Regularity 3 in edge ideals associated to bipartite graphs

We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step i in the minimal graded free resolution where there exists a minimal generator of degree > i+3, show that at this step the highest degree of a minimal generator is i+4, and determine the corresponding graded Betti number beta (i,i+4) in terms of the combinatorics of the graph. The results are then extended to the non-square-free case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.