Majority choosability of digraphs

TL;DR

This paper proves that every digraph can be majority 4-choosably colored, answering a recent open question, and extends the concept to broader coloring conditions, including a 3-coloring conjecture.

Contribution

The paper establishes that all digraphs are majority 4-choosable and introduces a more general theorem extending the majority coloring concept.

Findings

Every digraph is majority 4-choosable.

Every digraph admits a coloring from lists of size three with at most 2/3 out-neighbors sharing the color.

Supports the conjecture that every digraph is majority 3-colorable.

Abstract

A \emph{majority coloring} of a digraph is a coloring of its vertices such that for each vertex $v$, at most half of the out-neighbors of $v$ has the same color as $v$. A digraph $D$ is \emph{majority $k$-choosable} if for any assignment of lists of colors of size $k$ to the vertices there is a majority coloring of $D$ from these lists. We prove that every digraph is majority $4$-choosable. This gives a positive answer to a question posed recently by Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain this result as a consequence of a more general theorem, in which majority condition is profitably extended. For instance, the theorem implies also that every digraph has a coloring from arbitrary lists of size three, in which at most $2/3$ of the out-neighbors of any vertex share its color. This solves another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture stating that every digraph is majority $3$-colorable.