Depth and Krull dimension of Binomial edge ideals

Abstract

Let $J_G$ denote the binomial edge ideal of a finite graph $G$ in the polynomial ring $S$. We determine all triples $(n,t,d)$ with $n\ge 3$ for which there exists a finite connected graph $G$ on $n$ vertices with $\mathrm{depth}(S/J_G)=t$ and $\dim (S/J_G)=d$.