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

TL;DR

This work investigates edge rings of triangular cactus graphs, focusing on those with diameter $4$. It shows that the edge ring $\, ext{K}[G]$ is not normal, since the graph fails the odd cycle condition, but it satisfies Serre's condition $(S_2)$ by applying Katthän's hole-decomposition criterion: every family of holes in the corresponding affine semigroup has dimension $d-1$, where $d$ is the number of vertices. The authors divide the diameter-$4$ case into two graph types, analyze exceptional pairs and fundamental sets, and explicitly describe the associated hole structures, ultimately proving $(S_2)$ via the unique hole-decomposition description. These results provide evidence for a broader conjecture that the edge rings of all triangular cactus graphs are $(S_2)$ (and potentially Cohen–Macaulay), linking combinatorial graph properties to depth conditions in affine semigroup rings.

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 Katthän for non-normal affine semigroup rings.

Figures (6)

• Figure 1: Differentiating regular vertex and single element fundamental set
• Figure 2: A general triangular cactus graph of diameter $3$
• Figure 3: A general form of a triangular cactus graph of diameter $4$
• Figure 4: Illustration of an exceptional pair
• Figure 5: An illustration of graph $\mathbf{G}^{\prime}$

Theorems & Definitions (21)

• Definition 2.1
• Example 2.2
• Theorem 2.3: Katt*Theorem 3.1
• Theorem 2.4: Katt*Theorem 5.2
• Theorem 2.5: OH*Theorem 1.7
• Theorem 2.6: OH*Theorem 2.2
• Theorem 2.7: OH*Theorem 2.2,svv*Theorem 1.1
• Conjecture 3.1
• Theorem 3.2
• Lemma 4.1