Abstract
Let G be a graph on the vertex set [n] and \(J_G\) the associated binomial edge ideal in the polynomial ring \(S=\mathbb {K}[x_1,\ldots ,x_n,y_1,\ldots ,y_n]\). In this paper, we investigate the depth of binomial edge ideals. More precisely, we first establish a combinatorial lower bound for the depth of \(S/J_G\) based on some graphical invariants of G. Next, we combinatorially characterize all binomial edge ideals \(J_G\) with \(\mathrm {depth} S/J_G=5\). To achieve this goal, we associate a new poset \(\mathscr {M}_G\) with the binomial edge ideal of G and then elaborate some topological properties of certain subposets of \(\mathscr {M}_G\) in order to compute some local cohomology modules of \(S/J_G\).
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References
Àlvarez Montaner, J.: Local cohomology of binomial edge ideals and their generic initial ideals. Collect. Math. 71, 331–348 (2020)
Àlvarez Montaner, J., Boix, A.F., Zarzuela, S.: On some local cohomology spectral sequences. Int. Math. Res. Not. 19, 6197–6293 (2020)
Banerjee, A., Núñez-Betancourt, L.: Graph connectivity and binomial edge ideals. Proc. Amer. Math. Soc. 145, 487–499 (2017)
Björner, A., Walker, J.W.: A homotopy complementation formula for partially ordered sets. European J. Combin. 4, 11–19 (1983)
Bolognini, D., Macchia, A., Strazzanti, F.: Binomial edge ideals of bipartite graphs. European J. Combin. 70, 1–25 (2018)
Bolognini, D., Macchia, A., Strazzanti, F.: Cohen-Macaulay binomial edge ideals and accessible graphs, (2021), arXiv:2101.03619v1
Ene, V., Herzog, J., Hibi, T.: Cohen-Macaulay binomial edge ideals. Nagoya Math. J. 204, 57–68 (2011)
Ene, V., Rinaldo, G., Terai, N.: Licci binomial edge ideals. J. Combin. Theory Ser. A. 175, 23 (2020). 105278
Ene, V., Rinaldo, G., Terai, N.: Powers of binomial edge ideals with quadratic Gröbner bases. Nagoya Math. J. (2021). https://doi.org/10.1017/nmj.2021.1
Ene, V., Zarojanu, A.: On the regularity of binomial edge ideals. Math. Nachr. 288(1), 19–24 (2015)
González-Martínez, R.: Gorenstein binomial edge ideals, (2021). arXiv:2102.11188v1
Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu.Macaulay2/
Herzog, J., Hibi, T., Hreinsdóttir, F., Kahle, T., Rauh, J.: Binomial edge ideals and conditional independence statements. Adv. Appl. Math. 45, 317–333 (2010)
Jayanthan, A.V., Kumar, A., Sarkar, R.: Regularity of powers of quadratic sequences with applications to binomial ideals. J. Algebra. 564, 98–118 (2020)
Kiani, D., Saeedi Madani, S.: The Castelnuovo-Mumford regularity of binomial edge ideals. J. Combin. Theory Ser. A. 139, 80–86 (2016)
Kumar, A.: Regularity bound of generalized binomial edge ideal of graphs. J. Algebra. 546, 357–369 (2020)
Kumar, A.: Lovász-Saks-Schrijver ideals and parity binomial edge ideals of graphs. European J. Combin. 93, 103274 (2021)
Kumar, A.: Ress algebra and special fiber ring of binomial edge ideals of closed graphs, (2021), arXiv:2102.03348v1
Kumar, A., Sarkar, R.: Depth and extremal Betti number of binomial edge ideals. Math. Nachr. 293(9), 1746–1761 (2020)
Matsuda, K., Murai, S.: Regularity bounds for binomial edge ideals. J. Commutative Algebra. 5(1), 141–149 (2013)
Ohtani, M.: Graphs and ideals generated by some 2-minors. Comm. Algebra. 39, 905–917 (2011)
Rouzbahani Malayeri, M., Saeedi Madani, S., Kiani, D.: Regularity of binomial edge ideals of chordal graphs, Collect. Math. (2020). https://doi.org/10.1007/s13348-020-00293-3
Rouzbahani Malayeri, M., Saeedi Madani, S., Kiani, D.: Binomial edge ideals of small depth. J. Algebra. 572, 231–244 (2021)
Rouzbahani Malayeri, M., Saeedi Madani, S., Kiani, D.: A proof for a conjecture on the regularity of binomial edge ideals. J. Combin. Theory Ser. A. 180, 9 (2021) 105432
Saeedi Madani, S., Kiani, D.: Binomial edge ideals of graphs. Electron. J. Combin. 19(2), \(\sharp \)P44 (2012)
Saeedi Madani, S., Kiani, D.: Binomial edge ideals of regularity 3. J. Algebra. 515, 157–172 (2018)
Sarkar, R.: Binomial edge ideals of unicyclic graphs, (2020). arXiv:1911.12677v2
Schenzel, P., Zafar, S.: Algebraic properties of the binomial edge ideal of a complete bipartite graph. An. St. Univ. Ovidius Constanta, Ser. Mat. 22(2), 217–237 (2014)
Walker, J.W.: Homotopy type and Euler characteristic of partially ordered sets. European J. Combin. 2, 373–384 (1981)
Zafar, S., Zahid, Z.: On the Betti numbers of some classes of binomial edge ideals. Electron. J. Combin. 20(4), \(\sharp \) P37 (2013)
Acknowledgements
The authors would like to thank the anonymous referees for their careful reading of the manuscript and for their valuable comments and suggestions. The authors would also like to thank Institute for Research in Fundamental Sciences (IPM) for financial support. The research of the second author was in part supported by a grant from IPM (No. 1400130113).
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Rouzbahani Malayeri, M., Saeedi Madani, S. & Kiani, D. On the depth of binomial edge ideals of graphs. J Algebr Comb 55, 827–846 (2022). https://doi.org/10.1007/s10801-021-01072-4
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DOI: https://doi.org/10.1007/s10801-021-01072-4