The circular chromatic numbers of planar digraphs

The circular chromatic number is a refinement of the chromatic number of a graph. It has been established in 3,6,7] that there exists planar graphs with circular chromatic number r if and only if r is a rational in the set {1} ∪ 2,4]. Recently, Mohar, in 1,2] has extended the concept of the circular chromatic number to digraphs and it is interesting to ask what the corresponding result is for digraphs. In this article, we shall prove the new result that there exist planar digraphs with circular chromatic number r if and only if r is a rational in the interval 1,4]. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 14–26, 2007