Mathematische Zeitschrift

, Volume 270, Issue 1–2, pp 1–18 | Cite as

Ehrhart theory, modular flow reciprocity, and the Tutte polynomial

  • Felix Breuer
  • Raman Sanyal
Open Access


Given an oriented graph G, the modular flow polynomial \({\phi_G(k)}\) counts the number of nowhere-zero \({\mathbb{Z}_k}\)-flows of G. We give a description of the modular flow polynomial in terms of (open) Ehrhart polynomials of lattice polytopes. Using Ehrhart–Macdonald reciprocity we give a combinatorial interpretation for the values of \({\phi_G}\) at negative arguments which answers a question of Beck and Zaslavsky (Adv Math 205:134–162, 2006). Our construction extends to \({\mathbb{Z}_{\l}}\)-tensions and we recover Stanley’s reciprocity theorem for the chromatic polynomial. Combining the combinatorial reciprocity statements for flows and tensions, we give an enumerative interpretation for positive evaluations of the Tutte polynomial t G (x, y) of G.


Nowhere-zero flows Modular flow polynomial Reciprocity theorems Lattice-point counting Ehrhart theory Tutte polynomial 

Mathematics Subject Classification (2000)

Primary 05C31 Graph polynomials Secondary 05C21 Flows in graphs 52B20 Lattice polytopes 52C35 Arrangements of points, flats, hyperplanes 



We would like to thank Matthias Beck for valuable conversations and comments on an earlier version of this paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Babson, E., Beck, M.: Minimal-distance chromatic and modular flow polynomials. (In preparation.)Google Scholar
  2. 2.
    Beck, M., Robins, S.: Computing the continuous discretely: integer-point enumeration in polyhedra. In: Undergraduate Texts in Mathematics. Springer, New York (2007)Google Scholar
  3. 3.
    Beck M., Zaslavsky T.: Inside-out polytopes. Adv. Math. 205, 134–162 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Beck M., Zaslavsky T.: The number of nowhere-zero flows on graphs and signed graphs. J. Comb. Theory Ser. B 96, 901–918 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Breuer, F.: Ham sandwiches, staircases and counting polynomials. PhD thesis, Freie Universität Berlin (2009)Google Scholar
  6. 6.
    Brylawski, T., Oxley, J.: The Tutte polynomial and its applications. In: Matroid Applications. Encyclopedia Math. Appl., vol. 40, pp. 123–225. Cambridge University Press, Cambridge (1992)Google Scholar
  7. 7.
    Chen B.: Orientations, lattice polytopes, and group arrangements I: chromatic and tension polynomials of graphs. Ann. Comb. 13, 425–452 (2010)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, B., Stanley, R.P.: Orientations, lattice polytopes, and group arrangements II: modular and integral flow polynomials of graphs. PreprintGoogle Scholar
  9. 9.
    Ehrhart, E.: Polynômes arithmétiques et méthode des polyèdres en combinatoire. International Series of Numerical Mathematics, vol. 35. Birkhäuser Verlag, Basel (1977)Google Scholar
  10. 10.
    Gioan E.: Enumerating degree sequences in digraphs and a cycle-cocycle reversing system. Eur. J. Comb. 28, 1351–1366 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Greene C., Zaslavsky T.: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. Trans. Am. Math. Soc. 280, 97–126 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kochol M.: Polynomials associated with nowhere-zero flows. J. Comb. Theory Ser. B 84, 260–269 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kook W., Reiner V., Stanton D.: A convolution formula for the Tutte polynomial. J. Comb. Theory Ser. B 76, 297–300 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Matoušek, J.: Lectures on discrete geometry. In: Graduate Texts in Mathematics, vol. 212. Springer-Verlag New York (2002)Google Scholar
  15. 15.
    Reiner V.: An interpretation for the Tutte polynomial. Eur. J. Comb. 20, 149–161 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Schrijver, A.: Theory of linear and integer programming. In: Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1986)Google Scholar
  17. 17.
    Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Volume A. In: Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)Google Scholar
  18. 18.
    Stanley R.P.: Acyclic orientations of graphs. Discrete Math. 5, 171–178 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Stanley R.P.: Combinatorial reciprocity theorems. Adv. Math. 14, 194–253 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Tutte W.T.: A ring in graph theory. Proc. Camb. Philos. Soc. 43, 26–40 (1947)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Tutte W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Institut für Mathematik, Freie Universität BerlinBerlinGermany
  2. 2.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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