PreprintCC BY 4.0arXiv
Quadratic Gröbner bases for cut ideals of cycles and ring graphs
Abstract
Let Cn be the cycle of length n≥3 and let ICn be its cut ideal. We show that ICn 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.
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BibTeX
@misc{ohsugi2026quadratic,
title = {{Quadratic Gröbner bases for cut ideals of cycles and ring graphs}},
author = {Hidefumi Ohsugi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11512},
eprint = {2609.11512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11512},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quadratic Gröbner bases for cut ideals of cycles and ring graphs
AU - Hidefumi Ohsugi
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11512
UR - https://arxiv.org/abs/2609.11512
AB - Let C_n be the cycle of length n≥3 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.
ER -