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arXiv · 2406.04189

A Note About Majority Colorings of Countable DAGs

Bartłomiej Bosek·Aleksander Katan
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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.

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Bartłomiej Bosek, Aleksander Katan. 2024-06-06. A Note About Majority Colorings of Countable DAGs. https://arxiv.org/abs/2406.04189

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