Colorings of oriented planar graphs avoiding a monochromatic subgraph
Abstract
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 (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.