  One can associate to a bipartite graph a so-called edge ring whose spectrum
is an affine normal toric variety. We characterize the faces of the (edge) cone
associated to this toric variety in terms of some independent sets of the
bipartite graph. By applying to this characterization the combinatorial study
of deformations of toric varieties by Altmann, we present certain criteria for
their rigidity purely in terms of graphs.
