Toric ideals of graphs minimally generated by a Gröbner basis
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arXiv
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| Format: | Preprint |
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2025
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| author | García-Marco, Ignacio Márquez-Corbella, Irene Tatakis, Christos |
| author_facet | García-Marco, Ignacio Márquez-Corbella, Irene Tatakis, Christos |
| contents | Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gröbner bases is a central topic in commutative algebra. In this paper, we address this problem in the context of toric ideals of graphs. We say that a graph $G$ is an MG-graph if its toric ideal $I_G$ is minimally generated by some Gröbner basis, and a UMG-graph if every reduced Gröbner basis of $I_G$ forms a minimal generating set. We prove that a graph $G$ is a UMG-graph if and only if its toric ideal $I_G$ is a generalized robust ideal (that is, its universal Gröbner basis coincides with its universal Markov basis). Although the class of MG-graphs is not closed under taking subgraphs, we prove that it is hereditary, that is, closed under taking induced subgraphs. In addition, we describe two families of bipartite MG-graphs: ring graphs (which correspond to complete intersection toric ideals, as shown by Gitler, Reyes, and Villarreal) and graphs in which all chordless cycles have the same length. The latter extends a result of Ohsugi and Hibi, which corresponds to graphs whose chordless cycles are all of length $4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06216 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toric ideals of graphs minimally generated by a Gröbner basis García-Marco, Ignacio Márquez-Corbella, Irene Tatakis, Christos Commutative Algebra Combinatorics 14M25, 20M14, 05C25, 13C05 Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gröbner bases is a central topic in commutative algebra. In this paper, we address this problem in the context of toric ideals of graphs. We say that a graph $G$ is an MG-graph if its toric ideal $I_G$ is minimally generated by some Gröbner basis, and a UMG-graph if every reduced Gröbner basis of $I_G$ forms a minimal generating set. We prove that a graph $G$ is a UMG-graph if and only if its toric ideal $I_G$ is a generalized robust ideal (that is, its universal Gröbner basis coincides with its universal Markov basis). Although the class of MG-graphs is not closed under taking subgraphs, we prove that it is hereditary, that is, closed under taking induced subgraphs. In addition, we describe two families of bipartite MG-graphs: ring graphs (which correspond to complete intersection toric ideals, as shown by Gitler, Reyes, and Villarreal) and graphs in which all chordless cycles have the same length. The latter extends a result of Ohsugi and Hibi, which corresponds to graphs whose chordless cycles are all of length $4$. |
| title | Toric ideals of graphs minimally generated by a Gröbner basis |
| topic | Commutative Algebra Combinatorics 14M25, 20M14, 05C25, 13C05 |
| url | https://arxiv.org/abs/2504.06216 |