Lexicographic and reverse lexicographic quadratic Gröbner bases of cut ideals

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. Nagel and Petrović claimed that a cut ideal of a cycle has a lexicographic quadratic Gröbner basis using the results of Chifman and Petrović. However, we point out that the results of Chifman and Petrović used by Nagel and Petrović 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.