Parity binomial edge ideals

Abstract

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.

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Acknowledgments

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.

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Correspondence to Thomas Kahle.

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Kahle, T., Sarmiento, C. & Windisch, T. Parity binomial edge ideals. J Algebr Comb 44, 99–117 (2016). https://doi.org/10.1007/s10801-015-0657-3

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Keywords

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

Mathematics Subject Classification

  • Primary 05E40
  • Secondary 13P10
  • 05C38