Let $I_A$ be a toric ideal. We prove that the degrees of the elements of the Graver basis of $I_A$ are not polynomially bounded by the true degrees of the circuits of $I_A$.
We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We pro
ve that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of $G$ except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally we characterize all complete intersection toric ideals of graphs which are normal.
The universal Gr{o}bner basis of $I$, is a Gr{o}bner basis for $I$ with respect to all term orders simultaneously. Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the elements of the universal Gr{o}bner basis o
f the toric ideal $I_G$. We provide a bound for the degree of the binomials in the universal Gr{o}bner basis of the toric ideal of a graph. Finally we give a family of examples of circuits for which their true degrees are less than the degrees of some elements of the Graver basis.
Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$.
