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