# Regularity of powers of $d$-sequence (parity) binomial edge ideals of
unicycle graphs

Marie Amalore Nambi  Department of Mathematics, IIT Hyderabad, Kandi,
Sangareddy - 502285<EMAIL_ADDRESS>and  Neeraj Kumar  Department
of Mathematics, IIT Hyderabad, Kandi, Sangareddy - 502285
<EMAIL_ADDRESS>
###### Abstract.

We classify all unicycle graphs whose edge-binomials form a $d$-sequence,
particularly linear type binomial edge ideals. We also classify unicycle
graphs whose parity edge-binomials form a $d$-sequence. We study the
regularity of powers of (parity) binomial edge ideals of unicycle graphs
generated by $d$-sequence (parity) edge-binomials.

###### Key words and phrases:

$d$-sequence, regularity, binomial edge ideal, parity binomial edge ideal

###### 2020 Mathematics Subject Classification:

05E40, 13D02

## Introduction

Let $G$ be a simple graph on $n$ vertices. Let
$S=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ be a polynomial ring on $2n$
variables over an infinite field $k$. For a given graph $G$, one can associate
binomial ideals in $S$; namely

$J_{G}=\langle f_{ij}\;|\;\\{i,j\\}\in E(G)\rangle\text{ and
}\mathcal{I}_{G}=\langle g_{ij}\;|\;\\{i,j\\}\in E(G)\rangle$

where $f_{ij}=x_{i}y_{j}-x_{j}y_{i}$ and $g_{ij}=x_{i}x_{j}-y_{j}y_{i}$. The
binomials $f_{ij}$ and $g_{ij}$ are called edge-binomial and parity edge-
binomial, respectively. The binomial ideals $J_{G}$ and $\mathcal{I}_{G}$ are
called _binomial edge ideal_ and _parity binomial edge ideal_ respectively.

The notion of the binomial edge ideal was introduced by Herzog et al. in [11]
and independently by Ohtani in [27]. Herzog et al. have shown that the
binomial edge ideal has natural connections to the study of conditional
independence ideals that are suitable to model robustness in contexts of
algebraic statistics (cf. [11], [31]). One may also view the binomial edge
ideal as a generalization of an ideal generated by a set of $2$-minors of
$2\times n$ matrix of indeterminates. The binomial edge ideal of path graphs
coincides with the ideal of adjacent minors of a $2\times n$ matrix of
indeterminates. In (cf. [5]), Diaconis, Eisenbud, and Sturmfels studied the
ideal generated by all the adjacent $2$-minors of a $2\times n$ generic
matrix. The authors also investigated the ideal generated by corner minors of
a $2\times n$ matrix of indeterminates which coincides with the binomial edge
ideal of star graphs (cf. [5]). For a complete graph $K_{n}$, the homogeneous
coordinate ring $S/J_{G}$ can be seen as a Segre variety given by the image of
Segre product $\mathbb{P}^{1}\times\mathbb{P}^{n-1}$ of projective spaces. The
parity binomial edge ideal was introduced by Kahle et al. in [18]. The authors
studied primary decomposition, mesoprimary decomposition, Markov bases, and
radicality of parity binomial edge ideals (cf. [18]). In ([3, Corollary 6.2]),
Bolognini et al. proved that if $G$ is a bipartite graph, then the binomial
edge ideal $J_{G}$ coincides with the parity binomial edge ideal
$\mathcal{I}_{G}$, Lovász–Saks–Schrijver ideal $L_{G}(2)$ (cf. [23]), and the
Permanental edge ideal $\Pi_{G}$ (cf. [10]). More results on the equality of
$J_{G}$ and $\mathcal{I}_{G}$ with $\Pi_{G}$, see [20, Remark 3.4].

The Rees algebra of an ideal encodes many asymptotic properties of that ideal.
An ideal in a commutative ring is said to be of _linear type_ if their Rees
and symmetric algebras are isomorphic; equivalently, the defining ideal of
Rees algebra is generated by linear forms. The notion of $d$-sequence was
introduced and initially studied by Huneke in [12, 14]. The author proved that
an ideal generated by $d$-sequence in a commutative ring is of linear type
(cf. [12]). Villarreal characterized graphs for which edge ideals are of the
linear type in [32], namely, edge ideals are of linear type if and only if the
graph is a tree or has a unique cycle of odd length. Researchers have recently
been interested in characterizing linear type (parity) binomial edge ideal.

Ene et al. have shown that the binomial edge ideal of a path graph is a
complete intersection ideal, hence of linear type [7]. Jayanthan et al.
characterized graphs whose binomial edge ideal is an almost complete
intersection in [17, Theorems $4.3$, and $4.4$]. One has the following strict
inclusions: _almost complete intersection ideal_ $\implies$ _d-sequence ideal_
$\implies$ _ideal of linear type_ [12, 17] and [13, page no. 341]. Jayanthan
et al. proposed the following conjecture.

######  Conjecture 0.1.

[17, Conjecture 4.17] If the given graph is a tree or a unicyclic graph, then
the binomial edge ideal is of linear type.

In [22], Kumar has shown that $J_{G}$ is of linear type for the closed graphs.
In [1], Amalore Nambi and Kumar have characterized all trees whose edge-
binomials form a $d$-sequence; in particular, $J_{G}$ is of a linear type. In
[20], the author characterized graphs whose parity binomial edge ideals are
complete intersection and almost complete intersection. By $\mathcal{T}_{m}$,
we denote the class of trees on $n$ vertices having degree sequences
$(m,2,\dots,1)$ or $(m,1,\dots,1)$, where $m\geq 2$. In Section 2, we classify
unicyclic graphs whose (parity) edge-binomials form a $d$-sequence. We
summarize the results below.

