  We study the family of graphs whose number of primitive cycles equals its
cycle rank. It is shown that this family is precisely the family of ring
graphs. Then we study the complete intersection property of toric ideals of
bipartite graphs and oriented graphs. An interesting application is that
complete intersection toric ideals of bipartite graphs correspond to ring
graphs and that these ideals are minimally generated by Groebner bases. We
prove that any graph can be oriented such that its toric ideal is a complete
intersection with a universal Groebner basis determined by the cycles. It turns
out that bipartite ring graphs are exactly the bipartite graphs that have
complete intersection toric ideals for any orientation.
