The circular chromatic number of a digraph

We introduce the circular chromatic number χ_c of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k-cycle has circular chromatic number k/(k - 1), for k ≥ 2, values of χ_c between 1 and 2 are possible. We show that in fact, χ_c takes on all rational values greater than 1. Furthermore, there exist digraphs of arbitrarlly large digirth and circular chromatic number. it is NP-complete to decide if a given digraph has χ_c at most 2.