof is 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.