Toric ideals and diagonal 2-minors

Abstract

Let $G$ be a simple graph on the vertex set $\{1,\dots ,n\}$ with $m$ edges. An algebraic object attached to $G$ is the ideal $P_G$ generated by diagonal 2-minors of an $n\times n$ 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 $\lfloor \frac{m+n+1}{2}\rfloor$. 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 Gr{ö}bner basis of $P_G$.

To appear in Acta Mathematica Hungarica

References in corpus (1)

• Two way subtable sum problems and quadratic Groebner bases