Majority choosability of 1-planar digraph

Abstract

A majority coloring of a digraph D with k colors is an assignment π: V(D) → {1, 2, …, k} such that for every v ∈ V(D) we have π(w) = π(v) for at most half of all out-neighbors w ∈ N⁺(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 if U(D) is a 1-planar graph without a 4-cycle, then D is majority 3-choosable. And we also prove that every NIC-planar digraph is majority 3-choosable.