On the depth of binomial edge ideals of graphs

TL;DR

This paper explores the depth of binomial edge ideals of graphs, providing a combinatorial lower bound and characterizing cases with depth exactly 5 using poset topology and local cohomology.

Contribution

It introduces a new combinatorial lower bound for the depth of binomial edge ideals and characterizes ideals with depth 5 through poset and topological analysis.

Findings

Established a combinatorial lower bound for depth based on graph invariants

Characterized all binomial edge ideals with depth exactly 5

Connected poset topology to local cohomology computations

Abstract

Let $G$ be a graph on the vertex set $[n]$ and $J_G$ the associated binomial edge ideal in the polynomial ring $S=\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$. In this paper we investigate the depth of binomial edge ideals. More precisely, we first establish a combinatorial lower bound for the depth of $S/J_G$ based on some graphical invariants of $G$. Next, we combinatorially characterize all binomial edge ideals $J_G$ with $\mathrm{depth}\hspace{1.2mm}S/J_G=5$. To achieve this goal, we associate a new poset $\mathcal{M}_G$ with the binomial edge ideal of $G$, and then elaborate some topological properties of certain subposets of $\mathcal{M}_G$ in order to compute some local cohomology modules of $S/J_G$.