Depth of edge rings arising from finite graphs

TL;DR

This paper constructs finite graphs with specific edge ring depths and dimensions, using Gr"obner bases and initial ideals, to explore algebraic properties of graph-associated rings.

Contribution

It demonstrates the existence of finite graphs with prescribed edge ring depth and dimension, advancing understanding of algebraic invariants in graph theory.

Findings

Existence of graphs with prescribed depth and dimension

Application of Gr"obner bases to graph edge rings

Construction methods for specific algebraic properties

Abstract

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