Regularity 3 in edge ideals associated to bipartite graphs

TL;DR

This paper characterizes bipartite graphs whose edge ideals have regularity 3 and analyzes the minimal free resolution structure for higher regularities, linking algebraic invariants to combinatorial properties.

Contribution

It provides a combinatorial characterization of edge ideals with regularity 3 and describes the first step where generators of higher degree appear in resolutions.

Findings

Characterization of bipartite graphs with regularity 3

Identification of the first step with generators of degree > i+3 for higher regularity

Explicit formulas for Betti numbers of certain ideals with regularity 4

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 $i$ 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 value of the corresponding graded Betti number $\beta_{i,i+4}$ 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.