#  $(S_{2})$-condition of edge rings for cactus graphs

Rodica Dinu  University of Konstanz, Fachbereich Mathematik und Statistik,
Fach D 197 D-78457 Konstanz, Germany, and Simion Stoilow Institute of
Mathematics of the Romanian Academy, Calea Grivitei 21, 010702, Bucharest,
Romania<EMAIL_ADDRESS>and  Nayana Shibu Deepthi  Department of
Pure and Applied Mathematics, Graduate School of Information Science and
Technology, Osaka University, Suita, Osaka 565-0871, Japan
<EMAIL_ADDRESS>
###### 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 satisfies
Serre’s condition $(S_{2})$. We use a criterion due to Kattän for non-normal
affine semigroup rings.

###### Key words and phrases:

Edge rings, Normality, Odd cycle condition, Cactus graphs, Serre’s condition
$(S_{2})$

###### 2020 Mathematics Subject Classification:

Primary : 13H10; Secondary: 52B20, 14M25

##  1\. Introduction

The study of the edge rings and edge polytopes arising from finite connected
graphs became of interest both from combinatorics and commutative algebra
perspectives. Being inspired by the article of Ohsugi and Hibi, [OH], we are
interested in studying the properties of the edge rings associated with finite
graphs. Some interesting results on the edge rings can be found in [HN, svv,
HHKO, HMT, Nayana].

Let $G$ be a finite connected graph on the vertex set $[d]=\\{1,2,\dots,d\\}$
with the edge set $E(G)=\\{e_{1},e_{2},\dots,e_{n}\\}$. We will always assume
that $G$ is simple, i.e., $G$ is without loops and multiple edges. We consider
the polynomial ring $\mathbb{K}[\mathbf{t}]:=\mathbb{K}[t_{1},\dots,t_{d}]$ in
$d$ variables over a field $\mathbb{K}$. For an edge $e=\\{i,j\\}\in E(G)$, we
define the quadratic monomial $\mathbf{t}^{e}:=t_{i}t_{j}$. The subring of
$\mathbb{K}[\mathbf{t}]$ generated by
$\mathbf{t}^{e_{1}},\dots,\mathbf{t}^{e_{n}}$ is called the edge ring of $G$,
denoted by $\mathbb{K}[G]$. By [OH]*Proposition 1.3, the edge ring
$\mathbb{K}[G]$ has dimension $d$ if $G$ is not a bipartite graph. For an edge
$e=\\{i,j\\}\in E(G)$, we define $\rho(e):=\mathbf{e}_{i}+\mathbf{e}_{j}$,
where $\mathbf{e}_{1},\dots,\mathbf{e}_{d}$ is the canonical unit coordinate
vectors of $\mathbb{R}^{d}$. We consider the affine semigroup generated by
$\rho(e_{1}),\dots,\rho(e_{n})$, which we denote by $S_{G}$. Let
$\mathbb{A}_{G}:=\\{\rho(e):e\in E(G)\\}$. We may regard the edge ring
$\mathbb{K}[G]$ as the affine semigroup ring of $S_{G}$, for which the cone
$\mathcal{C}_{G}:=\mathbb{Q}_{\geq 0}\mathbb{A}_{G}$ plays an essential role
its study. The problem of the normality of $\mathbb{K}[G]$ was studied by Hibi
and Ohsugi in [OH], and by Simis, Vasconcelos, and Villareal in [svv]. They
gave a combinatorial criterion for normality in terms of the graph $G$. It
turns out that the odd cycle condition in classical graph theory gives a
characterization of normal edge rings, see [OH]*Theorem 2.2 and [svv]*Theorem
1.1. By [Hochster]*Theorem 1, it is known that if $\mathbb{K}[G]$ is normal,
then $\mathbb{K}[G]$ is Cohen–Macaulay. Another characterization of the
normality of edge rings was given in [BH]*Theorem 2.2.22, namely
$\mathbb{K}[G]$ is normal if and only if $\mathbb{K}[G]$ satisfies Serre’s
conditions $(R_{1})$ and $(S_{2})$. Hence, Serre’s condition $(S_{2})$ is a
necessary condition for $\mathbb{K}[G]$ to be Cohen–Macaulay. We are
interested in understanding the $(S_{2})$-condition for $\mathbb{K}[G]$. We
recall here that, for a Noetherian ring $A$, the $(S_{2})$-condition requires
that
$\operatorname{depth}(A_{\mathfrak{p}})\geq\inf\\{\operatorname{height}(\mathfrak{p}),2\\}$
for any prime ideal $\mathfrak{p}$. A criterion in terms of the combinatorics
of the graph $G$ for edge rings to satisfy Serre’s condition $(R_{1})$ was
given in [HK]*Theorem 2.1. However, not much is known in general about the
Cohen–Macaulay property for non-normal edge rings. In the paper [Nayana], the
second named author provided a class of graphs for which their associated edge
rings are non-normal but satisfy $(S_{2})$-condition. It was also shown that
the respective class of graphs is the smallest with respect to those
properties: adding new edges breaks the non-normality of the edge ring or
violates $(S_{2})$-condition. Higashitani and Kimura [HK], provided a
combinatorial criterion to check if an edge ring satisfies
$(S_{2})$-condition, which was actually used in the paper [Nayana]. However,
in this article we will use a result of Kattän [Katt], which shows that
Serre’s conditions are hidden in the geometry of the set of holes of the
affine semigroup $S_{G}$. We show that for the class of triangular cactus
graphs of diameter 4, the associated edge ring is non-normal and it satisfies
$(S_{2})$-condition. Recall that a triangular cactus graph is a connected
graph in which every block is a cycle of length 3.

