Acyclic Homomorphisms and Circular Colorings of Digraphs

An acyclic homomorphism of a digraph $D$ into a digraph $F$ is a mapping $\phi\colon V(D) \to V(F)$ such that for every arc $uv\in E(D)$, either $\phi(u)=\phi(v)$ or $\phi(u)\phi(v)$ is an arc of $F$, and for every vertex $v\in V(F)$, the subgraph of $D$ induced on $\phi^{-1}(v)$ is acyclic. For each fixed digraph $F$ we consider the following decision problem: Does a given input digraph $D$ admit an acyclic homomorphism to $F$? We prove that this problem is NP-complete unless $F$ is acyclic, in which case it is polynomial time solvable. From this we conclude that it is NP-complete to decide if the circular chromatic number of a given digraph is at most $q$, for any rational number $q > 1$. We discuss the complexity of the problems restricted to planar graphs. We also refine the proof to deduce that certain $F$-coloring problems are NP-complete.