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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
##### Keywords:
Gröbner bases; cut ideals; finite graphs
CoCoA; Risa/Asir
Full Text:
##### References:
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