Binomial edge ideals of small depth

TL;DR

This paper studies the topological and algebraic properties of binomial edge ideals of graphs, focusing on the structure of their primary decompositions and characterizing graphs with a specific depth.

Contribution

It introduces a topological approach to analyze binomial edge ideals and characterizes graphs with depth exactly four in the polynomial ring.

Findings

The poset associated with the primary decomposition has contractible subposets.

Characterization of all graphs with depth exactly four.

Provides algebraic consequences from topological properties.

Abstract

Let $G$ be a graph on $[n]$ and $J_G$ be 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 paper we investigate some topological properties of a poset associated to the minimal primary decomposition of $J_G$. 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 $\mathrm{depth}\hspace{1.2mm} S/J_G=4$.