Regularity of powers of bipartite graphs

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all s≥ 1 , we obtain upper bounds for reg (I(G) s) for bipartite graphs. We then compare the properties of G and G′, where G′ is the graph associated with the polarization of the ideal (I(G) s+1: e1⋯ es) , where e1, ⋯ , es are edges of G. Using these results, we explicitly compute reg (I(G) s) for several subclasses of bipartite graphs..