Majority choosability of 1-planar digraph
Weihao Xia, Jihui Wang, Jiansheng Cai · Czechoslovak Mathematical Journal · 2023
summary:A majority coloring of a digraph $D$ with $k$ colors is an assignment $\pi \colon V(D) \rightarrow \{1,2,\cdots ,k\}$ such that for every $v\in V(D)$ we have $\pi (w)=\pi (v)$ for at most half of all out-neighbors $w\in N^+(v)$. 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.