  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.
