Majority Colourings of Digraphs

TL;DR

This paper proves that every directed graph can be colored with four colors so that each vertex shares its color with at most half of its out-neighbors, and explores related conjectures for three colors.

Contribution

It establishes a new vertex 4-coloring result for digraphs and investigates the conjecture for 3-colorings, advancing understanding in graph coloring.

Findings

Every digraph admits a 4-coloring with the specified property.

Several results related to the 3-coloring conjecture are obtained.

The paper advances the theory of majority colorings in digraphs.

Abstract

We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3.