Regularity 3 in edge ideals associated to bipartite graphs

2012/07/23 by Óscar Fernández‐Ramos, Fernández-Ramos, Oscar, Philippe Giménez +1
Mathematics · #05E40 #13D02 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1207.5553

openalex publication_date 2012/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 β_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.

Citations

• Restricting linear syzygies: algebra and geometry
• Dependence of Betti Numbers on Characteristic

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