Skip to main content

Parity binomial edge ideals


Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gröbner bases and are radical if and only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. Bayer, D., Popescu, S., Sturmfels, B.: Syzygies of unimodular Lawrence ideals. Journal für die Reine und Angewandte Mathematik 534, 169–186 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Conradi, C., Kahle, T.: Detecting binomiality. Adv. Appl. Math. 71, 52–67 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. De Loera, J.A., Hemmecke, R., Köppe, M.: Algebraic and Geometric Ideas in the Theory of Discrete Optimization. MPS-SIAM Series on Optimization, SIAM, Cambridge (2013)

    MATH  Google Scholar 

  4. Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics, Oberwolfach Seminars, A Birkhäuser book, vol. 39, Springer, Berlin (2009)

  5. Eisenbud, D., Sturmfels, B.: Binomial ideals. Duke Math. J. 84(1), 1–45 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry.

  7. Grossman, J.W., Kulkarni, D.M., Schochetman, I.E.: On the minors of an incidence matrix and its Smith normal form. Linear Algebra Appl. 218(1995), 213–224 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Herzog, J., Hibi, T., Hreinsdóttir, F., Kahle, T., Rauh, J.: Binomial edge ideals and conditional independence statements. Adv. Appl. Math. 45(3), 317–333 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Herzog, J., Macchia, A., Madani, S.S., Welker, V.: On the ideal of orthogonal representations of a graph. Adv. Appl. Math. 71, 146–173 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hoşten, S., Shapiro, J.: Primary decomposition of lattice basis ideals. J. Symb. Comput. 29(4–5), 625–639 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Hoşten, S., Sullivant, S.: Ideals of adjacent minors. J. Algebra 277(2), 615–642 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kahle, T.: Decompositions of binomial ideals. J. Softw. Algebra Geom. 4, 1–5 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kahle, T., Miller, E.: Decompositions of commutative monoid congruences and binomial ideals. Algebra Number Theory 8(6), 1297–1364 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kahle, T., Rauh, J., Sullivant, S.: Positive margins and primary decomposition. J. Commut. Algebra 6(2), 173–208 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Miller, E.: Theory and applications of lattice point methods for binomial ideals. In: Proceedings of the Abel Symposium held at Voss, Norway, 2009, pp. 99–154. Springer (2011)

  16. Miller, E.: Affine stratifications from finite misère quotients. J. Algebraic Comb. 37(1), 1–9 (2013)

    Article  MATH  Google Scholar 

  17. Pérez Millán, M., Dickenstein, A., Shiu, A., Conradi, C.: Chemical reaction systems with toric steady states. Bull. Math. Biol. 74, 1027–1065 (2012)

  18. Rauh, J., Sullivant, S.: Lifting Markov bases and higher codimension toric fiber products. J. Symb. Comput. 74, 276–307 (2016). doi:10.1016/j.jsc.2015.07.003

    Article  MathSciNet  MATH  Google Scholar 

  19. Swanson, I.: On the embedded primes of the Mayr–Meyer ideals. J. Algebra 275, 143–190 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Windisch, T.: BinomialEdgeIdeals, a Macaulay2 package for (parity) binomial edge ideals.

Download references


The authors would like to thank Rafael Villareal for posting the question of radicality of parity binomial edge ideals. We thank Fatemeh Mohammadi for pointing us at [9]. The authors appreciate the many comments and suggestions by Issac Burke and Mourtadha Badiane. T.K. and C.S. are supported by the Center for Dynamical Systems (CDS) at Otto-von-Guericke University Magdeburg. T.W. is supported by the German National Academic Foundation and TopMath, a graduate program of the Elite Network of Bavaria.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Thomas Kahle.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kahle, T., Sarmiento, C. & Windisch, T. Parity binomial edge ideals. J Algebr Comb 44, 99–117 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Binomial ideals
  • Primary decomposition
  • Mesoprimary decomposition
  • Binomial edge ideals
  • Markov bases

Mathematics Subject Classification

  • Primary 05E40
  • Secondary 13P10
  • 05C38