Non-normal Edge Rings Satisfying \((S_{2})\)-condition

Abstract

Let G be a finite simple connected graph on the vertex set \(V(G)=[d]=\{1,\dots ,d\}\) with edge set \(E(G)=\{e_{1},\dots , e_{n}\}\). Let \(\mathbb {K}[\textbf{t}]=\mathbb {K}[t_{1},\dots ,t_{d}]\) be the polynomial ring in d variables over a field \(\mathbb {K}\). The edge ring of G is the affine semigroup ring \(\mathbb {K}[G]\) generated by monomials \(\textbf{t}^{e}:=t_{i}t_{j}\), for \(e=\{i,j\} \in E(G)\). In this paper, we will prove that, given integers d and n, where \(d\ge 7\) and \(d+1\le n\le \frac{d^{2}-7d+24}{2}\), there exists a finite simple connected graph G with \(|V(G)|=d\) and \(|E(G)|=n\), such that \(\mathbb {K}[G]\) is non-normal and satisfies \((S_{2})\)-condition.