We present the structure of the article. Section 2 is devoted to recalling
fundamental definitions, notations, and results concerning classical graph
theory, edge rings, and tools from Kattän’s paper that will be needed. The key
ingredient will be Theorem 2.3, saying that, for a $d$-dimensional affine
semigroup $S$, its affine semigroup ring $\mathbb{K}[S]$ satisfies Serre’s
condition $(S_{2})$ if and only if every family of holes of $S$ is of
dimension $d-1$. In Section 3, we focus on the class of triangular cactus
graphs of diameter 4 and state in Theorem 3.2 that, in this case, the edge
ring $\mathbb{K}[G]$ is non-normal and satisfies $(S_{2})$-condition. Section
4 is entirely devoted to the proof of Theorem 3.2. The description of the
supporting hyperplanes of $\mathcal{C}_{G}$ depends on the regular vertices
and the fundamental sets of $G$, see Theorem 2.4. We shall consider certain
vertices and shall analyze, based on their configuration, whether they are
regular or non-regular cutpoints in $G$. Hence we distinguish the triangular
cactus graphs of diameter 4 in two types, and in both cases, we look for the
fundamental sets. In addition, as $\overline{S_{G}}$, the normalization of
$S_{G}$ can be expressed in terms of exceptional pairs in $G$, due to Theorem
2.5. Hence, another goal will be to understand the exceptional pairs in both
types. In Proposition 4.3 and Proposition 4.4, we show that, in both types,
the $\overline{S_{G}}\setminus S_{G}$ decomposes as a union of
$(d-1)$-dimension families of holes of $S$. Using Theorem 2.3, we conclude our
main result. In Section 5, we present some conclusions. We conjecture that the
edge ring of a triangular cactus graph satisfies $(S_{2})$-condition, and we
believe that it might be even Cohen–Macaulay.

##  2\. Preliminaries

We recall in this section some fundamental definitions and results from finite
graphs theory, edge rings, and an important criterion regarding Serre’s
condition $(S_{2})$ for the affine semigroup ring, which will be of use for
our paper.

###  2.1. Graph theory

Let $G$ be a finite simple graph on the vertex set $V(G)=[d]$, where $d\geq
2$, and $E(G)=\\{e_{1},e_{2},\dots,e_{n}\\}$, the set of edges of $G$. A path
between any two vertices is a sequence of distinct edges that joins the two
vertices. A path from vertex $u$ to $v$ is the shortest path if there is no
other path from $u$ to $v$ with a lower length. The distance between two
vertices is the length of the shortest path between those two vertices. The
shortest path length between a graph’s most distant vertices is known as its
diameter and for the graph $G$, let us denote it as $\operatorname{diam}{G}$.
A cycle of length $n$ will be called an $n$-cycle. A minimal cycle in $G$ is a
cycle with no chord. An odd cycle in a graph is a cycle whose length is odd.

Let $C$ and $C^{\prime}$ be two minimal cycles of $G$ with $V(C)\cap
V(C^{\prime})=\emptyset$, then a bridge between $C$ and $C^{\prime}$ is an
edge $e=\\{i,j\\}$ of $G$ with $i\in V(C)$ and $j\in V(C^{\prime})$. A pair of
odd cycles $(C,C^{\prime})$ is called exceptional if $C$ and $C^{\prime}$ are
minimal odd cycles in $G$ such that $V(C)\cap V(C^{\prime})=\emptyset$ and
there is no bridge connecting them.

We say that a graph $G$ satisfies the odd cycle condition if, for any two odd
cycles $C$ and $C^{\prime}$ of $G$, either $V(C)\cap
V(C^{\prime})\neq\emptyset$ or there exists a bridge between $C$ and
$C^{\prime}$. Equivalently, the graph $G$ has no exceptional pairs.

Let $U\subset V(G)$. The induced subgraph of $G$ on $U\subset[d]$ is the
subgraph $G_{U}$ of $G$ with $V(G_{U})=U$ and $E(G_{U})=\\{e\in E(G):e\subset
U\\}$. For $v\in V(G)$, we denote $G\setminus v$ as the induced subgraph of
$G$ on the vertex set $V(G\setminus v)=V(G)\setminus\\{v\\}$. For $T\subset
V(G)$, we denote the set of its neighbor vertices by

$N_{G}(T):=\\{v\in V(G):\\{v,w\\}\in E(G)\text{ for some }w\in T\\}.$

A non-empty subset of vertices $T\subset V(G)$ is called independent if no
edge of $G$ is of the form $e=\\{i,j\\}$ with $i\in T$ and $j\in T$. For an
independent set $T\subset V(G)$, we define a bipartite graph induced by $T$ in
the following way: its set of vertices is $T\cup N_{G}(T)$ and its set of
edges is given by $\\{\\{v,w\\}\in E(G):v\in T,w\in N_{G}(T)\\}$.

######  Definition 2.1.

Let $G$ be a finite connected simple graph and $V(G)$ its set of vertices. A
vertex $v\in V(G)$ is called regular in $G$ if every connected component of
$G\setminus v$ contains at least one odd cycle. A non-empty set $T\subset
V(G)$ is called fundamental in $G$ if the following conditions are satisfied:

  1. (1)

$T$ is an independent set;

  2. (2)

the bipartite graph induced by $T$ is connected;

  3. (3)

either $T\cup N_{G}(T)=V(G)$ or every connected components of the graph
$G_{V(G)\setminus(T\cup N_{G}(T))}$ contains at least one odd cycle.

Note that a regular vertex is not the same as a fundamental set with one
element.

###  2.2. A characterization of Serre’s condition $(S_{2})$

For an arbitrary affine semigroup $S\subset\mathbb{Z}^{d}_{\geq 0}$, let
$\\{s_{1},\dots,s_{t}\\}$ be the minimal finite subset of $S$ such that
$\displaystyle S=\left\\{\sum\limits_{i=1}^{t}z_{i}s_{i}\colon
z_{i}\in\mathbb{Z}_{\geq 0}\right\\}$. Then $\\{s_{1},\dots,s_{t}\\}\subset S$
is called the minimal generating system of $S$. Note that, for
$\mathbb{A}\subset\mathbb{R}^{d}$, we denote
$\displaystyle\mathbb{A}S=\left\\{\sum\limits_{i=1}^{t}a_{i}s_{i}:a_{i}\in\mathbb{A}\right\\}$.
Let $\mathbb{Z}S$ be the free abelian group generated by $S$ and
$\mathbb{Q}_{\geq 0}S$ be the rational polyhedral cone generated by $S$. Let
$\overline{S}=\mathbb{Q}_{\geq 0}S\cap\mathbb{Z}S$ be the normalization of
$S$. Analogous to the definition of normality of $S$, we say that $S$ is
normal if $S=\overline{S}$ holds. (See, e.g., [BH]*Section 6.1.) The set
$F\subset S$ is said to be a face of $S$, if the following holds:
$s,s^{\prime}\in S,\ s+s^{\prime}\in F$ if and only if $s\in F$ and
$s^{\prime}\in F$. And, the dimension of a face $F$ is defined to be the rank
of the free abelian group $\mathbb{Z}F$. Throughout our study, we consider
only positive affine semigroups, i.e., the minimal face of $S$ is $\\{0\\}$.

