Toric Ideals of Flow Polytopes

A referee found an error in the proof of the Main Theorem (“toric ideals of flow polytopes are generated in degree 3”) that we could not fix. More precisely, the proof of Lemma 4.2.(ii) is incorrect. The results on Gr"obner bases are untouched by this.

We show that toric ideals of flow polytopes are generated in degree 3. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gr"obner bases of the toric ideal of the Birkhoff polytope $B_n$ have at most degree $n$. We show that this bound is sharp for some revlex term orders. For $(m\times n)$-transportation polytopes, a similar result holds: they have Gr"obner bases of at most degree $\lfloor mn/2\rfloor$. We construct a family of examples, where this bound is sharp.