Splittings of toric ideals of graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910779963342848 |
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| author | Katsabekis, Anargyros Thoma, Apostolos |
| author_facet | Katsabekis, Anargyros Thoma, Apostolos |
| contents | Let $G$ be a simple graph on the vertex set $\{v_{1},\ldots,v_{n}\}$. An algebraic object attached to $G$ is the toric ideal $I_G$. We say that $I_G$ is subgraph splittable if there exist subgraphs $G_1$ and $G_2$ of $G$ such that $I_G=I_{G_1}+I_{G_2}$, where both $I_{G_1}$ and $I_{G_2}$ are not equal to $I_G$. We show that $I_G$ is subgraph splittable if and only if it is edge splittable. We also prove that the toric ideal of a complete bipartite graph is not subgraph splittable. In contrast, we show that the toric ideal of a complete graph $K_n$ is always subgraph splittable when $n \geq 4$. Additionally, we show that the toric ideal of $K_n$ has a minimal splitting if and only if $4 \leq n \leq 5$. Finally, we prove that any minimal splitting of $I_G$ is also a reduced splitting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Splittings of toric ideals of graphs Katsabekis, Anargyros Thoma, Apostolos Commutative Algebra 13F65, 14M25, 05C25, 05E40 Let $G$ be a simple graph on the vertex set $\{v_{1},\ldots,v_{n}\}$. An algebraic object attached to $G$ is the toric ideal $I_G$. We say that $I_G$ is subgraph splittable if there exist subgraphs $G_1$ and $G_2$ of $G$ such that $I_G=I_{G_1}+I_{G_2}$, where both $I_{G_1}$ and $I_{G_2}$ are not equal to $I_G$. We show that $I_G$ is subgraph splittable if and only if it is edge splittable. We also prove that the toric ideal of a complete bipartite graph is not subgraph splittable. In contrast, we show that the toric ideal of a complete graph $K_n$ is always subgraph splittable when $n \geq 4$. Additionally, we show that the toric ideal of $K_n$ has a minimal splitting if and only if $4 \leq n \leq 5$. Finally, we prove that any minimal splitting of $I_G$ is also a reduced splitting. |
| title | Splittings of toric ideals of graphs |
| topic | Commutative Algebra 13F65, 14M25, 05C25, 05E40 |
| url | https://arxiv.org/abs/2405.09836 |