Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T06:12:04.153Z Has data issue: false hasContentIssue false
  • English
  • Français

The binomial edge ideal of a pair of graphs

Published online by Cambridge University Press:  11 January 2016

Viviana Ene ,
Jürgen Herzog ,
Takayuki Hibi  and
Ayesha Asloob Qureshi
Viviana Ene
Affiliation:
Faculty of Mathematics and Computer Science Ovidius University, 900527 Constanta, Romania, vivian@univ-ovidius.ro
Jürgen Herzog
Affiliation:
Fachbereich Mathematik Universität Duisburg-Essen, Campus Essen 45117 Essen, Germany, juergen.herzog@uni-essen.de
Takayuki Hibi
Affiliation:
Department of Pure and Applied Mathematics Graduate School of Information Science and Technology Osaka University, Toyonaka Osaka 560-0043, Japan, hibi@math.sci.osaka-u.ac.jp
Ayesha Asloob Qureshi
Affiliation:
Abdus Salam School of Mathematical Sciences GC University, New Muslim Town Lahore 54600, Pakistan, ayesqi@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

13P1013C1313C1513P25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 213 , March 2014 , pp. 105 - 125
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Andrade, J. F., Regular sequences of minors, Comm. Algebra 9 (1981), 765–781. MR 0609222. DOI 10.1080/00927878108822617.Google Scholar
[2] Bruns, W. and Conca, A., “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. MR 2030262.Google Scholar
[3] Crupi, M. and Rinaldo, G., Binomial edge ideals with quadratic Gröbner bases, Electron. J. Combin. 18 (2011), paper 211. MR 2853068.Google Scholar
[4] Diaconis, P., Eisenbud, D., and Sturmfels, B., “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. MR 1627343.Google Scholar
[5] Ene, V., Herzog, J., and Hibi, T., Cohen–Macaulay binomial edge ideals, Nagoya Math. J. 204 (2011), 57–68. MR 2863365.Google Scholar
[6] Herzog, J. and Hibi, T., Ideals generated by adjacent 2-minors, J. Commut. Algebra 4 (2012), 525–549. MR 3053451. DOI 10.1216/JCA-2012-4-4-525.Google Scholar
[7] Herzog, J., Hibi, T., Hreinsdóottir, F., Kahle, T., and Rauh, J., Binomial edge ideals and conditional independence statements, Adv. in Appl. Math. 45 (2010), 317–333. MR 2669070. DOI 10.1016/j.aam.2010.01.003.CrossRefGoogle Scholar
[8] Hochster, M. and Eagon, J. A., Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058. MR 0302643.Google Scholar
[9] Hoşten, S. and Shapiro, J., “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. MR 1769658. DOI 10.1006/jsco.1999.0397.Google Scholar
[10] Hoşten, S. and Sullivant, S., Ideals of adjacent minors, J. Algebra 277 (2004), 615–642. MR 2067622. DOI 10.1016/j.jalgebra.2004.01.027.Google Scholar
[11] Ohtani, M., Graphs and ideals generated by some 2-minors, Comm. Algebra 39 (2011), 905–917. MR 2782571. DOI 10.1080/00927870903527584.Google Scholar
[12] Rauh, J. and Ay, N., Robustness and conditional independence ideals, preprint, arXiv:1110.1338 [math.AC].Google Scholar
[13] Sturmfels, B., Gröbner bases and Stanley decompositions of determinantal rings, Math. Z. 205 (1990), 137–144. MR 1069489. DOI 10.1007/BF02571229.Google Scholar