######  Theorem 0.1.

Let $S=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ be a polynomial ring. Let $G$
be a connected unicyclic graph and $H\in\mathcal{T}_{m}$ be a tree. Then the
edge-binomials of $G$ form a $d$-sequence in $S$ if and only if the
corresponding unicyclic graph has the following form

  1. (a)

$G$ is obtained by adding an edge between a pendant vertex of $H$ and the
center of $H$;

  2. (b)

$G$ is obtained by adding an edge between the center of $H$ and an internal
vertex of $H$;

  3. (c)

$G=C_{n}$, where $n\geq 3$.

Moreover, if char$(k)\neq 2$, and $G$ has one of the above forms, then the
parity edge-binomials form a $d$-sequence in $S$.

The above theorem supports Conjecture 0.1. Moreover, we expect that parity
binomial edge ideals of trees and unicyclic graphs are of linear type.

From a combinatorial commutative algebra point of view, one tries to
understand the Castelnuovo-Mumford regularity of (parity) binomial edge ideal
of a graph via combinatorial data of that graph. Matsuda and Murai obtained
the bounds for the regularity of binomial edge ideal of graphs in terms of the
number of the longest induced path and the number of vertices of that graph,
[25]. Several authors investigated the regularity of binomial edge ideal of
various class graphs (see [15, 24, 25, 28] for a partial list). Kumar [21]
obtained the lower bound for the regularity of parity binomial edge ideal of
connected graphs. We have computed the regularity of $\mathcal{I}_{G}$ of
graphs whose parity edge-binomials form a $d$-sequence, in Lemma 3.2 and Lemma
3.3. In Section 3, we obtain the regularity of the product of the parity
binomial edge ideal of disjoint union of paths and the binomial edge ideal of
a complete graph in Theorem 3.1.

Our primary interest is to study the regularity of powers of (parity) binomial
edge ideals. Cutkosky, Herzog, and Trung (and independently Kodiyalam) proved
that for any homogeneous ideal $I$, the regularity of $I^{s}$ is
asymptotically a linear function in $s$ (cf. [4, 19]). Raghavan in [30]
generalized the notion of $d$-sequence to quadratic sequence and studied the
depth of powers of an ideal generated by quadratic sequences. Note that
$d$-sequence implies quadratic sequences, but the converse need not be true
[30]. Jayanthan et al. in [16] obtained the bounds for the regularity of
powers of almost complete intersection binomial edge ideals using the
quadratic sequence and related ideals approach. Furthermore, the authors
provided the regularity of powers of binomial edge ideals of star graphs. Shen
and Zhu in [29] obtained explicit formulas for the regularity of powers of an
almost complete intersection (parity) binomial edge ideals. Amalore Nambi and
Kumar in [1] obtained explicit expression for the regularity of powers of
binomial edge ideals of trees whose edge-binomial forms a $d$-sequence. Ene et
al. in [8] obtained the regularity of powers of binomial edge ideals of closed
graphs.

A clique is a subset $U$ of $V(G)$ such that the induced subgraph $G[U]$ is a
complete graph. If a vertex $v$ of $G$ is contained in only one maximal
clique, it is called _free vertex_ of the graph; otherwise, _internal vertex_
of the graph. By $i(G)$, we denote the number of internal vertices of $G$. In
Sections 4 and 5, we obtained a precise expression for the regularity of
powers of (parity) binomial edge ideals of unicyclic graphs whose (parity)
edge-binomial forms a $d$-sequence. We summarize the results below.

######  Theorem 0.2.

Let $H\in\mathcal{T}_{m}$ be a tree, and $G$ be a unicyclic graph. Let
$S=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ be a polynomial ring and let
$J_{G}$ be the binomial edge ideal of $G$ and $\mathcal{I}_{G}$ be the parity
binomial edge ideal of $G$.

  * (i)

If $G$ is obtained by adding an edge between a pendant vertex of $H$ and the
center of $H$, then for all $s\geq 1$, one has

$\begin{split}\operatorname{reg}{S}/{J_{G}^{s}}&=\begin{cases}2s+i(G)-1&\text{if
girth}(G)=3\\\ 2s+i(G)-3&\text{if girth}(G)\geq 4\end{cases}\\\
\operatorname{reg}{S}/{\mathcal{I}_{G}^{s}}&=\begin{cases}2s+i(G)&\text{if
girth}(G)=3\\\ 2s+i(G)-2&\text{if odd-girth}(G)\geq 5.\end{cases}\\\
\end{split}$

  * (ii)

If $G$ is obtained by adding an edge between the center of $H$ and an internal
vertex of $H$, then one has

$\begin{split}\operatorname{reg}{S}/{J_{G}^{s}}&=\begin{cases}2s+i(G)-1&\text{if
girth}(G)=3\\\ 2s+i(G)-2&\text{if girth}(G)\geq 4\end{cases}\\\ \text{for all
}s\geq 2.\\\
\operatorname{reg}{S}/{\mathcal{I}_{G}^{s}}&=\begin{cases}2s+i(G)-1&\text{if
girth}(G)=3\\\ 2s+i(G)-2&\text{if odd-girth}(G)\geq 5\end{cases}\\\ \text{for
all }s\geq 1.\end{split}$

Acknowledgement. The first author is financially supported by the University
Grant Commission, India. The second author is partially funded by MATRICS
grant, project no. MTR/2020/000635, from Science and Engineering Research
Board (SERB), India.

##  1\. Preliminaries

In this section, we recall definitions from graph theory and commutative
algebra. Throughout this article, $S$ denotes the polynomial ring
$k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ over a field $k$ unless otherwise
stated.