In the article [Katt], Katthän gave a geometric description for the set of
holes $\overline{S}\backslash S$, and this is connected to the ring-
theoretical features of $S$, where the affine semigroups considered were not
necessarily positive. We recall it here as it will be of great importance for
our results.

######  Theorem 2.2 ([Katt]*Theorem 3.1).

Let $S$ be an affine semigroup. Then there exists a (not-necessarily disjoint)
decomposition

(2.1) $\overline{S}\backslash
S=\bigcup\limits_{i=1}^{l}(s_{i}+\mathbb{Z}F_{i})\cap\mathbb{Q}_{\geq 0}S$

with $s_{i}\in\overline{S}$ and faces $F_{i}$ of $S$. If no
$s_{i}+\mathbb{Z}F_{i}$ can be omitted from the union, then the decomposition
is unique.

The set $s_{i}+\mathbb{Z}F_{i}$ in (2.1) is called a $j$-dimensional family of
holes of $S$, where $j=\dim F_{i}$.

######  Theorem 2.3 ([Katt]*Theorem 5.2).

Let $S$ be an affine semigroup of dimension $d$. Then the affine semigroup
ring $\mathbb{K}[S]$ satisfies Serre’s condition $(S_{2})$ if and only if
every family of holes of $S$ is of dimension $d-1$.

###  2.3. Edge rings

Let $G$ be a finite connected simple graph on the vertex set $V(G)=[d]$ with
the edge set $E(G)=\\{e_{1},e_{2},\dots,e_{n}\\}$. We consider
$\mathbb{K}[\mathbf{t}]:=\mathbb{K}[t_{1},\dots,t_{d}]$, the polynomial ring
in $d$ variables over the field $\mathbb{K}$. If $e=\\{i,j\\}\in E(G)$, then
we define $\mathbf{t}^{e}\in\mathbb{K}[\mathbf{t}]$ for the quadratic monomial
$t_{i}t_{j}$. We write $\mathbb{\mathbb{K}}[G]$ for the toric ring
$\mathbb{K}[\\{\mathbf{t}^{e}:e\in E(G)\\}]$ and we call it the edge ring of
$G$. For more details, we refer to [OH, HHO].

We consider the canonical unit coordinate vectors of $\mathbb{R}^{d}$, denoted
by $\mathbf{e}_{1},\dots,\mathbf{e}_{d}$. For an edge $e=\\{i,j\\}$, we define
$\rho(e):=\mathbf{e}_{i}+\mathbf{e}_{j}$. Now we consider
$\mathbb{A}_{G}:=\\{\rho(e):e\in E(G)\\}$ and $S_{G}$ to be the affine
semigroup generated by $\rho(e_{1}),\dots,\rho(e_{n})$, hence
$S_{G}=\mathbb{Z}_{\geq 0}\mathbb{A}_{G}$. One can regard the edge ring
$\mathbb{K}[G]$ as the affine semigroup ring of $S_{G}$. Consider the convex
rational polyhedral cone spanned by $S_{G}$ in $\mathbb{Q}^{d}$, denoted by
$\mathcal{C}_{G}$. We may assume that $\mathcal{C}_{G}$ is $d$-dimensional.
The facets of $\mathcal{C}_{G}$ are given by the intersection of the half-
spaces defined by the supported hyperplanes of $\mathcal{C}_{G}$ which are
presented in the following result.

######  Theorem 2.4 ([OH]*Theorem 1.7).

Let $G$ be a finite connected simple graph with $V(G)=[d]$ containing at least
one odd cycle. Then all the supporting hyperplanes of $\mathcal{C}_{G}$ are
given by:

  1. (1)

$\mathcal{H}_{v}=\\{(x_{1},\dots,x_{d})\in\mathbb{R}^{d}:x_{v}=0\\}$, where
$v$ is a regular vertex in $G$;

  2. (2)

$\mathcal{H}_{T}=\\{(x_{1},\dots,x_{d})\in\mathbb{R}^{d}:\sum_{i\in
T}x_{i}=\sum_{j\in N_{G}(T)}x_{j}\\}$, where $T$ is a fundamental set in $G$.

We denote by $F_{v}$ and $F_{T}$ respectively the facets of $\mathcal{C}_{G}$
corresponding to the supporting hyperplanes $\mathcal{H}_{v}$ and
$\mathcal{H}_{T}$.

######  Theorem 2.5 ([OH]*Theorem 2.2).

The normalization of the edge ring $\mathbb{K}[G]$ can be expressed as

$\overline{S_{G}}=S_{G}+\mathbb{Z}_{\geq
0}\\{\mathbb{E}_{C}+\mathbb{E}_{C^{\prime}}:(C,C^{\prime})\text{ is an
exceptional pair in }G\\},$

where for any odd cycle $C$, we define $\mathbb{E}_{C}:=\sum_{i\in
V(C)}\mathbf{e}_{i}$.

The following result characterizes the normality of edge rings.

######  Theorem 2.6 ([OH]*Theorem 2.2,[svv]*Theorem 1.1).

Let G be a finite simple graph. Then the edge ring $\mathbb{K}[G]$ is normal
if and only if $G$ satisfies the odd cycle condition.

Observe that if $G$ satisfies the odd cycle condition, then $\mathbb{K}[G]$ is
a Cohen–Macaulay ring, as per the result of Hochster [Hochster]*Theorem 1.

##  3\. On triangular cactus graphs

Let $G$ be a finite connected simple graph. A vertex $v\in V(G)$ is called a
cutpoint if the subgraph $G\backslash v$ of $G$ has more connected components
than that of $G$. Recall that a vertex $v$ is regular in $G$ if every
connected component of $G\backslash v$ contains at least one odd cycle. If a
cutpoint $v$ is regular in $G$, then $v$ is called a regular cutpoint. We say
that a connected graph without a cutpoint is non-separable. A maximal non-
separable subgraph of $G$ is called a block of the graph $G$.

A cactus graph is a connected graph in which every block is either an edge or
a cycle. The cactus graph whose blocks are all $n$-cycles is defined as
$n$-cactus graph. For more details on cactus graphs, we refer to
[RinaldoCactus]. In this study, we will focus only on $3$-cactus graphs, which
will be called triangular cactus.

The $3$-cycles are triangular cactus graphs of diameter $1$. The friendship
graphs, formed from a collection of $3$-cycles joined together at a single
common vertex, are triangular cactus graphs of diameter $2$. For edge ring
related studies for this family of graphs, see [HN]. A generic illustration of
a triangular cactus of diameter $3$ is shown in Figure 1. Therefore, we
observe that any triangular cactus of diameter less than or equal to 3
satisfies the odd cycle condition, and by Theorem 2.6, it has a normal
Cohen–Macaulay edge ring.

