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 regularity3. When the regularity is strictly bigger than3, 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 ? 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.
edge ideal bipartite graph graded betti numbers castelnuovomumford regularity independence complex stanleyreisner ideal