On the depth of binomial edge ideals of graphs

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 $$\mathrm {depth} S/J_G=5$$ . To achieve this goal, we associate a new poset $$\mathscr {M}_G$$ with the binomial edge ideal of G and then elaborate some topological properties of certain subposets of $$\mathscr {M}_G$$ in order to compute some local cohomology modules of $$S/J_G$$ .