$\dots$$\dots$$\dots$ Figure 1. A general triangular cactus of diameter $3$

We focus on the non-normal edge rings associated to triangular cactus graphs
that satisfy $(S_{2})$-condition. A broader objective of this study is to
prove the following conjecture:

######  Conjecture 3.1.

The edge ring associated to a triangular cactus of diameter $\geq 4$ satisfies
$(S_{2})$-condition.

We will exclusively study the family of triangular cactus graphs whose
diameter is $4$ and prove Conjecture 3.1 for this specific family. Let
$\mathbf{G}$ be the triangular cactus of diameter $4$. For an illustration of
$\mathbf{G}$ in general, see Figure 2.

$\dots$$x_{i}$$\vdots$$w$$\vdots$$x_{i}^{2}$$x_{i}^{1}$$\dots$$x_{i}^{k-1}$$x_{i}^{k}$$\vdots$$\vdots$$\vdots$$\vdots$$\dots$
Figure 2. A general form of a triangular cactus of diameter $4$

The main theorem of our study is as follows.

######  Theorem 3.2.

Let $\mathbf{G}$ be a triangular cactus with
$\operatorname{diam}{\mathbf{G}}=4$. Then the edge ring
$\mathbb{K}[\mathbf{G}]$ is non-normal and satisfies $(S_{2})$-condition.

##  4\. Towards the proof of Theorem 3.2

This section centers on the proof of Theorem 3.2. Here, we will study the two
types of graph $\mathbf{G}$ and collect the necessary results about them to
finally prove our main theorem.

Let us consider that $|V(\mathbf{G})|=d$ and $n$ copies of $3$-cycles are
attached to the vertex $w$ in $\mathbf{G}$. Note that, $w$ is always a
cutpoint of $\mathbf{G}$. Moreover, all the non-cutpoints of $\mathbf{G}$ are
regular in $\mathbf{G}$.

Recall that, by Theorem 2.4, for any regular vertex $v$ and fundamental set
$T$ of $\mathbf{G}$, we denote by $F_{v}$ and $F_{T}$ the facets of
$\mathcal{C}_{G}=\mathbb{Q}_{\geq 0}A_{\mathbf{G}}$ corresponding to the
hyperplanes $\mathcal{H}_{v}$ and $\mathcal{H}_{T}$ respectively.

The exceptional pairs in $\mathbf{G}$ are of the form $(C,C^{\prime})$ where
$C=\\{x_{p},x_{p}^{k-1},x_{p}^{k}\\}$ and
$C^{\prime}=\\{x_{q},x_{q}^{k^{\prime}-1},x_{q}^{k^{\prime}}\\}$ for some
$p,q\in[2n]$ such that $x_{p}\neq x_{q}$ and $\\{x_{p},x_{q}\\}\notin
E(\mathbf{G})$. An illustration of an exceptional pair in $\mathbf{G}$ is
shown in Figure 3.

$\dots$$x_{p}$$\vdots$$w$$\vdots$$\dots$$x_{p}^{k-1}$$x_{p}^{k}$$\vdots$$\vdots$$\vdots$$\vdots$$x_{q}$$x_{q}^{k^{\prime}}$$x_{q}^{k^{\prime}-1}$$\dots$
Figure 3. Illustration of an exceptional pair

For any odd cycle $C$ in $\mathbf{G}$, recall that
$\mathbb{E}_{C}\coloneqq\sum_{i\in V(C)}\mathbf{e}_{i},$ where
$\mathbf{e}_{i}$ represents the $i$-th canonical vector of $\mathbb{R}^{d}$.

######  Lemma 4.1.

Let $(C_{i},C_{i^{\prime}})$ and $(C_{j},C_{j^{\prime}})$ be any two
exceptional pairs in $\mathbf{G}$. Then

$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}\in
S_{\mathbf{G}}$

if and only if both $(C_{i},C_{j})$ and $(C_{i^{\prime}},C_{j^{\prime}})$
$\Big{(}\text{ or }(C_{i},C_{j^{\prime}})$ and $(C_{i^{\prime}},C_{j})\Big{)}$
are not exceptional.

###### Proof.

Let us consider $C_{p}=\\{x_{p},x_{p}^{k_{p}-1},x_{p}^{k_{p}}\\}$, for
$p\in\\{i,i^{\prime},j,j^{\prime}\\}$. We know that, the pair $(C_{p},C_{q})$
exceptional in $\mathbf{G}$ implies that $x_{p}\neq x_{q}$ and
$\\{x_{p},x_{q}\\}\notin E(\mathbf{G})$.

($\implies$) Let us assume that,
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}\in
S_{\mathbf{G}}$. This implies that
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}$
is a linear combination of $\rho(e)$ for some $e\in E(\mathbf{G})$.

According to the structure of the graph $\mathbf{G}$,
$\\{x_{u}^{k},x_{v}^{k^{\prime}}\\}\notin E(\mathbf{G})$, for any distinct
$u,v\in[2n]$. Moreover, for the exceptional pairs $(C_{i},C_{i^{\prime}})$ and
$(C_{j},C_{j^{\prime}})$, we know that $x_{i}\neq x_{i^{\prime}}$, $x_{j}\neq
x_{j^{\prime}}$, $\\{x_{i},x_{i^{\prime}}\\}\notin E(\mathbf{G})$, and
$\\{x_{j},x_{j^{\prime}}\\}\notin E(\mathbf{G})$. Therefore,
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}$
can be expressed as a linear combination of $\rho(e)$ for some $e\in
E(\mathbf{G})$ only if both of the vertices $x_{i}$ and $x_{i^{\prime}}$ are
such that

$x_{i},x_{i^{\prime}}\in\\{x_{j},x_{j^{\prime}}\\}\cup
N_{\mathbf{G}}\big{(}\\{x_{j},x_{j^{\prime}}\\}\big{)}.$

This also implies $x_{j},x_{j^{\prime}}\in\\{x_{i},x_{i^{\prime}}\\}\cup
N_{\mathbf{G}}\big{(}\\{x_{i},x_{i^{\prime}}\\}\big{)}.$ That is, either both
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{j}},\
\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j^{\prime}}}\in S_{\mathbf{G}}$ or
both $\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{j^{\prime}}},\
\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}\in S_{\mathbf{G}}$. Therefore,
in the given odd cycles $C_{p}$ ($p=i,i^{\prime},j,j^{\prime}$), there exists
two pairs of odd cycles that are not exceptional.

