Regularity of powers of bipartite graphs

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all (s ge 1), we obtain upper bounds for ({text {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} : e_1cdots e_s)), where (e_1,cdots , e_s) are edges of G. Using these results, we explicitly compute ({text {reg}}(I(G)^s)) for several subclasses of bipartite graphs.