Partitioning problems via random 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 non-empty 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 Erdős–Rényi 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.