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 = K [ x 1 , … , x n , y 1 , … , 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.
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