Abstract
An acyclic coloring of a digraph as defined by V. Neumann-Lara is a vertex-coloring such that no monochromatic directed cycles occur. Counting the number of such colorings with k colors can be done by counting so-called Neumann-Lara-coflows (NL-coflows), which build a polynomial in k. We will present a representation of this polynomial using totally cyclic subdigraphs, which form a graded poset Q. Furthermore we will decompose our NL-coflow polynomial, which becomes the chromatic polynomial of a digraph by multiplication with the number of colors to the number of components, using the geometric structure of the face lattices of a class of polyhedra that corresponds to Q. This decomposition leads to a representation using certain subsets of edges of the underlying undirected graph and will confirm the equality of our chromatic polynomial of a digraph and the chromatic polynomial of the underlying undirected graph in the case of symmetric digraphs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aigner, M.: Combinatorial Theory. Springer, Berlin (1980)
Altenbokum, B., Hochstättler, W., Wiehe, J.: The NL-flow polynomial. Discrete Appl. Math. (2021 in press). https://doi.org/10.1016/j.dam.2020.02.011
Beck, M., Sanyal, R.: Combinatorial Reciprocity Theorems. American Math. Society, Providence (2018)
Bokal, D., Fijavz, G., Juvan, M., Kayll, P.M., Mohar, B.: The circular chromatic number of a digraph. J. Graph Theory 46(3), 227–240 (2004)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, London (2008)
Ellis, P., Soukup, D.T.: Cycle reversions and dichromatic number in tournaments. Eur. J. Comb. 77, 31–48 (2019)
Erdős, P.: Problems and results in number theory and graph theory. In: Proc. Ninth Manitoba Conf. on Numerical Math. and Computing, pp. 3–21 (1979)
Erdős, P., Gimbel, J., Kratsch, D.: Some extremal results in cochromatic and dichromatic theory. J. Graph Theory 15(6), 579–585 (1991)
Hochstättler, W.: A flow theory for the dichromatic number. Eur. J. Comb. 66, 160–167 (2017)
Li, Z., Mohar, B.: Planar digraphs of digirth four are 2-colorable. SIAM J. Discret. Math. 31, 2201–2205 (2017)
Mohar, B., Wu, H.: Dichromatic number and fractional chromatic number. Forum Math. Sigma 4, e32 (2016)
Neumann-Lara, V.: The dichromatic number of a digraph. J. Comb. Theory Ser. B 33, 265–270 (1982)
Neumann-Lara, V.: Vertex colourings in digraphs. Some problems. Tech. rep., University of Waterloo (1985)
Neumann-Lara, V.: The 3 and 4-dichromatic tournaments of minimum order. Discrete Math. 135(1), 233–243 (1994)
Rota, G.C.: On the foundations of combinatorial theory. Z. Wahrscheinlichkeitstheorie Verw. Geb. 2, 340–368 (1964)
Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2011)
Tutte, W.T.: A contribution to the theory of chromatic polynomials. Canad. J. Math. 6, 80–91 (1954)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hochstättler, W., Wiehe, J. (2021). The Chromatic Polynomial of a Digraph. In: Gentile, C., Stecca, G., Ventura, P. (eds) Graphs and Combinatorial Optimization: from Theory to Applications. AIRO Springer Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-63072-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-63072-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-63071-3
Online ISBN: 978-3-030-63072-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)