Arithmetical rank of toric ideals associated to graphs

TL;DR

This paper investigates the arithmetical ranks of toric ideals associated with graphs, showing that for bipartite graphs with quadratic binomial generators, these ranks equal the minimal number of generators.

Contribution

It establishes the equality of binomial arithmetical rank, G-homogeneous arithmetical rank, and minimal generators for specific classes of graph-associated toric ideals.

Findings

Binomial arithmetical rank equals minimal generators for bipartite graphs.

G-homogeneous arithmetical rank coincides with binomial arithmetical rank.

Results apply to toric ideals generated by quadratic binomials.

Abstract

Let $I_{G} \subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the $G$-arithmetical rank coincide with the minimal number of generators of $I_G$.