Majority choosability of countable graphs
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 , the number of bad edges incident to does not exceed the number of good edges incident to . A well known result of Lov\'{a}sz \cite{Lovasz} asserts that every finite graph has a majority -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 -choosable} if it has a majority coloring from any lists of size assigned arbitrarily to the vertices. We prove that every countable graph is majority -choosable. We also consider a natural analog of majority coloring for directed graphs. We prove that every countable digraph is also majority -choosable. We pose list and directed analogs of the Unfriendly Partition Conjecture, stating that every countable graph is majority -choosable and every countable digraph is majority -choosable.
Cite
@article{arxiv.2003.02883,
title = {Majority choosability of countable graphs},
author = {Marcin Anholcer and Bartłomiej Bosek and Jarosław Grytczuk},
journal= {arXiv preprint arXiv:2003.02883},
year = {2020}
}