($\impliedby$) Let us assume that both the pairs $(C_{i},C_{j})$ and
$(C_{i^{\prime}},C_{j^{\prime}})$, are not exceptional. Therefore, we have one
of the following cases:

  * •

$V(C_{i})\cap V(C_{j})\neq\emptyset$ and $V(C_{i^{\prime}})\cap
V(C_{j^{\prime}})\neq\emptyset$;

  * •

there exists a bridge between $C_{i},C_{j}$ and between
$C_{i^{\prime}},C_{j^{\prime}}$;

  * •

$V(C_{i})\cap V(C_{j})\neq\emptyset$ and there exists a bridge between
$C_{i^{\prime}},C_{j^{\prime}}$ or vice versa.

In any of these cases, the method of the proof remains the same and therefore,
without loss of generality, we assume that $C_{i},C_{j}$ shares a common
vertex, i.e., $x_{i}=x_{j}$ and there exists a bridge between
$C_{i^{\prime}},C_{j^{\prime}}$, i.e., $\\{x_{i^{\prime}},x_{j^{\prime}}\\}\in
E(\mathbf{G})$. Hence,

$\begin{array}[]{lcl}\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}&=&(\mathbf{e}_{x_{i}}+\mathbf{e}_{x_{i}^{k_{i}}})+(\mathbf{e}_{x_{i}}+\mathbf{e}_{x_{i}^{k_{i}-1}})\\\
&&+(\mathbf{e}_{x_{j}^{k_{j}}}+\mathbf{e}_{x_{j}^{k_{j}-1}})+(\mathbf{e}_{x_{i^{\prime}}}+\mathbf{e}_{x_{j^{\prime}}})\\\
&&+(\mathbf{e}_{x_{i^{\prime}}^{k_{i^{\prime}}}}+\mathbf{e}_{x_{i^{\prime}}^{k_{i^{\prime}}-1}})+(\mathbf{e}_{x_{j^{\prime}}^{k_{j^{\prime}}}}+\mathbf{e}_{x_{j^{\prime}}^{k_{j^{\prime}}-1}}).\end{array}$

This implies,
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}$
is a linear combination of $\rho(e)$ for some $e\in E(\mathbf{G})$. Thus,
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}\in
S_{\mathbf{G}}$. ∎

######  Lemma 4.2.

Let $(C,C^{\prime})$ be exceptional in $\mathbf{G}$. Then, for any
$\\{v,w\\}\in E(\mathbf{G})$ with $v\in V(C)\cup V(C^{\prime})\cup
N_{\mathbf{G}}\big{(}V(C)\cup V(C^{\prime})\big{)}$, we have

$\mathbb{E}_{C}+\mathbb{E}_{C^{\prime}}+\mathbf{e}_{v}+\mathbf{e}_{w}\in
S_{\mathbf{G}}.$

###### Proof.

Since $(C,C^{\prime})$ is exceptional in $\mathbf{G}$, we have $V(C)\cap
V(C^{\prime})=\emptyset$ and $N_{\mathbf{G}}\big{(}V(C)\big{)}\cap
N_{\mathbf{G}}\big{(}V(C^{\prime})\big{)}=\\{w\\}$ (see, Figure 3). Hence both
$\mathbb{E}_{C}+\mathbf{e}_{w}$ and $\mathbb{E}_{C^{\prime}}+\mathbf{e}_{w}$
can be expressed as a linear combination of $\rho(e)$ for some $e\in
E(\mathbf{G})$. Now, without loss of generality, we assume that $v\in V(C)\cup
N_{\mathbf{G}}\big{(}V(C)\big{)}$. Thus we can write
$\mathbb{E}_{C}+\mathbf{e}_{v}=\sum_{e\in E(\mathbf{G})}\rho(e)$. Hence,
$\mathbb{E}_{C}+\mathbb{E}_{C^{\prime}}+\mathbf{e}_{v}+\mathbf{e}_{w}=(\mathbb{E}_{C}+\mathbf{e}_{v})+(\mathbb{E}_{C^{\prime}}+\mathbf{e}_{w})$,
can be expressed as a linear combination of $\rho(e)$ for some $e\in
E(\mathbf{G})$. Therefore, we have
$\mathbb{E}_{C}+\mathbb{E}_{C^{\prime}}+\mathbf{e}_{v}+\mathbf{e}_{w}\in
S_{\mathbf{G}}$. ∎

Based on the fact whether $w$ in $\mathbf{G}$ is a regular cutpoint or not, we
classify the graph $\mathbf{G}$ into two types. Let us look at these two cases
in-depth in the following subsections.

###  4.1. Type 1: $w$ is a regular cutpoint of $\mathbf{G}$

Let $\mathbf{G}^{\prime}$ be the triangular cactus $\mathbf{G}$ where any
$3$-cycle in $\mathbf{G}^{\prime}$ containing the vertex $w$ is such that at
least one of its remaining vertices $x_{i}$ will have at least one $3$-cycle
$\\{x_{i},x_{i}^{k-1},x_{i}^{k}\\}$ attached to it. This is the first type of
triangular cactus $\mathbf{G}$ that we will be studying.

Let us denote the vertices $x_{i}$ with no $3$-cycles
$\\{x_{i},x_{i}^{k-1},x_{i}^{k}\\}$ attached to them as $\zeta_{i}$ and let
there be $l$ such $\zeta_{i}$ in $\mathbf{G}^{\prime}$. That is, $\zeta_{i}\in
V(\mathbf{G}^{\prime})$ has no $3$-cycles attached to each of them, for all
$1\leq i\leq l$. Note that as per the description of $\mathbf{G}^{\prime}$,
for any $\\{x_{i},x_{j}\\}\in E(\mathbf{G}^{\prime})$, at most one of the
vertices ($x_{i}$ or $x_{j}$) can be a vertex $\zeta_{i}$. Therefore we have,
$0\leq l\leq n$. An illustration of the graph $\mathbf{G}^{\prime}$ is shown
in Figure 4.

