Depth of edge rings arising from finite graphs

Abstract

Let $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the technique of Gröbner bases and initial ideals, it will be proved that, given integers $f$ and $d$ with $7\le f\le d$, there exists a finite graph $G$ on $[d]=1,...,d$ with $\depth K[G] = f$ and with $\Krull-dim K[G] = d$.

7 pages