Majority Choosability of Digraphs
A 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 $v$ have the same color as $v$. A digraph $D$ is 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 (2017). 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 by the same authors, and supports an intriguing conjecture stating that every digraph is majority $3$-colorable.