A Note About Majority Colorings of Countable DAGs

TL;DR

This paper investigates majority colorings of countable directed acyclic graphs (DAGs), demonstrating that some require three colors for a majority coloring, thus disproving a conjecture that two colors suffice.

Contribution

It provides a counterexample showing that not all countable DAGs can be majority colored with only two colors, establishing the necessity of three colors in some cases.

Findings

Counterexample DAG requiring three colors

Disproof of the conjecture that two colors suffice

Extension of known results from undirected graphs to DAGs

Abstract

A majority coloring of an undirected graph is a vertex coloring in which for each vertex there are at least as many bi-chromatic edges containing that vertex as monochromatic ones. It is known that for every countable graph a majority 3-coloring always exists. The Unfriendly Partition Conjecture states that every countable graph admits a majority 2-coloring. Since the 3-coloring result extends to countable DAGs, a variant of the conjecture states that 2 colors are enough to majority color every countable DAG. We show that this is false by presenting a DAG for which 3 colors are necessary. Presented construction is strongly based on a StackExchange conversation regarding labellings of infinite graphs that is linked in the references.