$\zeta_{i}$$x_{j}$$w$$\vdots$$x_{j}^{2}$$x_{j}^{1}$$x_{j}^{k-1}$$x_{j}^{k}$$\vdots$$\vdots$$\vdots$$\vdots$$\dots$
Figure 4. An illustration of graph $\mathbf{G}^{\prime}$

Now, let us look at the regular vertices and fundamental sets in
$\mathbf{G}^{\prime}$. The vertex $w$ and all the $x_{i}$ of
$\mathbf{G}^{\prime}$ with at least one $3$-cycle
$\\{x_{i},x_{i}^{k-1},x_{i}^{k}\\}$ attached to them, are cutpoints of
$\mathbf{G}^{\prime}$. That is, the set of cutpoints in $\mathbf{G}^{\prime}$
is $\\{w\\}\cup\big{(}\\{x_{i}\colon 1\leq i\leq
2n\\}\backslash\bigcup_{i=1}^{l}\\{\zeta_{i}\\}\big{)}.$ The vertex $w$ and
all the non-cutpoints of $\mathbf{G}^{\prime}$ are regular in
$\mathbf{G}^{\prime}$.

Let us now focus on the fundamental sets in $\mathbf{G}^{\prime}$. For
$i\in[2n]$, let $s_{i}$ number of $3$-cycles, not containing the vertex $w$,
be attached to $x_{i}\in V(\mathbf{G}^{\prime})$. Then the fundamental set in
$\mathbf{G}^{\prime}$, containing the vertex $w$, is of the form
$\\{w\\}\cup\bigcup_{1\leq i\leq
2n}\\{x_{i}^{k_{1}},x_{i}^{k_{2}},\dots,x_{i}^{k_{s_{i}}}\\}$, where
$k_{u}\in[2s_{i}]$ and none of the vertices $x_{i}^{k_{u}}$ are adjacent to
each other. Note that, for the fundamental set $T$ with $w\in T$, and any
exceptional pair $(C,C^{\prime})$ in $\mathbf{G}^{\prime}$, $\big{(}T\cup
N_{\mathbf{G}^{\prime}}(T)\big{)}\cap\big{(}V(C)\cup
V(C^{\prime})\big{)}\neq\emptyset$.

For graph $\mathbf{G}^{\prime}$, and for any $i,j\in[2n]$, the building blocks
of any fundamental set not containing $w$ are as follows.

  1. (i)

Suppose $s$ is the number of $3$-cycles that are attached to the vertex
$x_{i}$ but do not contain the vertex $w$. Then, the set
$\\{x_{i}^{k_{1}},x_{i}^{k_{2}},\dots,x_{i}^{k_{s}}\\}$, where $k_{u}\in[2s]$,
and none of the vertices $x_{i}^{k_{u}}$ adjacent to each other, is
fundamental in $\mathbf{G}^{\prime}$.

  2. (ii)

Let $\\{x_{i},x_{j}\\}\in E(\mathbf{G}^{\prime})$, and let $t$ be the number
of $3$-cycles attached to the vertex $x_{j}$ that do not contain the vertex
$w$. Then, the set
$\\{x_{i},x_{j}^{k_{1}},x_{j}^{k_{2}},\dots,x_{j}^{k_{t}}\\}$, where none of
the vertices $x_{j}^{k_{v}}$ are adjacent to each other for $k_{v}\in[2t]$,
forms a fundamental set in $\mathbf{G}^{\prime}$.

Note that any fundamental set $T$ in $\mathbf{G}^{\prime}$ with $w\notin T$ is
either one of the listed building blocks or can be expressed as their unions.
Moreover, for any fundamental set $T$ in $\mathbf{G}^{\prime}$ consisting of
building blocks listed in (ii), we have $w\in N_{\mathbf{G}^{\prime}}(T)$.

######  Proposition 4.3.

For the triangular cactus $\mathbf{G}^{\prime}$ (Figure 4), the set of holes
$\overline{S_{\mathbf{G}^{\prime}}}\backslash S_{\mathbf{G}^{\prime}}$ is as
follows:

(4.1) $\overline{S_{\mathbf{G}^{\prime}}}\backslash
S_{\mathbf{G}^{\prime}}=\bigcup\limits_{\begin{subarray}{c}i;\\\
T\in\mathcal{T}_{i}\end{subarray}}(q_{i}+F_{T})\cup\bigcup\limits_{i}(q_{i}+F_{w}),$

where the index $i$ is taken over all possible exceptional pairs
$(C_{i},C_{i}^{\prime})$ of $\mathbf{G}^{\prime}$, for which we define
$q_{i}=\sum_{i}(\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i}^{\prime}})\notin
S_{\mathbf{G}^{\prime}};$ and $\mathcal{T}_{i}$ is the set of all fundamental
sets $T$ in $\mathbf{G}^{\prime}$ such that $w\in N_{\mathbf{G}^{\prime}}(T)$
and $\big{(}T\cup N_{\mathbf{G}^{\prime}}(T)\big{)}\cap\big{(}V(C_{i})\cup
V(C_{i}^{\prime})\big{)}=\emptyset$ for any exceptional pair
$(C_{i},C_{i}^{\prime})$ considered in $q_{i}$.

###### Proof.

Recall that, by Theorem 2.5, the normalization of edge ring
$\mathbb{K}[\mathbf{G}^{\prime}]$ can be expressed as

$\overline{S_{\mathbf{G}^{\prime}}}=S_{\mathbf{G}^{\prime}}\cup\mathbb{Z}_{\geq
0}\\{\mathbb{E}_{C}+\mathbb{E}_{C^{\prime}}\colon(C,C^{\prime})\text{ is
exceptional in }\mathbf{G}^{\prime}\\}.$

Also note that in (4.1), we choose $i$ in accordance with Lemma 4.1.

($\supset$) Let $\alpha\in q_{i}+F_{w}$ for some $q_{i}$. This implies that
$\alpha$ does not have any contribution from any edges adjacent to $w$. Thus,
$\alpha\in\overline{S_{\mathbf{G}^{\prime}}}\backslash
S_{\mathbf{G}^{\prime}}$. Note that, all $T\in\mathcal{T}_{i}$ is such that
$\big{(}T\cup N_{\mathbf{G}}(T)\big{)}\cap\big{(}V(C_{i})\cup
V(C_{i}^{\prime})\big{)}=\emptyset$. Therefore, any combination of edges of
the facet $F_{T}$ combined to $q_{i}$, never belong to
$S_{\mathbf{G}^{\prime}}$. Hence,
$q_{i}+F_{T}\in\overline{S_{\mathbf{G}^{\prime}}}\backslash
S_{\mathbf{G}^{\prime}}$, for all $T\in\mathcal{T}_{i}$.

