Arithmetical rank of toric ideals associated to graphs

Let $I_{G} \subset K[x_{1},\ldots,x_{m}]$ 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:

1. $G$ is bipartite,
2. $I_G$ is generated by quadratic binomials.

In both cases we prove that the binomial arithmetical rank and the $G$-homogeneous arithmetical rank coincide with the minimal number of generators of $I_G$.