Let $M$ be a finitely generated graded $S$-module. Let $\mathbf{F}_{\bullet}$
be a minimal graded $S$-free resolution of $M$:

$\mathbf{F}_{\bullet}:0\longrightarrow\bigoplus_{j\in\mathbb{Z}}S(-p-j)^{\beta_{p,p+j}(M)}\stackrel{{\scriptstyle\phi_{p}}}{{\longrightarrow}}\cdots\longrightarrow\bigoplus_{j\in\mathbb{Z}}S(-j)^{\beta_{0,j}(M)}\stackrel{{\scriptstyle\phi_{0}}}{{\longrightarrow}}M\longrightarrow
0.$

Where, $S(-i-j)$ denotes the graded free module of rank $1$ obtained by
shifting the degrees in $S$ by $i+j$, and $\beta_{i,i+j}(M)$ denotes the
$(i,i+j)$-th graded Betti number of $M$ over $S$. From the minimal free
resolution, one can obtain an important invariant called _Castelnuovo-Mumford
regularity_ (or simply _regularity_) of $M$ over $S$, denoted by
$\operatorname{reg}_{S}M$, is defined as

$\operatorname{reg}_{S}M\coloneqq\max\\{j\mid\beta_{i,i+j}\neq 0\text{ for
some }i\\}.$

For convenience, we shall use $\operatorname{reg}M$ instead of
$\operatorname{reg}_{S}M$. We state a regularity lemma [6, Corollary 20.19]
that will be used throughout the article.

######  Lemma 1.1 (Regularity lemma).

Let $0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0$ be a
short exact sequence of finitely generated graded $S$-modules. Then the
following holds.

  1. (a)

$\operatorname{reg}{M_{1}}\leq\max\\{\operatorname{reg}M_{2},\operatorname{reg}M_{3}+1\\}$.
The equality holds if $\operatorname{reg}M_{2}\neq\operatorname{reg}M_{3}$.

  2. (b)

$\operatorname{reg}{M_{2}}\leq\max\\{\operatorname{reg}M_{1},\operatorname{reg}M_{3}\\}$.
The equality holds if $\operatorname{reg}M_{1}\neq\operatorname{reg}M_{3}+1$.

  3. (c)

$\operatorname{reg}{M_{3}}\leq\max\\{\operatorname{reg}M_{1}-1,\operatorname{reg}M_{2}\\}$.
The equality holds if $\operatorname{reg}M_{1}\neq\operatorname{reg}M_{2}$.

Let $G$ be a finite simple graph on the vertex set $V(G)$ and edge set $E(G)$.
For $U\subseteq V(G)$, an _induced subgraph_ on vertex set $U$ is denoted by
$G[U]$, is for $i,j\in U$, $\\{i,j\\}\in E(G[U])$ if and only if $\\{i,j\\}\in
E(G)$. A _complete_ graph on the vertex set $V(G)=[n]$ is denoted by $K_{n}$,
is a graph with $\\{i,j\\}\in E(G)$ for all $i,j\in V(G)$. A graph $G$ is
called a _block graph_ if every block of $G$ is a complete graph. A graph $G$
is said to be bipartite if there exists a bipartition $V(G)=V_{1}\sqcup V_{2}$
such that no two vertices of $V_{i}$ are adjacent for each $i=1,2$. A star
graph on $[n+1]$ vertices is denoted by $K_{1,n}$, is the graph with one
internal vertex and $n$ leaves. A cycle on $[n]$ is denoted by $C_{n}$, is a
graph with every vertex has degree $2$. A unicyclic graph is a graph
containing exactly one cycle as a subgraph.

######  Notation 1.1.

Let $G$ be a simple graph. For an edge $e^{\prime}$ in $G$, $G\setminus
e^{\prime}$ is the graph on the vertex set $V(G)$ and edge set
$E(G)\setminus{e}$. An edge $e^{\prime}$ is called a _bridge_ if
$c(G)<c(G\setminus e^{\prime})$, where $c(G)$ is the number of components of
$G$. For a vertex $v$,

$N_{G}(v)=\\{u\in V(G)\mid\\{u,v\\}\in E(G)\\}$

denotes the _neighborhood_ of $v$ in $G$. Let $e=\\{i,j\\}\notin E(G)$ be an
edge in $G\cup\\{e\\}$. Then $G_{e}$ (cf. [26, Definition 3.1]) is the graph
on vertex set $V(G)$ and edge set

$E(G_{e})=E(G)\cup\left\\{\\{k,l\\}:k,l\in N_{G}(i)\textnormal{ or }k,l\in
N_{G}(j)\right\\}.$

The following result provides equality of the binomial edge ideal and parity
binomial edge ideal for a bipartite graph.

######  Remark 1.1.

[3, Corollary 6.2] Let $G$ be a bipartite graph with a partition
$V(G)=V_{1}\sqcup V_{2}$. Let $\phi:S\rightarrow S$ be a map defined by

$\phi(x_{i})=\begin{cases}x_{i}\text{ if }i\in V_{1}\\\ y_{i}\text{ if }i\in
V_{2}\end{cases}\text{ and }\phi(y_{i})=\begin{cases}y_{i}\text{ if }i\in
V_{1}\\\ x_{i}\text{ if }i\in V_{2}.\end{cases}$

Then $\phi$ is an isomorphism and $\phi(J_{G})=\mathcal{I}_{G}$. We will
reserve $\phi$ throughout the article.

The following results describe the colon ideal operation on the binomial edge
ideal due to Mohammadi and Sharifan and the parity binomial edge ideal due to
Kumar.

######  Remark 1.2.

Let $G$ be a simple graph.

  1. (a)

[26, Theorem 3.4] Let $e=\\{i,j\\}\notin E(G)$ be a bridge in $G\cup\\{e\\}$.
Then $J_{G}:f_{e}=J_{G_{e}}$.

  2. (b)

[26, Theorem 3.7] Let $e=\\{i,j\\}\notin E(G)$. Then

$J_{G}:f_{e}=J_{G_{e}}+(g_{P,t}\mid\quad P:i,i_{1},\ldots,i_{s},j\text{ is a
path between }i,j\text{ and }0\leq t\leq s),$

where $g_{P,0}=x_{i_{1}}\ldots x_{i_{s}}$ and for each $1\leq t\leq
s,g_{P,t}=y_{i_{1}}\ldots y_{i_{t}}x_{i_{t+1}}\ldots x_{i_{s}}$.

