Partitioning problems via random processes

TL;DR

This paper investigates graph partitioning problems using random processes, proving the majority colouring conjecture for Erdős-Rényi random directed graphs and an approximate version of the internal partition conjecture, employing innovative probabilistic techniques.

Contribution

It introduces new probabilistic methods, including a personality-changing scheme, to analyze and prove conjectures related to graph partitioning in random graphs.

Findings

Majority colouring conjecture holds for Erdős-Rényi random directed graphs.

Internal partition conjecture holds with a few exceptional vertices.

Develops novel techniques for analyzing random recolouring processes.

Abstract

There are a number of well-known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3-colouring such that for every vertex $v$, at most half of the out-neighbours of $v$ have the same colour as $v$. As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every $d$, all but finitely many $d$-regular graphs have a partition into two nonempty parts such that for every vertex $v$, at least half of the neighbours of $v$ lie in the same part as $v$. We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for Erd\H{o}s-R\'enyi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of "exceptional vertices". Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a "personality-changing" scheme: we "forget" certain information based on the state of a Markov chain, giving us more independence to work with.