Colorings of oriented planar graphs avoiding a monochromatic subgraph

For a fixed simple digraph F and given D , an - free k -coloring of is vertex-coloring in which no induced copy monochromatic. We study the complexity deciding for whether admits -free -coloring. Our main focus on restriction problem to planar input digraphs, where it only interesting cases ? { 2 3 } . From known results follows that every whose underlying graph not forest, 2-coloring, with ? ( ) ? oriented 3-coloring. show contrast, • if orientation path length at least 2, then NP -hard decide acyclic 2-coloring. 1,