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 outneighbors 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. ∗Partially supported by NCN grant 2013/11/D/ST6/03100. †Partially supported by NCN grant 2015/17/B/ST1/02660. the electronic journal of combinatorics 24(3) (2017), #P3.57 1