Abstract
In this chapter we consider classes of binomial ideals which are naturally attached to finite simple graphs. The first of these classes are the binomial edge ideals. These ideals may also be viewed as ideals generated by a subset of 2-minors of a (2 × n)-matrix of indeterminates. Their Gröbner bases will be computed. Graphs whose binomial edge ideals have a quadratic Gröbner basis are called closed graphs. A full classification of closed graphs is given. For an arbitrary graph the initial ideal of the binomial edge ideal (for a suitable monomial order) is a squarefree monomial ideal. This has the pleasant consequence that the binomial edge ideal itself is a radical ideal. Its minimal prime ideals are determined in terms of cut point properties of the underlying graph. Based on this information, the closed graphs whose binomial edge ideal is Cohen–Macaulay are classified. In the subsequent sections, the resolution of binomial edge ideals is considered and a bound for the Castelnuovo–Mumford regularity of these ideals is given. Finally, the Koszul property of binomial edge ideals is studied. Intimately related to binomial edge ideals are permanental edge ideals and Lovász, Saks, and Schrijver edge ideals. Their primary decomposition will be studied.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Badiane, M., Burke, I., Sködberg, E.: The universal Gröbner basis of a binomial edge ideal. Electron. J. Comb. 24, #P4.11 (2017)
Banerjee, A., Núñez-Betancout, L.: Graph connectivity and binomial edge ideals. Proc. Am. Math. Soc. 145, 487–499 (2017)
Bruns, W., Conca, A.: Gröbner bases, initial ideals and initial algebras. In: Avramov, L.L., et al. (eds.) Homological Methods in Commutative Algebra, IPM Proceedings, Tehran (2004)
Chaudhry, F., Dokuyucu, A., Irfan, R.: On the binomial edge ideals of block graphs. An. Ştiinţ. Univ. Ovidius Constanţa 24, 149–158 (2016)
Conca, A, De Negri, E., Gorla, E.: Cartwright-Sturmfels ideals associated to graphs and linear spaces. arXiv:1705.00575
Conca, A., Welker, V.: Lovasz-Saks-Schrijver ideals and coordinate sections of determinantal varieties. arXiv:1801.07916
Corso, A., Nagel, U.: Monomial and toric ideals associated to Ferrers graphs. Trans. Am. Math. Soc. 361, 1371–1395 (2009)
Cox, D.A., Erskine, A.: On closed graphs I. Ars Combinatorica 120, 259–274 (2015)
Crupi, M., Rinaldo, G.: Binomial edge ideals with quadratic Gröbner bases. Electron. J. Comb. 18, ♯ P211 (2011)
Crupi, M., Rinaldo, G.: Closed graphs are proper interval graphs. An. Ştiinţ. Univ. Ovidius Constanţa 22, 37–44 (2014)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H. Singular 4-1-0 – A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2016)
Dirac, G.A.: On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 38, 71–76 (1961)
Dokuyucu, A.: Extremal Betti numbers of some classes of binomial edge ideals. Math. Rep. 17, 359–367 (2015)
Eisenbud, D., Sturmfels, B.: Binomial ideals. Duke Math. J. 84, 1–45 (1996)
Ene, V., Herzog, J., Hibi, T.: Cohen–Macaulay binomial edge ideals. Nagoya Math. J. 204, 57–68 (2011)
Ene, V., Herzog, J., Hibi, T.: Koszul binomial edge ideals. In: Ibadula, D., Veys, W. (eds.) Bridging Algebra, Geometry, and Topology. Springer Proceedings in Mathematics and Statistics, vol. 96, pp. 125–136. Springer, Switzerland (2014)
Ene, V., Herzog, J., Hibi, T.: Linear flags and Koszul filtrations. Kyoto J. Math. 55, 517–530 (2015)
Ene, V., Herzog, J., Hibi, T., Mohammadi, F.: Determinantal facet ideals. Mich. Math. J. 62, 39–57 (2013)
Ene, V., Herzog, J., Hibi, T., Qureshi, A.A.: The binomial edge ideal of a pair of graphs. Nagoya Math. J. 213, 105–125 (2014)
Ene, V., Zarojanu, A.: On the regularity of binomial edge ideals. Math. Nachr. 288, 19–24 (2015)
