Journal of Algebraic Combinatorics

, Volume 33, Issue 3, pp 465–482 | Cite as

Bounds on the coefficients of tension and flow polynomials



The goal of this article is to obtain bounds on the coefficients of modular and integral flow and tension polynomials of graphs. To this end we use the fact that these polynomials can be realized as Ehrhart polynomials of inside-out polytopes. Inside-out polytopes come with an associated relative polytopal complex and, for a wide class of inside-out polytopes, we show that this complex has a convex ear decomposition. This leads to the desired bounds on the coefficients of these polynomials.


Ehrhart polynomial Inside-out polytope Convex ear decomposition Regular subdivision Polytopal complex M-vector 


  1. 1.
    Beck, M., Robins, S.: Computing the Continuous Discretely. Undergraduate Texts in Mathematics. Springer, New York (2007) MATHGoogle Scholar
  2. 2.
    Beck, M., Zaslavsky, T.: Inside-out polytopes. Adv. Math. 205(1), 134–162 (2006) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beck, M., Zaslavsky, T.: The number of nowhere-zero flows on graphs and signed graphs. J. Comb. Theory Ser. B 96(6), 901–918 (2006) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Billera, L.J., Björner, A.: Face Numbers of Polytopes and Complexes, 2nd edn., pp. 407–430. Chapman & Hall/CRC Press, London/Boca Raton (2004). Chap. 18 Google Scholar
  5. 5.
    Breuer, F.: Ham sandwiches, staircases and counting polynomials. Ph.D. thesis, Freie Universität, Berlin (2009) Google Scholar
  6. 6.
    Breuer, F., Dall, A.: Viewing counting polynomials as Hilbert functions via Ehrhart theory. In: 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010). DMTCS, pp. 413–424 (2010) Google Scholar
  7. 7.
    Breuer, F., Sanyal, R.: Ehrhart theory, modular flow reciprocity, and the Tutte polynomial. arXiv:0907.0845v1 (2009)
  8. 8.
    Bruns, W., Gubeladze, J.: Polytopes, Rings, and k-Theory. Springer Monographs in Mathematics. Springer, Berlin (2009) MATHGoogle Scholar
  9. 9.
    Bruns, W., Römer, T.: h-Vectors of Gorenstein polytopes. J. Combin. Theory, Ser. A 114(1), 65–76 (2007) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chari, M.K.: Two decompositions in topological combinatorics with applications to matroid complexes. Trans. Am. Math. Soc. 349(10), 3925–3943 (1997) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chen, B.: Orientations, lattice polytopes, and group arrangements i: Chromatic and tension polynomials of graphs. Ann. Combin. 13(4), 425–452 (2010) CrossRefGoogle Scholar
  12. 12.
    Dall, A.: The flow and tension complexes. Master’s thesis, San Francisco State University (2008) Google Scholar
  13. 13.
    Felsner, S., Knauer, K.: Distributive lattices, polyhedra, and generalized flow. arXiv:0811.1541, November 2008
  14. 14.
    Haase, C., Paffenholz, A.: Quadratic Gröbner bases for smooth 3×3 transportation polytopes. J. Algebraic Combin. 30(4), 477–489 (2009) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hersh, P., Swartz, E.: Coloring complexes and arrangements. J. Algebraic Combin. 27(2), 205–214 (2008) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hibi, T.: Dual polytopes of rational convex polytopes. Combinatorica 12(2), 237–240 (1992) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hultman, A.: Link complexes of subspace arrangements. Eur. J. Combin. 28(3), 781–790 (2007) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Jonsson, J.: The topology of the coloring complex. J. Algebraic Combin. 21, 311–329 (2005) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kochol, M.: Polynomials associated with nowhere-zero flows. J. Combin. Theory Ser. B 84(2), 260–269 (2002) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lee, C.W.: Subdivisions and Triangulations of Polytopes, 2nd edn., pp. 383–406. Chapman & Hall/CRC Press, London/Boca Raton (2004). Chap. 17 Google Scholar
  21. 21.
    Ohsugi, H., Hibi, T.: Convex polytopes all of whose reverse lexicographic initial ideals are squarefree. Proc. Am. Math. Soc. 129(9), 2541–2546 (2001) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1986) MATHGoogle Scholar
  23. 23.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences.
  24. 24.
    Stanley, R.P.: A monotonicity property of h-vectors and h -vectors. Eur. J. Combin. 14(3), 251–258 (1993) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Stanley, R.P.: Combinatorics and Commutative Algebra, 2nd edn. Birkhäuser, Basel (1996) MATHGoogle Scholar
  26. 26.
    Steingrímsson, E.: The coloring ideal and coloring complex of a graph. J. Algebraic Combin. 14, 73–84 (2001) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. Am. Math. Soc., Providence (1996) MATHGoogle Scholar
  28. 28.
    Swartz, E.: g-elements, finite buildings and higher Cohen–Macaulay connectivity. J. Combin. Theory Ser. A 113, 1305–1320 (2006) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    West, Douglas B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Berlin (2001) Google Scholar
  30. 30.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany
  2. 2.Universitat Politècnica de CatalunyaBarcelonaSpain

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