Splittings of toric ideals of graphs

Anargyros Katsabekis; Apostolos Thoma

paper

Splittings of toric ideals of graphs

arXiv:2405.09836

Abstract

Let $G$ be a simple graph on the vertex set ${v_{1}, \dots, v_{n}}$ . 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.