Gorenstein homogeneous subrings of graphs

TL;DR

This paper characterizes the properties of graphs whose associated homogeneous monomial subrings are Gorenstein and normal, revealing structural conditions like being unmixed, having specific cover numbers, and bipartiteness for even-sized graphs.

Contribution

It establishes necessary and sufficient conditions linking Gorenstein normal subrings to graph properties such as being unmixed, having a certain cover number, and bipartiteness for even graphs.

Findings

If the subring is normal and Gorenstein, then the graph is unmixed with a specific cover number.

For even n, the graph must be bipartite under these conditions.

Provides sufficient conditions for the subring to be Gorenstein when the graph is unmixed with a certain cover number.

Abstract

Let $G=(V,E)$ be a connected simple graph, with $n$ vertices such that $S$ is its homogeneous monomial subring. We prove that if $S$ is normal and Gorenstein, then $G$ is unmixed with cover number $\lceil\frac{n}{2}\rceil$ and $G$ has a strong $\lceil\frac{n}{2}\rceil$-$\tau$-reduction. Furthermore, if $n$ is even, then we show that $G$ is bipartite. Finally, if $S$ is normal and $G$ is unmixed whose cover number is $\lceil\frac{n}{2}\rceil$, we give sufficient conditions for $S$ to be Gorenstein.