Levelness versus almost Gorensteinness of edge rings of complete multipartite graphs

TL;DR

This paper characterizes when the edge rings of complete multipartite graphs are level or almost Gorenstein, providing complete criteria based on the graph parameters.

Contribution

It offers a full characterization of levelness and almost Gorensteinness for edge rings of complete multipartite graphs, a significant extension in the study of these properties.

Findings

Complete characterization of level edge rings

Complete characterization of almost Gorenstein edge rings

Criteria expressed in terms of graph parameters

Abstract

Levelness and almost Gorensteinness are well-studied properties on graded rings as a generalized notion of Gorensteinness. In the present paper, we study those properties for the edge rings of the complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$ with $1 \leq r_1 \leq \cdots \leq r_n$. We give the complete characterization of which $\Bbbk[K_{r_1,\ldots,r_n}]$ is level in terms of $n$ and $r_1,\ldots,r_n$. Similarly, we also give the complete characterization of which $\Bbbk[K_{r_1,\ldots,r_n}]$ is almost Gorenstein in terms of $n$ and $r_1,\ldots,r_n$.