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

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 $S=K[x_1,\dots ,x_n]$. 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 $\alpha =3$.

arXiv comment9 pages, 1 figure

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• A Characterization of Triangle-free Gorenstein graphs and Cohen-Macaulayness of second powers of edge ideals
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Cited by in corpus (2)

• Gorenstein and Cohen-Macaulay Matching Complexes
• On Gorenstein Circulant Graphs and Gorenstein SQC Graphs