Let I G be the toric ideal of a graph G. We characterize in graph theoretical terms primitive, minimal, indispensable and fundamental binomials of the toric ideal I G .

The universal Gröbner basis of an ideal is a Gröbner basis with respect to all term orders simultaneously. We characterize in graph theoretical terms the elements of the universal Gröbner basis of the toric ideal of a graph. We also provide a new degree bound. Finally, we give examples of graphs for which the true degrees of their circuits are less than the degrees of some elements of the Graver basis.

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 prove 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.

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Let I A be a toric ideal. We prove that the degrees of the elements of the Graver basis of I A are not bounded above by a polynomial on the maximal true degree of the circuits of I A .