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arXiv · 2402.17413

On the $(S_2)$-condition of edge rings for cactus graphs

Rodica Dinu·Nayana Shibu Deepthi
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Abstract

A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we will examine cactus graphs where all the blocks are $3$-cycles, i.e., triangular cactus graphs, of diameter $4$. Our main focus is to prove that the corresponding edge ring of this family of graphs is not normal and satisfies Serre's condition $(S_2)$. We use a criterion due to Katth\"an for non-normal affine semigroup rings.

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Rodica Dinu, Nayana Shibu Deepthi. 2024-02-27. On the $(S_2)$-condition of edge rings for cactus graphs. https://doi.org/10.1080/00927872.2025.2451726

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