The circular chromatic number of a digraph
Drago Bokal, Gašper Fijavž, Martin Juvan, P. Mark Kayll, Bojan Mohar · Journal of Graph Theory · 2004
Abstract 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 arbitrarily large digirth and circular chromatic number. It is NP‐complete to decide if a given digraph has χ c at most 2. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 227–240, 2004