  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 $I_G$.
