Lexicographic and reverse lexicographic quadratic Gr\"obner bases of cut ideals

Hibi conjectured that if a toric ideal has a quadratic Gr\"obner basis, then the toric ideal has either a lexicographic or a reverse lexicographic quadratic Gr\"obner 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 Gr\"obner basis of a cut ideal of a cycle. Nagel and Petrovi\'c claimed that a cut ideal of a cycle has a lexicographic quadratic Gr\"obner basis using the results of Chifman and Petrovi\'c. However, we point out that the results of Chifman and Petrovi\'c used by Nagel and Petrovi\'c are incorrect for cycles of length greater than or equal to 6. Hence the existence of a quadratic Gr\"obner basis for the cut ideal of a cycle (a ring graph) is an open question. We also provide a lexicographic quadratic Gr\"obner basis of a cut ideal of a cycle of length less than or equal to 7.