Circular Coloring of Planar Digraphs

Abstract

Let D be a digraph. The circular chromatic number \({\chi_c(D)}\) and chromatic number \({\chi(D)}\) of D were proposed recently by Bokal et al. Let \({\vec{\chi_c}(G)={\rm max}\{\chi_c(D)| D\, {\rm is\, an\, orientation\, of} G\}}\). Let G be a planar graph and n ≥ 2. We prove that if the girth of G is at least \({\frac{10n-5}{3},}\) then \({\vec{\chi_c}(G)\leq \frac{n}{n-1}}\). We also study the circular chromatic number of some special planar digraphs.