Bounds on the regularity of toric ideals of graphs

Let G be a finite simple graph. We give a lower bound for the Castelnuovo–Mumford regularity of the toric ideal IG associated to G in terms of the sizes and number of induced complete bipartite graphs in G. When G is a chordal bipartite graph, we find an upper bound for the regularity of IG in terms of the size of the bipartition of G. We also give a new proof for the graded Betti numbers of the toric ideal associated to the complete bipartite graph K2,n.