($\subset$) Now let us prove that $\overline{S_{\mathbf{G}^{\prime}}}\subset
S_{\mathbf{G}^{\prime}}\cup\bigcup_{i;T\in\mathcal{T}_{i}}(q_{i}+F_{T})\cup\bigcup_{i}(q_{i}+F_{w})$.
Let $\alpha\in\overline{S_{\mathbf{G}^{\prime}}}$. We can write
$\alpha=\beta+\gamma$, where $\beta\in S_{\mathbf{G}^{\prime}}$ and
$\gamma\in\mathbb{Z}_{\geq
0}\\{\mathbb{E}_{C}+\mathbb{E}_{C^{\prime}}\colon(C,C^{\prime})\text{ are
exceptional in }\mathbf{G}^{\prime}\\}$.

Case 1: Let $\alpha_{w}=0$. This implies $\beta_{w}=0$ and we have $\beta\in
F_{w}$. From the structure of $\mathbf{G}^{\prime}$, we can see that any
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i^{\prime}}}+\mathbb{E}_{C_{j}}+\mathbb{E}_{C_{j^{\prime}}}\in
S_{\mathbf{G}^{\prime}}$ (in Lemma 4.1) is always contained in the facet
$F_{w}$. Thus, we express $\alpha=q_{i}+F_{w}$, for some $i$.

Case 2: Let $\alpha_{w}>0$. This implies $\beta_{w}>0$ and indicates that
among the edges defining the vector $\beta$, there must be at least one edge
adjacent to $w$, say $\\{v,w\\}$. By Lemma 4.2, we have
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i}^{\prime}}+\mathbf{e}_{v}+\mathbf{e}_{w}\in
S_{\mathbf{G}^{\prime}}$ if $v\in V(C_{i})\cup V(C_{i}^{\prime})\cup
N_{\mathbf{G}^{\prime}}\big{(}V(C_{i})\cup V(C_{i}^{\prime})\big{)}$. Let us
consider that $v\notin V(C_{i})\cup V(C_{i}^{\prime})\cup
N_{\mathbf{G}^{\prime}}\big{(}V(C_{i})\cup V(C_{i}^{\prime})\big{)}$, and
therefore
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i}^{\prime}}+\mathbf{e}_{v}+\mathbf{e}_{w}\in\overline{S_{\mathbf{G}^{\prime}}}\backslash
S_{\mathbf{G}^{\prime}}$. By observing the structure of the fundamental sets,
the existence of edge $\\{v,w\\}$ in the formation of $\beta$ implies that
$\beta$ belongs to some facets corresponding to the fundamental sets with
building blocks of type (ii). Therefore, we see that $\\{v,w\\}$ is contained
in some facet $F_{T}$ corresponding to the fundamental set $T$ of
$\mathbf{G}^{\prime}$ such that $v\in T$, $w\in N_{\mathbf{G}^{\prime}}(T)$,
and $\big{(}\\{v\\}\cup
N_{\mathbf{G}^{\prime}}(\\{v\\})\big{)}\cap\big{(}V(C_{i})\cup
V(C_{i}^{\prime})\big{)}=\emptyset$. Hence, any
$\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i}^{\prime}}+\mathbf{e}_{v}+\mathbf{e}_{w}\in\overline{S_{\mathbf{G}^{\prime}}}\backslash
S_{\mathbf{G}^{\prime}}$ can be expressed as $q_{i}+F_{T}$ for some
$T\in\mathcal{T}_{i}$.

In both cases, we get the desired containment. ∎

###  4.2. Type 2: $w$ is not regular in $\mathbf{G}$

Let $\widetilde{\mathbf{G}}$ be our second type of triangular cactus
$\mathbf{G}$ such that there exists at least one $3$-cycle in
$\widetilde{\mathbf{G}}$ containing $w$, with the remaining vertices of this
cycle having no other $3$-cycles attached to them. Moreover,
$\operatorname{diam}{\widetilde{\mathbf{G}}}=4$.

As earlier, we denote the vertices $x_{i}$ with no $3$-cycles
$\\{x_{i},x_{i}^{k-1},x_{i}^{k}\\}$ attached to them as $\zeta_{i}$ and let
there be $l$ such $\zeta_{i}$ in $\widetilde{\mathbf{G}}$. Note that as per
the description of $\widetilde{\mathbf{G}}$, there exists at least a pair of
vertices $(\zeta_{i},\zeta_{j})$ in $\widetilde{\mathbf{G}}$ such that
$\\{\zeta_{i},\zeta_{j}\\}\in E(\widetilde{\mathbf{G}})$. Since
$\operatorname{diam}{\widetilde{\mathbf{G}}}=4$, the number of vertices
$\zeta_{i}\in V(\widetilde{\mathbf{G}})$ is at most $2(n-1)$. Therefore we
have, $2\leq l\leq 2(n-1)$. A generic diagram of $\widetilde{\mathbf{G}}$ is
shown in Figure 5.

$\zeta_{i}$$\zeta_{j}$$w$$\vdots$$\vdots$$\vdots$$\vdots$$\dots$ Figure 5. An
illustration of graph $\widetilde{\mathbf{G}}$

The set of cutpoints of $\widetilde{\mathbf{G}}$ is
$\\{w\\}\cup\big{(}\\{x_{i}\colon 1\leq i\leq
2n\\}\backslash\bigcup_{i=1}^{l}\\{\zeta_{i}\\}\big{)}.$ Since there exists at
least one $\\{\zeta_{i},\zeta_{j}\\}\in E(\widetilde{\mathbf{G}})$, we observe
that $w$ is not regular in $\widetilde{\mathbf{G}}$. Note that, the only
regular vertices in $\widetilde{\mathbf{G}}$ are the non-cutpoints of
$\widetilde{\mathbf{G}}$.

For $i\in[2n]$, let $s_{i}$ number of $3$-cycles, not containing the vertex
$w$, be attached to $x_{i}\in V(\widetilde{\mathbf{G}})$. Then
$\\{w\\}\cup\bigcup_{1\leq i\leq
2n}\\{x_{i}^{k_{1}},x_{i}^{k_{2}},\dots,x_{i}^{k_{s_{i}}}\\}$, where
$k_{u}\in[2s_{i}]$ and none of the vertices $x_{i}^{k_{u}}$ adjacent to each
other, is the only fundamental set in $\widetilde{\mathbf{G}}$ that contains
the vertex $w$.

