Almost complete intersection binomial edge ideals and their Rees algebras

Let G be a simple graph on n vertices and JG denote the binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn]. In this article, we compute the second graded Betti numbers of JG, and we obtain a minimal presentation of it when G is a tree or a unicyclic graph. We classify all graphs wh...

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Bibliographic Details
Published in:Journal of pure and applied algebra 2021-06, Vol.225 (6), p.106628, Article 106628
Main Authors: Jayanthan, A.V., Kumar, Arvind, Sarkar, Rajib
Format: Article
Language:English
Subjects:
Betti number
Binomial edge ideal
Rees algebra
Syzygy
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Summary:Let G be a simple graph on n vertices and JG denote the binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn]. In this article, we compute the second graded Betti numbers of JG, and we obtain a minimal presentation of it when G is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a d-sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2020.106628