Some Combinatorial Characterizations of Gorenstein Graphs with Independence Number Less than Four

TL;DR

This paper characterizes Gorenstein graphs with independence number less than four, providing conditions and properties for such graphs, especially focusing on those with independence numbers 2 and 3, including triangle-free cases.

Contribution

It offers a new combinatorial characterization of Gorenstein graphs with small independence numbers, extending understanding of their structure and properties.

Findings

Characterization of Gorenstein graphs with independence number 2.

Properties of Gorenstein graphs with independence number 3.

Characterization of triangle-free Gorenstein graphs with independence number 3.

Abstract

Let $\alpha=\alpha(G)$ be the independence number of a simple graph $G$ with $n$ vertices and $I(G)$ be its edge ideal in $S=K[x_1,\ldots, x_n]$. If $S/I(G)$ is Gorenstein, the graph $G$ is called Gorenstein over $K$ and if $G$ is Gorenstein over every field, then we simply say that $G$ is Gorenstein. In this article, first we state a condition equivalent to $G$ being Gorenstein and using this we give a characterization of Gorenstein graphs with $\alpha=2$. Then we present some properties of Gorenstein graphs with $\alpha=3$ and as an application of these results we characterize triangle-free Gorenstein graphs with $\alpha=3$.