Let us consider that there are $m$ pair of vertices
$(\zeta_{p},\zeta_{p^{\prime}})$ in $\widetilde{\mathbf{G}}$ such that
$\\{\zeta_{p},\zeta_{p^{\prime}}\\}\in E(\widetilde{\mathbf{G}})$. For such a
pair of vertices $(\zeta_{p},\zeta_{p^{\prime}})$, we define $\omega_{p}$ as
$\omega_{p}\in\\{\zeta_{p},\zeta_{p^{\prime}}\\}$, $1\leq p\leq m$. In
$\widetilde{\mathbf{G}}$, and for any $i,j\in[2n]$, the building blocks of any
fundamental set not containing $w$, are as follows:

  1. (a)

The set $\bigcup_{p=1}^{m}\\{\omega_{p}\\}$ is fundamental in
$\widetilde{\mathbf{G}}$.

  2. (b)

Suppose that $s$ number of $3$-cycles that do not contain the vertex $w$, are
attached to $x_{i}$. Then the set
$\\{x_{i}^{k_{1}},x_{i}^{k_{2}},\dots,x_{i}^{k_{s}}\\}$, where $k_{u}\in[2s]$
such that none of the vertices $x_{i}^{k_{u}}$ adjacent to each other, is a
fundamental set in $\widetilde{\mathbf{G}}$.

  3. (c)

Let $\\{x_{i},x_{j}\\}\in E(\widetilde{\mathbf{G}})$ and $t$ number of
$3$-cycles that do not contain $w$ be attached to $x_{j}$. Then,
$\bigcup_{p=1}^{m}\\{\omega_{p}\\}\cup\\{x_{i},x_{j}^{k_{1}},x_{j}^{k_{2}},\dots,x_{j}^{k_{t}}\\}$
with none of the vertices $x_{j}^{k_{v}}$, $k_{v}\in[2t]$, adjacent to each
other, is a fundamental set in $\widetilde{\mathbf{G}}$.

Any fundamental set $T$ in $\widetilde{\mathbf{G}}$ with $w\notin T$ is either
one of the listed building blocks or can be expressed as their unions. For any
fundamental set $T$ consisting of the building blocks in (a) or (c), we
observe that $w\in N_{\widetilde{\mathbf{G}}}(T)$.

######  Proposition 4.4.

For the triangular cactus $\widetilde{\mathbf{G}}$, the set of holes
$\overline{S_{\widetilde{\mathbf{G}}}}\backslash S_{\widetilde{\mathbf{G}}}$
can be expressed as:

(4.2) $\overline{S_{\widetilde{\mathbf{G}}}}\backslash
S_{\widetilde{\mathbf{G}}}=\bigcup\limits_{\begin{subarray}{c}i;\\\
T\in\mathcal{T}_{i}\end{subarray}}(q_{i}+F_{T}),$

where the index $i$ is taken over all possible exceptional pair
$(C_{i},C_{i}^{\prime})$ of $\widetilde{\mathbf{G}}$, for which we define
$q_{i}=\sum_{i}(\mathbb{E}_{C_{i}}+\mathbb{E}_{C_{i}^{\prime}})\notin
S_{\widetilde{\mathbf{G}}};$ and $\mathcal{T}_{i}$ is the set of all
fundamental sets $T$ in in $\widetilde{\mathbf{G}}$ such that $w\in
N_{\widetilde{\mathbf{G}}}(T)$ and $\big{(}T\cup
N_{\widetilde{\mathbf{G}}}(T)\big{)}\cap\big{(}V(C_{i})\cup
V(C_{i}^{\prime})\big{)}=\emptyset$ for any exceptional pair
$(C_{i},C_{i}^{\prime})$ considered in $q_{i}$.

###### Proof.

For our graph $\widetilde{\mathbf{G}}$, we do not have any facet $F_{w}$ since
the vertex $w$ is not regular. Therefore note that, for any $\beta\in
S_{\widetilde{\mathbf{G}}}$, we have $\beta_{w}>0$. Further, we can proceed
with a similar proof as that of Proposition 4.3 and prove (4.2). ∎

We combine all the above results and proceed to the proof of our main theorem.

###### Proof of Theorem 3.2.

Since $\operatorname{diam}{\mathbf{G}}=4$, there exists at least one pair of
odd cycles of the form $(\\{x_{p},x_{p}^{k-1},x_{p}^{k}\\},\
\\{x_{q},x_{q}^{k^{\prime}-1},x_{q}^{k^{\prime}}\\})$ in $\mathbf{G}$ such
that the distance between $x_{p}$ and $x_{q}$ is $2$. All such pairs of odd
cycles are exceptional in $\mathbf{G}$. Therefore, the graph $\mathbf{G}$ does
not satisfy the odd cycle condition and by Theorem 2.6, we conclude that the
edge ring $\mathbb{K}[\mathbf{G}]$ is non-normal irrespective of the two types
of $\mathbf{G}$.

In Sections 4.1 and 4.2, we explored the two types of graph $\mathbf{G}$. Let
us assume that both $\mathbf{G}^{\prime}$ and $\widetilde{\mathbf{G}}$ are
triangular cactus graphs on $d$ vertices. We know that for a graph
$\mathbf{G}$ on $d$ vertices, all the facets $F_{T}$ and $F_{v}$ correspond to
hyperplanes $\mathcal{H}_{T}$ and $\mathcal{H}_{v}$ respectively and therefore
are of dimension $d-1$. Now, we compare (4.1) and (4.2) with the description
of holes given in (2.1) and observe that, all $q_{i}+F_{T}$ and $q_{i}+F_{w}$
in both (4.1) and (4.2) correspond to the holes of dimension $d-1$. Hence by
Theorem 2.3, we say that the edge ring $\mathbb{K}[\mathbf{G}]$ always
satisfies $(S_{2})$-condition. ∎

##  5\. Conclusions

We have seen that the associated edge ring of a triangular cactus graph of
diameter $4$ is always non-normal and satisfies $(S_{2})$-condition. Moreover,
our primary theorem, Theorem 3.2 provides supporting evidence to Conjecture
3.1.

######  Remark 5.1.

By further computations using Macaulay2, we believe that every non-normal edge
ring of triangular cactus graphs satisfying $(S_{2})$-condition is always
Cohen–Macaulay. To the best of our knowledge, no triangular cactus has been
identified whose edge ring does not satisfy $(S_{2})$-condition. Hence, we
expect that the edge ring of every triangular cactus graph is Cohen–Macaulay.

## Acknowledgement

The first named author was supported by the Alexander von Humboldt Foundation
and by a grant of the Ministry of Research, Innovation and Digitization, CNCS
- UEFISCDI, project number PN-III-P1-1.1-TE-2021-1633, within PNCDI III. The
second named author would like to thank Professor Akihiro Higashitani for his
valuable comments that greatly improved the manuscript.

## References