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A Note About Majority Colorings of Countable DAGs

Bartłomiej Bosek, Aleksander Katan

TL;DR

The paper investigates whether every countable DAG admits a majority $2$-coloring. It introduces a gadget-based construction using OR$^T$ gadgets on an infinite path to produce a countable acyclic digraph $G$ that cannot be majority $2$-colored, thus requiring at least $3$ colors. The argument relies on translating colorings into truth assignments via $ ho^T(v)$ and applying an induction over gadgets to derive a contradiction. This result closes the gap on the 2-color conjecture for countable DAGs, and the authors discuss open problems linking to undirected multigraphs and the Unfriendly Partition Conjecture.

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.

A Note About Majority Colorings of Countable DAGs

TL;DR

The paper investigates whether every countable DAG admits a majority -coloring. It introduces a gadget-based construction using OR gadgets on an infinite path to produce a countable acyclic digraph that cannot be majority -colored, thus requiring at least colors. The argument relies on translating colorings into truth assignments via and applying an induction over gadgets to derive a contradiction. This result closes the gap on the 2-color conjecture for countable DAGs, and the authors discuss open problems linking to undirected multigraphs and the Unfriendly Partition Conjecture.

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.
Paper Structure (6 sections, 3 theorems, 2 equations, 2 figures)

This paper contains 6 sections, 3 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The counterexample
  3. Construction of the counterexample
  4. Proof that 3 colors are required for any majority coloring of $G$.
  5. Conclusions and open problems
  6. Acknowledgments

Key Result

Lemma 1

Given a graph $H = (V, E)$ of any cardinality and any majority 2-coloring $\phi$ of $G$, if $U$ is a valid OR$^T(u_1, \dots, u_k)$ gadget in $H$ with an output vertex $o$, then $\phi^T(u_1) \lor \dots \lor \phi^T(u_k) = \phi^T(o)$.

Figures (2)

  • Figure 1: An OR$^T(v_1, v_2)$ gadget and an OR$^T(v_1, v_2, v_3)$ gadget.
  • Figure 2: A countable acyclic digraph $G$ that we claim not to be majority 2-colorable.

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Conjecture 1