# Generalised Majority Colourings of Digraphs

## TL;DR

This paper investigates the minimum number of colours needed to colour directed graphs so that each vertex shares its colour with at most a 1/k proportion of its out-neighbours, providing bounds and supporting conjectures.

## Contribution

It establishes bounds for the minimal colours needed in majority colourings of digraphs and extends results to majority choosability, advancing understanding in graph colouring.

## Key findings

- m(k) is either 2k-1 or 2k
- Supports the conjecture that m(2)=3
- Provides bounds for majority choosability

## Abstract

The purpose of this note is to draw attention to problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised a problem of determining, for a natural number $k$, the smallest number $m=m(k)$ such that every digraph can be coloured with $m$ colours where each vertex has the same colour as at most $1/k$ proportion of its out-neighbours. We show that $m(k)\in\{2k-1,2k\}$. We also prove a result supporting the conjecture that $m(2)=3$. Moreover, we prove similar results for a more general concept called majority choosability.