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Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals. (English) Zbl 1457.13051
Summary: Hibi conjectured that if a toric ideal has a quadratic Gröbner basis, then the toric ideal has either a lexicographic or a reverse lexicographic quadratic Gröbner basis. In this paper, we present a cut ideal of a graph that serves as a counterexample to this conjecture. We also discuss the existence of a quadratic Gröbner basis of a cut ideal of a cycle. U. Nagel and S. Petrović [J. Commut. Algebra 1, No. 3, 547–565 (2009; Zbl 1190.13016)] claimed that a cut ideal of a cycle has a lexicographic quadratic Gröbner basis using the results of J. Chifman and S. Petrović [Lect. Notes Comput. Sci. 4545, 307–321 (2007; Zbl 1127.92030)]. However, we point out that the results of Chifman and Petrović [loc. cit.] used by Nagel and Petrović [loc. cit.] are incorrect for cycles of length greater than or equal to 6. Hence the existence of a quadratic Gröbner basis for the cut ideal of a cycle (a ring graph) is an open question. We also provide a lexicographic quadratic Gröbner basis of a cut ideal of a cycle of length less than or equal to 7.

MSC:
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13A70 General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.)
05E40 Combinatorial aspects of commutative algebra
Software:
CoCoA; Risa/Asir
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References:
[1] Abbott, J.; Bigatti, A. M.; Robbiano, L., CoCoA: a system for doing computations in commutative algebra, Available at
[2] Aoki, S.; Hibi, T.; Ohsugi, H.; Takemura, A., Gröbner bases of nested configurations, J. Algebra, 320, 6, 2583-2593 (2008) · Zbl 1152.14047
[3] Aoki, S.; Hibi, T.; Ohsugi, H.; Takemura, A., Markov basis and Gröbner basis of Segre-Veronese configuration for testing independence in group-wise selections, Ann. Inst. Stat. Math., 62, 299-321 (2010) · Zbl 1440.62219
[4] Aramova, A.; Herzog, J.; Hibi, T., Finite lattices and lexicographic Gröbner bases, Eur. J. Comb., 21, 431-439 (2000) · Zbl 0969.13010
[5] Chifman, J.; Petrović, S., Toric ideals of phylogenetic invariants for the general group-based model on claw trees \(K_{1 , n}\), (Anai, H.; Horimoto, K.; Kutsia, T., Proceedings of the Second International Conference on Algebraic Biology. Proceedings of the Second International Conference on Algebraic Biology, Springer LNCS, vol. 4545 (2007), Springer-Verlag), 307-321 · Zbl 1127.92030
[6] D’Alì, A., Toric ideals associated with gap-free graphs, J. Pure Appl. Algebra, 219, 9, 3862-3872 (2015) · Zbl 1435.13022
[7] Deza, M.; Laurent, M., Geometry of Cuts and Metrics (1997), Springer-Verlag: Springer-Verlag Berlin · Zbl 0885.52001
[8] Engström, A., Cut ideals of \(K_4\)-minor free graphs are generated by quadratics, Mich. Math. J., 60, 3, 705-714 (2011) · Zbl 1234.14036
[9] Hibi, T.; Nishiyama, K.; Ohsugi, H.; Shikama, A., Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases, J. Algebra, 408, 138-146 (2014) · Zbl 1304.13040
[10] Nagel, U.; Petrović, S., Properties of cut ideals associated to ring graphs, J. Commut. Algebra, 1, 547-565 (2009) · Zbl 1190.13016
[11] Noro, N., Risa/Asir, a computer algebra system · Zbl 1027.68152
[12] Ohsugi, H.; Herzog, J.; Hibi, T., Combinatorial pure subrings, Osaka J. Math., 37, 745-757 (2000) · Zbl 1096.13525
[13] Ohsugi, H.; Hibi, T., Toric ideals generated by quadratic binomials, J. Algebra, 218, 509-527 (1999) · Zbl 0943.13014
[14] Ohsugi, H.; Hibi, T., Koszul bipartite graphs, Adv. Appl. Math., 22, 25-28 (1999) · Zbl 0916.05046
[15] Ohsugi, H.; Hibi, T., Compressed polytopes, initial ideals and complete multipartite graphs, Ill. J. Math., 44, 391-406 (2000) · Zbl 0943.13016
[16] Ohsugi, H.; Hibi, T., Quadratic initial ideals of root systems, Proc. Am. Math. Soc., 130, 1913-1922 (2002) · Zbl 1012.13012
[17] Ohsugi, H.; Hibi, T., Two way subtable sum problems and quadratic Gröbner bases, Proc. Am. Math. Soc., 137, 5, 1539-1542 (2009) · Zbl 1162.13012
[18] Ohsugi, H.; Hibi, T., Toric rings and ideals of nested configurations, J. Commut. Algebra, 2, 187-208 (2010) · Zbl 1237.13053
[19] Shibata, K., Strong Koszulness of the toric ring associated to a cut ideal, Comment. Math. Univ. St. Pauli, 64, 71-80 (2015) · Zbl 1330.13017
[20] Sturmfels, B., Gröbner Bases and Convex Polytopes (1996), American Mathematical Society · Zbl 0856.13020
[21] Sturmfels, B.; Sullivant, S., Toric geometry of cuts and splits, Mich. Math. J., 57, 689-709 (2008) · Zbl 1180.13040
[22] Sullivant, S., Toric fiber products, J. Algebra, 316, 560-577 (2007) · Zbl 1129.13030
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