  3. (c)

[20, Lemma 3.3] Let $G$ be a non-bipartite graph. Assume that there exists an
edge $e=\\{u,v\\}$ such that $G\setminus e$ is a bipartite graph. Then

$\mathcal{I}_{G\setminus e}:g_{e}=\mathcal{I}_{G\setminus e}+\langle
f_{i,j}\mid\\{i,j\\}\in N_{G\setminus e}(u)\text{ or }\\{i,j\\}\in
N_{G\setminus e}(v)\rangle=\phi(J_{(G\setminus e)_{e}}).$

######  Remark 1.3.

Let $G$ be a bipartite graph and $e=\\{i,j\\}\notin E(G)$ be a bridge in
$G\cup\\{e\\}$ such that $V(G\cup e)=V_{1}\sqcup V_{2}$. Then one has
$\mathcal{I}_{G}:g_{e}=\mathcal{I}_{G}+\langle f_{kl}\mid k,l\in
N_{G}(i)\textnormal{ or }k,l\in N_{G}(j)\rangle$.

###### Proof.

The proof follows from Remarks 1.1 and 1.2(a). ∎

The following lemma follows from [11, Corollary 2.2], [18, Theorem 5.5] and
[17, Lemma 4.1].

######  Lemma 1.2.

Let $G$ be a simple graph and $g\in S$, then

  1. (a)

$J_{G}:g=J_{G}:g^{n}$ for any $n\geq 2$,

  2. (b)

$\mathcal{I}_{G}:g=\mathcal{I}_{G}:g^{n}$ for any $n\geq 2$, if char$(k)\neq
2$.

######  Remark 1.4.

[15, Corollary 3.2] Let $G$ be a new graph obtained by gluing finitely many
graphs at free vertices. Let $G=L_{1}\cup\cdots\cup L_{k}$ be a graph
satisfying the properties

  1. (a)

For $i\neq j,$ if $L_{i}\cap L_{j}\neq\emptyset$, then $L_{i}\cap
L_{j}=\\{v_{ij}\\}$, for some vertex $v_{ij}$ which is free vertex in $L_{i}$
as well as $L_{j}$;

  2. (b)

For distinct $i,j,k,L_{i}\cap L_{j}\cap L_{k}=\emptyset.$

Then $\operatorname{reg}S/J_{G}=\sum_{i=1}^{k}\operatorname{reg}S/J_{L_{i}}$.

######  Remark 1.5.

[1, Theorem 0.2] If $G\in\mathcal{T}_{m}$ (see, Notation 2.2), then
$\operatorname{reg}S/J_{G}^{s}=2s+i(G)-1$, for all $s\geq 1$ and $m\geq 2$.

A _flower graph_ $F_{h,k}(v)$ (cf. [24, Definition 3.1]) is a connected block
graph obtained by identifying $h$ copies of $C_{3}$ and $k$ copies of
$K_{1,3}$ with a common vertex $v$, where $v$ is one of the free vertices of
$C_{3}$ and of $K_{1,3}$, and $cdeg(v)\geq 3$. $G$ is called flower-free if
$G$ has no ﬂower graphs as induced subgraphs.

######  Remark 1.6.

[24, Corollary 3.2] Let $G$ be a connected block graph that does not have an
isolated vertex. If $G$ is a flower-free graph, then
$\operatorname{reg}S/J_{G}=i(G)+1$.

######  Definition 1.1.

[14, Definition 1.1] Let $R$ be a commutative ring. Set $a_{0}=0$. A sequence
of elements $a_{1},\ldots,a_{m}$ in $R$ is said to be a $d$-sequence if it
satisfies the conditions: $a_{1},\ldots,a_{r}$ is a minimal system of
generators of the ideal $(a_{1},\ldots,a_{r})$; and $(\langle
a_{0},\ldots,a_{i}\rangle:a_{i+1}a_{j})=(\langle
a_{0},\ldots,a_{i}\rangle:a_{j})$ for all $0\leq i\leq m-1$, and $j\geq i+1$.

######  Lemma 1.3.

[29, Observation 2.4] Let $R$ be a commutative ring. Suppose that
$a_{1},\ldots,a_{n}$ form a $d$-sequence in $R$ and $I=\langle
a_{1},\ldots,a_{n}\rangle$ is the ideal in $R$. Set $a_{0}=0$. Then, one has

$((a_{0},a_{1},\ldots,a_{i-1})+I^{s}:a_{i}=((a_{0},a_{1},\ldots,a_{i-1}:a_{i})+I^{s-1},$

for $s\geq 1$ and $i=1,\ldots,n$.

##  2\. d-sequence (parity) edge-binomials

In this section, we characterize connected unicyclic graphs whose edge-
binomials form a $d$-sequence. In addition, we show that parity edge-binomials
of these connected unicyclic graphs form a $d$-sequence. First, we set up
notations that we use thought out this article.

######  Notation 2.1.

Let $a_{1},\ldots,a_{n}$ be a sequence of (parity) edge-binomials of $G$. We
denote $J_{i}$ for an ideal generated by $\langle a_{1},\ldots,a_{i}\rangle$
and $H_{i}$ for the graph associated to (parity) edge-binomials
$a_{1},\ldots,a_{i}$. We denote an edge $\\{\alpha_{a_{i}},\beta_{a_{i}}\\}\in
E(G)$ for an associated (parity) edge-binomial $a_{i}$.

The following describes combinatorial characterization for edge-binomials of a
unicyclic graph to form a $d$-sequence.

######  Lemma 2.1.

Let $G$ be a unicyclic graph on $[n]$. Assume that $a_{1},\ldots,a_{n}$ form a
$d$-sequence edge-binomials of $G$. Then an edge associated with $a_{n}$ forms
a cycle in $G$, i.e., the graph $H_{n-1}$ is a tree.

