Splittings of toric ideals of graphs

TL;DR

This paper investigates the conditions under which toric ideals of graphs can be decomposed into simpler components, establishing equivalences and exploring specific cases like complete and bipartite graphs.

Contribution

It characterizes subgraph splittability of toric ideals, proves equivalence with edge splittability, and analyzes splittability in complete and bipartite graphs.

Findings

Toric ideal is subgraph splittable iff it is edge splittable.

Toric ideal of complete bipartite graphs is not subgraph splittable.

Toric ideal of complete graphs $K_n$ is always subgraph splittable for $n \\geq 4$.

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.