Let G be a graph on [n] and JG be the binomial edge ideal of G in the polynomial ring S = K[x1, . . . , xn, y1, . . . , yn]. In this paper we investigate some topological properties of a poset associated to the minimal primary decomposition of JG. We show that this poset admits some specific subposets which are contractible. This in turn, provides some interesting algebraic consequences. In particular, we characterize all graphs G for which depthS/JG = 4.

中文翻译：

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小深度的二项式边理想

设 G 是 [n] 上的图，JG 是多项式环 S = K[x1, . 中 G 的二项边理想。. . , xn, y1, . . . , yn]。在本文中，我们研究了与 JG 的最小初级分解相关联的偏序集的一些拓扑特性。我们证明这个偏序允许一些可收缩的特定子集。这反过来又提供了一些有趣的代数结果。特别是，我们表征了 depthS/JG = 4 的所有图 G。