###### Proof.

Let $i<n$ be the smallest integer such that $H_{i}$ has a cycle. Then from
Remark 1.2 (a) and (b) it follows that $J_{i-1}:a_{i}a_{j}\neq J_{i-1}:a_{j}$
for any $j>i$. Since the graph $H_{i-1}\cup e_{j}$ is a tree and $e_{i}$ is
not a bridge in $H_{i}$, where $e_{i}$ and $e_{j}$ corresponds edge of edge-
binomials $a_{i}$ and $a_{j}$ respectively. ∎

The authors in [1] proved the following lemma for trees. We prove the result
for unicyclic graphs.

######  Lemma 2.2.

Let $G$ be a unicyclic graph on $[n]$. Assume that $a_{1},\ldots,a_{n}$ are
$d$-sequence edge-binomials of $G$, where $a_{k}$ corresponds to an edge
$\\{\alpha_{a_{k}},\beta_{a_{k}}\\}\in E(G)$ for all $k$.

If there exists a smallest integer $i$ such that $J_{i}:a_{i+1}\neq J_{i}$,
then
$\\{\alpha_{a_{i+1}},\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$,
for all $j>i+1$. In particular, for all $j>i+1$, one has
$\\{\alpha_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$ or
$\\{\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$.

###### Proof.

Assume that $i<n-1$. From Lemma [1, Lemma 2.1] and 2.1, it is enough to prove
that
$\\{\alpha_{a_{i+1}},\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{n}},\beta_{a_{n}}\\}\neq\emptyset$.
Suppose that
$\\{\alpha_{a_{i+1}},\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{n}},\beta_{a_{n}}\\}=\emptyset$
and $i$ be the smallest integer such that $J_{i}:a_{i+1}\neq J_{i}$. Then from
Remark 1.2 one has $f_{kl}\in J_{i}:a_{i+1}$ such that $f_{kl}\notin J_{i}$,
where $\\{k,l\\}\in N_{H_{i}}(\alpha_{a_{i+1}})$ or $\\{k,l\\}\in
N_{H_{i}}(\beta_{a_{i+1}})$. If $\\{k,l\\}\in N_{H_{i}}(\alpha_{a_{i+1}})$
then clearly $\\{k,l\\}\notin N_{H_{i}}(\beta_{a_{i+1}})$, otherwise $G$ has
two cycles. So one can further assume that $\\{k,l\\}\in
N_{H_{i}}(\alpha_{a_{i+1}})$. We have $J_{i}:a_{i+1}a_{n}=J_{i}:a_{n}$ by
hypothesis. But, if $f_{kl}\in J_{i}:a_{n}$ then $\\{k,l\\}\in
N_{H_{i}}(\alpha_{a_{n}})$ or $\\{k,l\\}\in N_{H_{i}}(\beta_{a_{n}})$, by
Remark 1.2. In either case, $H_{i}$ has a cycle. For instance, if
$\\{k,l\\}\in N_{H_{i}}(\alpha_{a_{n}})$ then edges
$\\{\\{\alpha_{a_{i+1}},k\\},\\{\alpha_{a_{i+1}},l\\},\\{\alpha_{a_{n}},k\\},\\{\alpha_{a_{n}},l\\}\\}\in
E(H_{i})$ form a cycle. This is a contradiction by Lemma 2.1. Thus
$\\{\alpha_{a_{i+1}},\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{n}},\beta_{a_{n}}\\}\neq\emptyset$
as desired. ∎

######  Lemma 2.3.

Let $G=G_{1}\sqcup\cdots\sqcup G_{n}$ be a unicyclic graph. Then edge-
binomials of $G$ form a $d$-sequence if and only if edge-binomials of $G_{i}$
form a $d$-sequence and for $j\neq i$, $J_{G_{j}}$ is a complete intersection.

###### Proof.

Assume that edge-binomials of $G_{i}$ form a $d$-sequence and for $j\neq i$,
$J_{G_{j}}$ is a complete intersection. Note that if $J_{G}$ is a complete
intersection, then edge-binomials of $G$ form a regular sequence. Thus edge-
binomials associated with $G_{j}$’s for all $j\neq i$ form a condition
independent $d$-sequence (independent of order). We take a sequence of edge-
binomials of $G_{j}$ for all $j\neq i$ (independent of order), then edge-
binomials of $G_{i}$ in the same as it forms a $d$-sequence. Then the above
sequence forms a $d$-sequence in $S$. Thus, $J_{G}$ is generated by a
$d$-sequence.

It follows from [7, Corollary 1.2] that if $G$ is not a complete intersection,
then $G$ has a vertex of degree $3$ or all the vertices of $G$ have degree
$2$. Conversely, Assume that none of the edge-binomials of $G_{i}$ form a
$d$-sequence or there exists $j$ and $k$ such that $J_{G_{j}}$ and $J_{G_{k}}$
are not complete intersection for some $j\neq k$. It is enough to prove that
edge-binomials of $J_{G_{j}\sqcup G_{k}}$ do not form a $d$-sequence. There
are two cases for $G_{j}\sqcup G_{k}$

  1. (i)

both $G_{j}$ and $G_{j}$ has at least one vertex of degree $3$,

  2. (ii)

$G_{j}$ has a vertex of degree $3$, and all the vertices of $G_{k}$ have
degree $2$.

From Lemma 2.2, it follows that any sequence of edge-binomials of graph (i) or
graph (ii) do not form a $d$-sequence. ∎

Recall that $\mathcal{T}_{m}$ denotes the class of trees on $[n]$ vertices
having degree sequence $(m,2,\dots,1)$ or $(m,1,\dots,1)$, where $m\geq 2$,
[1]. The following version of $\mathcal{T}_{m}$ will be useful in proving
results.

