Regularity 3 in edge ideals associated to bipartite graphs
arXiv:1207.5553
Abstract
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 in the minimal graded free resolution where there exists a minimal generator of degree , show that at this step the highest degree of a minimal generator is , and determine the value of the corresponding graded Betti number in terms of the combinatorics of the associated bipartite graph. The results can then be easily extended to the non-squarefree 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.
19 pages