Fishburn, P.C.: Interval graphs and interval order. Discret. Math. 55, 135–149 (1985)
Gardi, F.: The Roberts characterization of proper and unit interval graphs. Discret. Math. 307, 2906–2908 (2007)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic, New York (1980)
Hajós, G.: Über eine Art von Graphen. Internationale Mathematische Nachrichten 11, 65–65 (1957)
Heggernes, P., Meister, D., Papadopoulos, C.: A new representation of proper interval graphs with an application to clique-width. DIMAP workshop on algorithmic graph theory 2009. Electron. Notes Discret. Math. 32, 27–34 (2009)
Herzog, J., Hibi, T.: Distributive lattices, bipartite graphs and Alexander duality. J. Algebraic Comb. 22, 289–302 (2005)
Herzog, J., Hibi, T., Hreinsdóttir, F., Kahle, T., Rauh, J.: Binomial edge ideals and conditional independence statements. Adv. Appl. Math. 45, 317–333 (2010)
Herzog, J., Kiani, D., Madani, S.S.: The linear strand of determinantal facet ideals. Mich. Math. J. 66, 10–123 (2017)
Herzog, J., Macchia, A., Madani, S.S., Welker, V.: On the ideal of orthogonal representations of a graph in \(\mathbb {R}^2\). Adv. Appl. Math. 71, 146–173 (2015)
Kahle, T., Sarmiento, C. Windisch, T.: Parity binomial edge ideals. J. Algebr. Comb. 44, 99–117 (2016)
Kiani, D., Madani, S.S.: Binomial edge ideals with pure resolutions. Collect. Math. 65, 331–340 (2014)
Kiani, D., Madani, S.S.: The Castelnuovo-Mumford regularity of binomial edge ideals. J. Comb. Theory Ser. A. 139, 80–86 (2016)
Laubenbacher, R.C., Swanson, I.: Permanental ideals. J. Symb. Comput. 30, 195–205 (2000)
Looges, P.J., Olariu, S.: Optimal greedy algorithms for indifference graphs. Comput. Math. Appl. 25, 15–25 (1993)
Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25, 1–7 (1979)
Lovász, L., Saks, M., Schrijver, A.: Orthogonal representations and connectivity of graphs. Linear Algebra Appl. 114/115, 439–454 (1989)
Matsuda, M., Murai, S.: Regularity bounds for binomial edge ideals. J. Commut. Algebra 5, 141–149 (2013)
Milne, J.S.: Algebraic Geometry, Version 6.00, (2014), Available at www.jmilne.org/math/
Mohammadi, F., Sharifan, L.: Hilbert function of binomial edge ideals. Commun. Algebra 42, 688–703 (2014)
Ohtani, M.: Graphs and Ideals generated by some 2-minors. Commun. Algebra 39, 905–917 (2011)
Rauf, A., Rinaldo, G.: Construction of Cohen-Macaulay binomial edge ideals. Commun. Algebra 42, 238–252 (2014)
Rauh, J.: Generalized binomial edge ideals. Adv. Appl. Math. 50, 409–414 (2013)
Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic, New York (1969)
Roberts, F.S.: Graph Theory and Its Applications to Problems of Society. SIAM Press, Philadelphia (1978)
Saeedi Madani, S., Kiani, D.: Binomial edge ideals of graphs. Electron. J. Combin. 19, ♯ P44 (2012)
Saeedi Madani, S., Kiani, D.: On the binomial edge ideal of a pair of graphs. Electron. J. Combin. 20, ♯ P48 (2013)
Schenzel, P., Zafar, S.: Algebraic properties of the binomial edge ideal of a complete bipartite graph. An. St. Univ. Ovidius Constanţa 22, 217–238 (2014)
Woodroofe, R: Matching, coverings, and Castelnuovo-Mumford regularity. J. Commut. Algebra 6, 287–304 (2014)
Zahid, Z., Zafar, S.; On the Betti numbers of some classes of binomial edge ideals. Electron. J. Combin. 20, ♯ P37 (2013)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Herzog, J., Hibi, T., Ohsugi, H. (2018). Binomial Edge Ideals and Related Ideals. In: Binomial Ideals. Graduate Texts in Mathematics, vol 279. Springer, Cham. https://doi.org/10.1007/978-3-319-95349-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-95349-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95347-2
Online ISBN: 978-3-319-95349-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)