Nagoya Mathematical Journal

The binomial edge ideal of a pair of graphs

Viviana Ene, Jürgen Herzog, Takayuki Hibi, and Ayesha Asloob Qureshi

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We introduce a class of ideals generated by a set of 2-minors of an (m×n)-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gröbner basis and characterize unmixedness and Cohen–Macaulayness.

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Nagoya Math. J., Volume 213 (2014), 105-125.

First available in Project Euclid: 6 November 2013

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Primary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13C13: Other special types
Secondary: 13C15: Dimension theory, depth, related rings (catenary, etc.) 13P25: Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)


Ene, Viviana; Herzog, Jürgen; Hibi, Takayuki; Qureshi, Ayesha Asloob. The binomial edge ideal of a pair of graphs. Nagoya Math. J. 213 (2014), 105--125. doi:10.1215/00277630-2389872.

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  • [1] J. F. Andrade, Regular sequences of minors, Comm. Algebra 9 (1981), 765–781.
  • [2] W. Bruns and A. Conca, “Gröbner bases and determinantal ideals” in Commutative Algebra, Singularities and Computer Algebra (Sinaia, 2002), NATO Sci. Ser. II Math. Phys. Chem. 115, Kluwer, Dordrecht, 2003, 9–66.
  • [3] M. Crupi and G. Rinaldo, Binomial edge ideals with quadratic Gröbner bases, Electron. J. Combin. 18 (2011), paper 211.
  • [4] P. Diaconis, D. Eisenbud, and B. Sturmfels, “Lattice walks and primary decomposition” in Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, Mass., 1996), Progr. Math. 161, Birkhäuser, Boston, 1998, 173–193.
  • [5] V. Ene, J. Herzog, and T. Hibi, Cohen–Macaulay binomial edge ideals, Nagoya Math. J. 204 (2011), 57–68.
  • [6] J. Herzog and T. Hibi, Ideals generated by adjacent $2$-minors, J. Commut. Algebra 4 (2012), 525–549.
  • [7] J. Herzog, T. Hibi, F. Hreinsdóttir, T. Kahle, and J. Rauh, Binomial edge ideals and conditional independence statements, Adv. in Appl. Math. 45 (2010), 317–333.
  • [8] M. Hochster and J. A. Eagon, Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058.
  • [9] S. Hoşten and J. Shapiro, “Primary decomposition of lattice basis ideals” in Symbolic Computation in Algebra, Analysis, and Geometry (Berkeley, 1998), J. Symbolic Comput. 29, Elsevier, Amsterdam, 2000, 625–639.
  • [10] S. Hoşten and S. Sullivant, Ideals of adjacent minors, J. Algebra 277 (2004), 615–642.
  • [11] M. Ohtani, Graphs and ideals generated by some $2$-minors, Comm. Algebra 39 (2011), 905–917.
  • [12] J. Rauh and N. Ay, Robustness and conditional independence ideals, preprint, arXiv:1110.1338 [math.AC].
  • [13] B. Sturmfels, Gröbner bases and Stanley decompositions of determinantal rings, Math. Z. 205 (1990), 137–144.