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Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the binomial edge ideal of $G$ in the polynomial ring $S = mathbb{K}[x_1, ldots, x_n, y_1, ldots, y_n].$ 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.
The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(ntimes n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omi
This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Grobner basis can be computed by studying paths in
In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $bz^noplus T$ with no invertible elements, where $T$ is a finite abelian group. We also ch
We give an explicit formula for the Hilbert-Poincar{e} series of the parity binomial edge ideal of a complete graph $K_{n}$ or equivalently for the ideal generated by all $2times 2$-permanents of a $2times n$-matrix. It follows that the depth and Cas
Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks whether its