Majority choosability of countable graphs

TL;DR

This paper proves that every countable graph and digraph can be majority colored from any lists of size four, extending known results and posing new conjectures for smaller list sizes.

Contribution

It establishes that all countable graphs and digraphs are majority 4-choosable, introducing list variants and analogs for directed graphs, and proposing related conjectures.

Findings

Every countable graph is majority 4-choosable.

Every countable digraph is majority 4-choosable.

Poses conjectures that countable graphs/digraphs are majority 2- or 3-choosable.

Abstract

In any vertex coloring of a graph some edges have differently colored ends (\emph{good} edges) and some are monochromatic (\emph{bad} edges). In a proper coloring all edges are good. In a \emph{majority coloring} it is enough that for every vertex $v$, the number of bad edges incident to $v$ does not exceed the number of good edges incident to $v$. A well known result of Lov\'{a}sz \cite{Lovasz} asserts that every finite graph has a majority $2$-coloring. A similar statement for countably infinite graphs is a challenging open problem, known as the \emph{Unfriendly Partition Conjecture}. We consider a natural list variant of majority coloring. A graph is \emph{majority $k$-choosable} if it has a majority coloring from any lists of size $k$ assigned arbitrarily to the vertices. We prove that every countable graph is majority $4$-choosable. We also consider a natural analog of majority coloring for directed graphs. We prove that every countable digraph is also majority $4$-choosable. We pose list and directed analogs of the Unfriendly Partition Conjecture, stating that every countable graph is majority $2$-choosable and every countable digraph is majority $3$-choosable.