Splittings of toric ideals of graphs

Let G be a simple graph on the vertex set ₁,…,vₙ\. An algebraic object attached to G is the toric ideal IG. We say that IG is splittable if there exist subgraphs G₁ and G₂ of G such that IG=IG₁+IG₂, where both IG₁ and IG₂ are not equal to IG. We show that IG is splittable if and only if it is edge splittable. We also prove that the toric ideal of a complete bipartite graph is not splittable. In contrast, we show that the toric ideal of a complete graph Kₙ is always splittable when n ≥ 4. Additionally, we show that the toric ideal of Kₙ has a minimal splitting if and only if 4 ≤ n ≤ 5. Finally, we prove that any minimal splitting of IG is also a reduced splitting.