Majority Colorings of Sparse Digraphs

A majority coloring of a directed graph is vertex-coloring in which every vertex has the same color as at most half its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that digraph $4$-coloring conjectured admits $3$-coloring. They observed Local Lemma implies conjecture for digraphs large enough minimum out-degree if, crucially, maximum in-degree bounded by a(n exponential) function out-degree.
 Our goal this paper to develop alternative methods allow verification natural, broad classes, without any restriction on in-degrees. Among others, we prove 1) with chromatic number $6$ or dichromatic $3$, thus all planar digraphs; 2) $4$. The benchmark case $r$-regular remains open $r \in [5,143]$.
 inductive proofs depend loaded statements about precoloring extensions list-colorings. This approach also gives rise stronger conclusions, involving choosability version coloring.
 We give further evidence towards existence majority-$3$-colorings showing fractional 3.9602-coloring. Moreover show $(2+\varepsilon)$-coloring.