Year: 2026
Type: preprint
Abstract: Let $C_n$ be the cycle of length $n\ge3$ and let $I_{C_n}$ be its cut ideal. We show that $I_{C_n}$ has a quadratic Gröbner basis with respect to an explicit weight order. Since the defining configuration consists of $(0,1)$-vectors, the initial monomials of such a basis are automatically squarefree. As the cut polytope of a cycle is the parity polytope, the result gives a regular unimodular flag triangulation of this classical polytope. Together with the known tree case and the clique-sum theorem for cut ideals, the cycle result also yields a quadratic Gröbner basis for the cut ideal of every connected ring graph with at least one edge, thereby supplying the missing cycle input and establishing the result for connected ring graphs.
Source: arXiv (Cornell University)
Author: Ohsugi, Hidefumi
Institution: Kwansei Gakuin University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212312312