Toric ideals and diagonal 2-minors

Abstract

Abstract. Let G be a simple graph on the vertex set {1, . . . , n} with m edges. An algebraic object attached to G is the ideal P G generated by diagonal 2-minors of an n × n matrix of variables. In this paper we prove that if G is bipartite, then every initial ideal of P G is generated by squarefree monomials of degree at most m+n+1 2. Furthermore, we completely characterize all connected graphs G for which P G is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Gröbner basis of P G .