######  Notation 2.2.

$\mathcal{T}_{m}$ denote the class of graphs with vertex set and edge set as
below:

$V(G)=\\{k_{0},p_{1_{1}},\ldots,p_{1_{s(1)+1}},p_{2_{1}},\ldots,p_{2_{s(2)+1}},\ldots,p_{m_{1}},\ldots,p_{m_{s(m)+1}}\\}$

with $s(i)\geq 0$ for all $1\leq i\leq m$, and edge set

$E(G)=\\{\\{k_{0},p_{i_{1}}\mid
i=1,\ldots,m\\}\cup\bigcup_{i=1}^{m}\\{p_{i_{j}},p_{i_{j+1}}\mid
j=1,\ldots,s(i)\\}\\}.$

######  Remark 2.1.

[1, Theorem 2.1] If $G\in\mathcal{T}_{m}$ then edge-binomials of $G$ form a
$d$-sequence.

In the following theorem, we construct unicyclic graphs whose edge-binomials
form a $d$-sequence.

######  Theorem 2.1.

Let $H\in\mathcal{T}_{m}$ be a tree. Consider a unicyclic graph $G$
constructed by adding

  1. (a)

an edge between a pendant vertex of $H$ and the center of $H$, or

  2. (b)

an edge between two pendant vertices of $H$, or

  3. (c)

an edge between the center of $H$ and an internal vertex of $H$.

Then edge-binomials of $G$ form a $d$-sequence. In particular, $J_{G}$ is of
linear type.

###### Proof.

Proof of (a): Suppose $G$ is obtained by adding an edge between a pendant
vertex of $H$ and the center of $H$. Let $e_{1}=\\{k_{0},p_{k_{s(k)}}\\}$, for
some $k$ with $s(k)\geq 1$, be an edge of $G$. Set $d_{0}=0\in S$. Consider
the following sequence of edge-binomials $d_{1},\ldots,d_{n}$ of $G$, where
the first $n-1$ elements are edge-binomials of $H$ with same order as in [1,
Theorem 2.1] and $d_{n}=f_{e_{1}}$. From Remark 2.1, it follows that the
binomial edge ideal of $H$ is generated by $d$-sequence. Thus, it is enough to
prove that
$(d_{0},d_{1},\ldots,d_{i}):d_{i+1}d_{n}=(d_{0},d_{1},\ldots,d_{i}):d_{n}$ for
all $i\leq n-1$. For $i=n-1$, the equality follows from Lemma 1.2. For $i\leq
n-2$, the equality follows from Remark 1.2 and Lemma 2.2.

Proof of (b): Suppose $G$ is obtained by adding an edge between two pendant
vertices of $H\in\mathcal{T}_{m}$ with $m\geq 3$. Then, $G$ can be viewed as a
graph obtained by adding an edge between a pendant vertex of $H^{\prime}$ and
the center of $H^{\prime}$, where $H^{\prime}\in\mathcal{T}_{m-1}$. If $m=2$,
then G is a cycle. From [17, Theorem 4.4] it follows that edge-binomials of
$C_{n}$ forms a $d$-sequence. Thus $J_{G}$ is generated by $d$-sequence.

Proof of (c): Suppose $G$ is obtained by an edge between the center of $H$ and
an internal vertex of $H$. Let $e_{2}=\\{u,v\\}$ be an edge of $G$ such that
$\deg_{G}(u)=m+1$ and $\deg_{G}(v)=3$. Clearly, $G\setminus\\{e\\}=H$. From
Remark 2.1 it follows that edge-binomials of $H$ form a $d$-sequence. Consider
the following sequence of edge-binomials $d_{1},\ldots,d_{n}$ of $G$, where
the first $n-1$ elements are edge-binomials of $H$ with same order as in [1,
Theorem 2.1] and $d_{n}=f_{e}$. The $d$-sequence conditions hold in a similar
way as (a). ∎

######  Remark 2.2.

Let $H\in\mathcal{T}_{m}$ be a tree.

  1. (a)

If $G$ is a unicyclic graph as mentioned in Theorem 2.1(a), then one has

$i(G)=\begin{cases}\sum_{i=1}^{m}{s_{(i)}}&\text{if girth}(G)=3\\\
2+\sum_{i=1}^{m}{s_{(i)}}&\text{if girth}(G)\geq 4.\end{cases}$

  2. (b)

If $G$ is a unicyclic graph as mentioned in Theorem 2.1(c), then one has

$i(G)=\begin{cases}\sum_{i=1}^{m}{s_{(i)}}&\text{if girth}(G)=3\\\
1+\sum_{i=1}^{m}{s_{(i)}}&\text{if girth}(G)\geq 4.\end{cases}$

Below we illustrated two types of graphs obtained from $G\in\mathcal{T}_{2}$
whose (parity) edge-binomials form a $d$-sequence. $G_{1}$ is a graph obtained
by adding an edge $e_{1}$ between a pendant vertex of $G$ and the center of
$G$. $G_{2}$ is a graph obtained by adding an edge $e_{2}$ between an internal
vertex of $G$ and the center of $G$.

$k_{0}$$p_{1_{1}}$$p_{1_{2}}$$p_{1_{3}}$$p_{2_{1}}$$k_{0}$$p_{1_{1}}$$p_{1_{2}}$$p_{1_{3}}$$p_{2_{1}}$$k_{0}$$p_{1_{1}}$$p_{1_{2}}$$p_{1_{3}}$$p_{2_{1}}$$G_{2}$$e_{2}$$G_{1}$$e_{1}$$G$

######  Remark 2.3.

The authors in [17, Theorem 4.4.] characterized unicyclic graphs whose
binomial edge ideals are almost complete intersections. Also, the same authors
in [16] classified into two types of graphs called $G_{1}$-type and
$G_{2}$-type. One can see that the $G_{1}$-type graphs are a subgraph of the
collection of graphs $G$ as in Theorem 2.1(a), and the $G_{2}$-type graphs are
a subgraph of the collection of graphs $G$ as in Theorem 2.1(c).

