The circular chromatic number of a digraph

We introduce the circular chromatic number χc 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 χc between 1 and 2 are possible. We show that in fact, χc takes on all rational values greater than 1. Furthermore, there exist digraphs of arbitrarily large digirth and circular chromatic number. It is NP-complete to decide if a given digraph has χc at most 2. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 227–240, 2004