Symmetric Exchange Graphs of Polymatroids

Abstract

White has conjectured that the toric ideal of a matroid is generated by quadric binomials corresponding to symmetric basis exchanges. Herzog and Hibi made a similar conjecture for discrete polymatroids. In this paper we study symmetric exchange graphs of polymatroids. We show that for polymatroids on at most seven variables, the degree two part of the associated toric ideal is generated by symmetric exchange binomials.