Almost complete intersection binomial edge ideals and their Rees algebras

Abstract

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the binomial edge ideal of $G$ in the polynomial ring $S=\mathbb{K}[x_1,\dots ,x_n,y_1,\dots ,y_n].$ In this article, we compute the second graded Betti numbers of $J_G$, 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.

20 Pages; Lemma 4.2 and Proposition 4.10 has been added. Accepted for publication in Journal of Pure and Applied Algebra