On the universal Gr\"obner bases of toric ideals of graphs

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 of 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.