Splittings of toric ideals of graphs
Anargyros Katsabekis, Apostolos Thoma
TL;DR
The paper investigates when toric ideals of graphs $I_G$ can be decomposed as sums of toric ideals of subgraphs, showing subgraph splittability is equivalent to edge splittability. It provides complete classifications in key cases: $I_{K_n}$ is subgraph splittable for $n\ge4$, with minimal splittings existing only for $n=4,5$ and none for $n\ge6$, and $I_G$ is not subgraph splittable for complete bipartite graphs. It further introduces minimal and reduced splittings, proving every minimal splitting is reduced and describing the canonical form of reduced splittings via $I_G=I_{G_S^F}+I_{G\setminus F}$. These results illuminate how toric ideals of graphs can be decomposed and how such splittings relate to the underlying graph structure, with implications for algebraic properties and invariants tied to these ideals.
Abstract
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.