Regularity 3 in edge ideals associated to bipartite graphs

By Oscar Fernández-Ramos, Philippe Gimenez. 2012

52 citations recorded by the source corpus.

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 βi,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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b6df1b736f8885dd861d64f0353b74edf2facaa2

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Powers of edge ideals of regularity three bipartite graphs (2014, methodology): “For our proof, we use the combinatorial characterization of regularity three bipartite graphs by Oscar Fernández-Ramos and Phillippe Gilmenez proved in [2] and the techniques introduced by the second author in [1].”
Regularity of Edge Ideals and Their Powers (2017, background): “in the latter case. A combinatorial characterization of edge ideals with regularity 3 is still not known. However, a partial result for bipartite graphs is achieved by Fernandez-Ramos and Gimenez in [23]. Recall that a graph Gis bipartite if the vertices V can be partitioned into disjoint subsets V ”