Skip to main content
SpringerLink
Log in

Regularity of powers of (parity) binomial edge ideals

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

In this paper, we provide exact formulas for the regularity of powers of an almost complete intersection ideal I which is generated by a homogeneous d-sequence. As applications, when I is an almost complete intersection, taking the form of a (parity) binomial edge ideal of a connected graph, we can describe explicitly \({\text {reg}}(I^t)\) for \(t\ge 2\). The only exception is when I is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

The data used to support the findings of this study are included within the article.

References

  1. Banerjee, A., Beyarslan, S.K., Huy Tai, H.: Regularity of edge ideals and their powers. In: Advances in Algebra, pp. 17–52, Springer Proceedings in Mathematics & Statistics, vol. 277. Springer, Cham (2019)

  2. Berlekamp, D.: Regularity defect stabilization of powers of an ideal. Math. Res. Lett. 19, 109–119 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beyarslan, S., Ha, H.T., Trung, T.N.: Regularity of powers of forests and cycles. J. Algebraic Combin. 42, 1077–1095 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bolognini, D., Macchia, A., Strazzanti, F.: Binomial edge ideals of bipartite graphs. Europ. J. Combin. 70, 1–25 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cutkosky, S.D., Herzog, J., Trung, N.V.: Asymptotic behaviour of the Castelnuovo–Mumford regularity. Compositio Math. 118, 243–261 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eisenbud, D., Ulrich, B.: Notes on regularity stabilization. Proc. Amer. Math. Soc. 140, 1221–1232 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ene, V., Herzog, J., Hibi, T.: Cohen–Macaulay binomial edge ideals. Nagoya Math. J. 204, 57–68 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ene, V., Rinaldo, G., Terai, N.: Powers of binomial edge ideals with quadratic Grobner bases. Nagoya Math. J. 246, 1–23 (2021)

    MathSciNet  MATH  Google Scholar 

  9. Ene, V., Zarojanu, A.: On the regularity of binomial edge ideals. Math. Nachr. 288, 19–24 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grayson, D. R., Stillman, M. E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/

  11. Ha, H.T.: Asymptotic linearity of regularity and a* -invariant of powers of ideals. Math. Res. Lett. 18, 1–9 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Herzog, J., Macchia, A., Saeedi Madani, S., Welker, V.: On the ideal of orthogonal representations of a graph in R\(^{ 2 }\). Adv. Appl. Math. 71, 146–173 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hoa, L.T., Tam, N.D.: On some invariants of a mixed product of ideals. Arch. Math. (Basel) 94, 327–337 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huneke, C.: The theory of d-sequences and powers of ideals. Adv. Math. 46, 249–279 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jayanthan, A.V., Kumar, A.: Regularity of binomial edge ideals of Cohen–Macaulay bipartite graphs. Comm. Algebra 47, 4797–4805 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jayanthan, A.V., Kumar, A., Sarkar, R.: Regularity of powers of quadratic sequences with applications to binomial ideals. J. Algebra 564, 98–118 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jayanthan, A.V., Kumar, A., Sarkar, R.: Almost complete intersection binomial edge ideals and their Rees algebras. J. Pure Appl. Algebra 225, 106628 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jayanthan, A.V., Narayanan, N., Raghavendra Rao, B.V.: Regularity of binomial edge ideals of certain block graphs. Proc. Indian Acad. Sci. Math. Sci. 129, Paper No. 36 (2019)

  20. Jayanthan, A.V., Narayanan, N., Selvaraja, S.: Regularity of powers of bipartite graphs. J. Algebraic Combin. 47, 17–38 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kahle, T., Sarmiento, C., Windisch, T.: Parity binomial edge ideals. J. Algebraic Combin. 44, 99–117 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kiani, D., Saeedi Madani, S.: Some Cohen–Macaulay and unmixed binomial edge ideals. Comm. Algebra 43, 5434–5453 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kodiyalam, V.: Asymptotic behaviour of Castelnuovo–Mumford regularity. Proc. Amer. Math. Soc. 128, 407–411 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kumar, A.: Lovasz–Saks–Schrijver ideals and parity binomial edge ideals of graphs. Europ. J. Combin. 93, 103274 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kumar, A.: Regularity of parity binomial edge ideals. Proc. Amer. Math. Soc. 149, 2727–2737 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lovasz, L., Saks, M., Schrijver, A.: Orthogonal representations and connectivity of graphs. Linear Algebra Appl. 114(115), 439–454 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Matsuda, K., Murai, S.: Regularity bounds for binomial edge ideals. J. Commut. Algebra 5, 141–149 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mohammadi, F., Sharifan, L.: Hilbert function of binomial edge ideals. Comm. Algebra 42, 688–703 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nguyen, H.D., Vu, T.: Powers of sums and their homological invariants. J. Pure Appl. Algebra 223, 3081–3111 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ohtani, M.: Graphs and ideals generated by some 2-minors. Comm. Algebra 39, 905–917 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Raghavan, K.: Powers of ideals generated by quadratic sequences. Trans. Amer. Math. Soc. 343, 727–747 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rauf, A., Rinaldo, G.: Construction of Cohen–Macaulay binomial edge ideals. Comm. Algebra 42, 238–252 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rinaldo, G.: Cohen–Macaulay binomial edge ideals of cactus graphs. J. Algebra Appl. 18, 1950072 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rouzbahani Malayeri, M., Saeedi Madani, S., Kiani, D.: Regularity of binomial edge ideals of chordal graphs. Collect. Math. 72, 411–422 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Saeedi Madani, S., Kiani, D.: Binomial edge ideals of graphs, Electron. J. Combin. 19: 44, 6 (2012)

  36. Saeedi Madani, S., Kiani, D.: Binomial edge ideals of regularity 3. J. Algebra 515, 157–172 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sarkar, R.: Binomial edge ideals of unicyclic graphs. Internat. J. Algebra Comput. 31(7), 1293–1318 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhu, G., Xu, L., Wang, H., Zhang, J.: Regularity and projective dimension of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. J. Algebra Appl. 19, 2050233 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of Jiangsu Province (No. BK20221353). The first author is also partially supported by the “Anhui Initiative in Quantum Information Technologies” (No. AHY150200) and the second author is also supported by the foundation of the Priority Academic Program Development of Jiangsu Higher Education Institutions. Meanwhile, the authors are grateful to the software system Macaulay2 [10], for serving as an excellent source of inspiration. Finally, the authors would like to thank the referees who read carefully the manuscript and gave very helpful comments, which improved the paper both in mathematics and presentation.

Author information

Authors and Affiliations

  1. CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Yi-Huang Shen

  2. School of Mathematical Sciences, Soochow University, Suzhou, 215006, Jiangsu, People’s Republic of China

    Guangjun Zhu

Authors
  1. Yi-Huang Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guangjun Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangjun Zhu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shen, YH., Zhu, G. Regularity of powers of (parity) binomial edge ideals. J Algebr Comb 57, 75–100 (2023). https://doi.org/10.1007/s10801-022-01163-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-022-01163-w

Keywords

Mathematics Subject Classification

Search

Navigation