######  Lemma 2.4.

Let $G$ be a connected unicyclic graph on $[n]$ and $H\in\mathcal{T}_{m}$ be a
tree. If edge-binomials of $G$ form a $d$-sequence, then a unicyclic graph is
one of the following forms:

  1. (a)

$G$ is obtained by adding an edge between a pendant vertex of $H$ and the
center of $H$,

  2. (b)

$G$ is obtained by adding an edge between the center of $H$ and an internal
vertex of $H$,

  3. (c)

$G=C_{n}$, where $n\geq 3$.

###### Proof.

Let $d_{1},\ldots,d_{n}$ be a sequence of edge-binomials of $G$ such that
$d_{1},\ldots,d_{n}$ form a $d$-sequence. Let $e$ be an edge associate to the
edge-binomial $d_{n}$. From Lemma 2.1 it follows that $G\setminus\\{e\\}$ is a
tree. From [1, Theorem 0.1] it follows that
$G\setminus\\{e\\}\in\\{\mathcal{T}_{m},\mathcal{H}_{m}\\}$. Clearly
$d_{1},\ldots,d_{n-1}$ form a $d$-sequence. Now we consider possible cases for
an edge $e$:

Case 1: If $e$ is an edge between two pendant vertices of a graph in
$\mathcal{T}_{m}$. For $m\geq 3$, $G$ is isomorphic to a graph in (a). For
$m=2$, $G$ is isomorphic to $C_{n}$.

Case 2: If $e$ is an edge between a pendant vertex and the center of a graph
in $\mathcal{T}_{m}$. Then $G$ is isomorphic to a graph in (a).

Case 3: If $e$ is an edge between a pendant vertex and an internal vertex of a
graph in $\mathcal{T}_{m}$ such that the cycle of $G$ has a vertex of degree
$m$. Then $G$ is isomorphic to a graph in (b).

Case 4: If $e$ is an edge between two pendant vertices of a graph in
$\mathcal{H}_{m}$ such that all the vertices of the cycle in $G$ have degree
$2$ except two vertices. Then $G$ is isomorphic to a graph in (b).

Case 5: If $e$ is an edge between a pendant vertex and the center of a graph
in $\mathcal{H}_{m}$ such that all the vertices of the cycle in $G$ have
degree $2$ except two vertices. Then $G$ is isomorphic to a graph in (b).

Case 6: From Lemma 2.2 and Lemma 2.1, it follows that if $G$ has a vertex of
degree at least three, which is not a vertex of the cycle of $G$, then any
sequence of edge-binomials of $G$ does not satisfy the $d$-sequence condition.
∎

In the following lemma, we prove parity edge-binomials of the graphs
considered in Theorem 2.1 form a $d$-sequence.

######  Theorem 2.2.

Let $H\in\mathcal{T}_{m}$ be a tree. Assume that char$(k)\neq 2$. Let $G$ be a
unicyclic graph obtained by adding

  1. (a)

an edge between a pendant vertex of $H$ and the center of $H$, or

  2. (b)

an edge between two pendant vertices of $H$, or

  3. (c)

an edge between the center of $H$ and an internal vertex of $H$.

Then parity edge-binomials of $G$ forms a $d$-sequence. In particular,
$\mathcal{I}_{G}$ is of linear type.

###### Proof.

If $G$ has an even girth, the statement follows from Remark 1.1. If $G$ has an
odd girth, then we take a sequence of parity edge-binomials in the same order
as edge-binomials taken in Theorem 2.1. It is enough to prove that
$(d_{0},d_{1},\ldots,d_{i}):d_{i+1}d_{n}=(d_{0},d_{1},\ldots,d_{i}):d_{n}$ for
all $i$. Since the graph associated with parity edge-binomials
$d_{1},\ldots,d_{n-1}$ is a bipartite graph. For $i=n-1$, the statement
follows from Lemma 1.2. For $i<n-1$ the statement follows from Remark 1.1 and
Lemma 1.2. ∎

######  Remark 2.4.

Let $G$ be a unicyclic graph on $[6]$ with edge set
{{1,2},{2,3},{2,3},{1,4},{4,5},{4,6}}. Then by using Macaulay2 [9] one can
check that a sequence of parity edge-binomials in the following order
$g_{12},g_{23},g_{13},g_{45},g_{46},g_{14}$ form a $d$-sequence. Moreover, one
can observe that $G$ is not isomorphic to any graphs mentioned in Lemma 2.4.

######  Question 2.1.

Classify all finite simple graphs such that their parity edge-binomials form a
$d$-sequence.

### Conclusion

###### Proof of Theorem 0.1.

The first statement follows from Theorem 2.1 and Lemma 2.4. The second
statement follows from Theorem 2.2. ∎

##  3\. Regularity of (parity) binomial edge ideal of d-sequence graphs

In this section, we obtain the regularity of the parity binomial edge ideal of
$d$-sequence unicyclic graphs. We obtained the regularity of the product of
the binomial edge ideal of a complete graph and the parity binomial edge ideal
of a disjoint union of paths. The regularity of binomial edge ideals of
unicyclic graphs is studied in [28]. For the sake of completeness, first, we
provided the regularity of the binomial edge ideals of connected unicyclic
graphs whose edge-binomials form a $d$-sequence.

