# Linear Bound for Majority Colourings of Digraphs

## TL;DR

This paper proves a linear bound on the majority colourings of directed graphs with list assignments, extending previous results and establishing optimal bounds for even and odd list sizes.

## Contribution

It introduces a new linear bound for majority colourings of digraphs with list sizes, generalizing prior specific cases and proving optimality for even and odd list sizes.

## Key findings

- For even k, the bound 2/k is optimal.
- For odd k, the bound 2/k cannot be improved below 2/(k+1).
- Generalizes previous results for k=3 and 4.

## Abstract

Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$, for any $v \in V(G)$. We show that every digraph $G$ equipped with an assignment of lists $L$, each of size at least $k$, has a $2/k$-majority $L$-colouring. For even $k$ this is best possible, while for odd $k$ the constant $2/k$ cannot be replaced by any number less than $2/(k+1)$. This generalizes a result of Anholcer, Bosek and Grytczuk, who proved the cases $k=3$ and $k=4$ and gave a weaker result for general $k$.