Acyclic Homomorphisms and Circular Colorings of Digraphs

An acyclic homomorphism of a digraph D into a digraph F is a mapping φ : V (D) → V (F ) such that for every arc uv ∈ E(D), either φ(u) = φ(v) or φ(u)φ(v) is an arc of F , and for every vertex v ∈ V (F ), the subgraph of D induced on φ(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 NPcomplete 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. ∗Supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1–0502–0101–01.