Toric ideals and diagonal 2-minors

Let G be a simple graph on the vertex set with m edges. An algebraic object attached to G is the ideal P (G) generated by diagonal 2-minors of an matrix of variables. We prove that if G is bipartite, then every initial ideal of P (G) is generated by squarefree monomials of degree at most . 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 Grobner basis of P (G) .