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.
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References
Bokal D., Fijavž G., Juvan M., Kayll P.M., Mohar B.: The circular chromatic number of a digraph. J. Graph Theory 46(3), 227–240 (2004)
Bokal, D., Fijavž, G., Juvan, M., Mohar, B., Vodopivec, A.: Circular chromatic number of hexagonal chains with orientations (Personal Communications)
Borodin O.V., Kim S.-J., Kostochka A.V., West D.B.: Homomorphisms from sparse graphs with large girth. J. Comb. Theory Ser. B 90, 147–159 (2004)
McDiarmid, C., Mohar, B.: Brooks theorem for digraphs (in press)
Raspaud A., Wang W.: On the vertex-arboricity of planar graphs. Eur. J. Comb. 29, 1064–1075 (2008)
Vince A.: Star chromatic number. J. Graph Theory 12(4), 551–559 (1988)
Zhu X.: Circular chromatic number: a survey. Discret. Math. 229(1–3), 371–410 (2001)
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This research was supported by NSFC Grants (61070230, 11026184, 11101243), IIFSDU (2009hw001), RFDP (20100131120017) and SRF for ROCS.
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Wang, G., Liu, B., Yu, J. et al. Circular Coloring of Planar Digraphs. Graphs and Combinatorics 28, 889–900 (2012). https://doi.org/10.1007/s00373-011-1084-4
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DOI: https://doi.org/10.1007/s00373-011-1084-4