Acyclic homomorphisms and circular colorings of digraphs

An acyclic homomorphism of a digraph D into a digraph F is a mapping phi: V (D) --> V (F) such that for every arc uv is an element of E(D), either phi(u) = phi(v) or phi(u) phi(v) is an arc of F, and for every vertex v is an element of 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 re. ne the proof to deduce that certain F-coloring problems are NP-complete.