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arXiv · 2003.02883

Majority choosability of countable graphs

Marcin Anholcer·Bartłomiej Bosek·Jarosław Grytczuk
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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.

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Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk. 2020-03-05. Majority choosability of countable graphs. https://arxiv.org/abs/2003.02883

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