######  Lemma 3.1.

Let $H\in\mathcal{T}_{m}$ be a tree. Consider a unicyclic graph $G$
constructed by adding

  1. (a)

an edge between a pendant vertex of $H$ and the center of $H$, then

$\operatorname{reg}{S}/{J_{G}}=2+\sum_{i=1}^{m}{s_{(i)}}-1.$

  2. (b)

an edge between the center of $H$ and an internal vertex of $H$, then one has

$\operatorname{reg}{S}/{J_{G}}=\begin{cases}1+\sum_{i=1}^{m}{s_{(i)}}&\text{if
girth}(G)=3\\\ \sum_{i=1}^{m}{s_{(i)}}&\text{if girth}(G)\geq 4.\end{cases}$

###### Proof.

Proof of (a): If $G$ has a girth greater than or equal to $4$, then the
statement follows from [28, Corollary 4.10] and Remark 1.4. If $G$ has girth
equal to $3$, then $G$ is a flower-free connected graph. Thus the statement
follows from Remark 1.6. ∎

In [21] Kumar obtained an upper bound for parity binomial edge ideals of a
non-bipartite graph $G$ such that $G\setminus e$ is a bipartite graph, where
$e\in E(G)$. We compute the regularity of parity binomial edge ideals of
connected unicyclic graphs whose parity edge-binomials form a $d$-sequence. By
Remark 1.1, one can consider only unicyclic graphs with odd girth.

######  Lemma 3.2.

Let $H\in\mathcal{T}_{m}$ be a tree. Let $G$ be a unicyclic graph with an odd
girth. If $G$ is obtained by adding an edge between a pendant vertex of $H$
and the center of $H$. Then one has

$\operatorname{reg}{S}/{\mathcal{I}_{G}}=2+\sum_{i=1}^{m}{s_{(i)}}.$

###### Proof.

Let $e=\\{u,v\\}$ be an edge of the cycle in $G$ such that
$\deg_{G}(u)=\deg_{G}(v)=2$. Consider the following short exact sequence

$0\longrightarrow\frac{S}{(\mathcal{I}_{G\setminus
e}):g_{e}}(-2)\longrightarrow\frac{S}{\mathcal{I}_{G\setminus
e}}\longrightarrow\frac{S}{\mathcal{I}_{G}}\longrightarrow 0.$ (3.1)

Clearly, $G\setminus e$ is a tree. From Remarks 1.2(c) and 1.5 it follows
that,

$\operatorname{reg}{S}/{((\mathcal{I}_{G\setminus
e}):g_{e})}=\operatorname{reg}{S}/{{J}_{(G\setminus
e)_{e}}}=\operatorname{reg}{S}/{J_{G\setminus
e}}=2+\sum_{i=1}^{m}{s_{(i)}}-1.$

Thus applying Lemma 1.1(c) to short exact sequence (3.1) gives
$\operatorname{reg}{S}/{\mathcal{I}_{G}}=2+\sum_{i=1}^{m}{s_{(i)}}.$ ∎

######  Lemma 3.3.

Let $H\in\mathcal{T}_{m}$ be a tree. Let $G$ be a unicyclic graph with an odd
girth. If $G$ is obtained by adding an edge between the center of $H$ and an
internal vertex of $H$, then

$\operatorname{reg}{S}/{\mathcal{I}_{G}}=2+\sum_{i=1}^{m}{s_{(i)}}-1.$

###### Proof.

Let $G$ have the girth greater than or equal to $4$ and $e=\\{u,v\\}$ be an
edge of the cycle in $G$ such that $\deg_{G}(u)=\deg_{G}(v)=2$. Then from
Remark 1.2(c) and [1, Lemma 3.1] it follows that

$\operatorname{reg}{S}/{((\mathcal{I}_{G\setminus
e}):g_{e})}=\operatorname{reg}{S}/{{J}_{(G\setminus
e)_{e}}}=\operatorname{reg}{S}/{J_{G\setminus e}}=\sum_{i=1}^{m}{s_{(i)}}.$

Therefore, applying Lemma 1.1(c) to exact sequence (3.1) yields
$\operatorname{reg}{S}/{\mathcal{I}_{G}}=2+\sum_{i=1}^{m}{s_{(i)}}-1.$

If $G$ have girth equal to $3$ then choose an edge of the cycle in $G$,
$e=\\{u,v\\}$ such that $\deg_{G}(u)=m+1$ and $\deg_{G}(v)=2$. $G\setminus e$
is a graph in $\mathcal{T}_{m+1}$, thus from Remark 1.1 and Remark 1.5 it
follows that

$\operatorname{reg}{S}/{\mathcal{I}_{G\setminus
e}}=\operatorname{reg}{S}/{J_{G\setminus e}}=1+\sum_{i=1}^{m}{s_{(i)}}.$

By Remark 1.2(c) we get $J_{(G\setminus e)_{e}}$ is a graph obtained by gluing
$K_{m}$ and gluing paths at free vertices. By Remark 1.4 one obtains

$\operatorname{reg}{S}/{((\mathcal{I}_{G\setminus
e}):g_{e})}=\sum_{i=1}^{m}{s_{(i)}}.$

Therefore, applying Lemma 1.1(c) to exact sequence (3.1) yields
$\operatorname{reg}{S}/{\mathcal{I}_{G}}=1+\sum_{i=1}^{m}{s_{(i)}}.$ Hence we
conclude the proof. ∎

Next, we obtain the regularity of the product of parity binomial edge of
disjoint union of paths and binomial edge ideal a complete graph.

######  Theorem 3.1.

Let $K_{m}$ be a complete graph on $[m]$. Let $H=\\{P_{1}^{\prime},\ldots
P_{t}^{\prime}\\}$ be a disjoint union of paths such that

  1. (1)

for any $i$, if $K_{m}\cap P_{i}^{\prime}\neq\emptyset$, then $V(K_{m})\cap
V(P_{i}^{\prime})=v_{i}$, for some $v_{i}$ which is free vertex in
$P_{i}^{\prime}$

  2. (2)

$V(K_{m})\cap V(P_{i}^{\prime})\cap V(P_{j}^{\prime})=\emptyset$, for all
distinct $i$ and $j$.

Let $n$ be the number of edges in $H$. Then, for any $n\geq 1$ and for any
$m\geq 2$, we have

$\operatorname{reg}\frac{S}{\mathcal{I}_{H}J_{K_{m}}}=2+n.$

###### Proof.

Since $H$ is a bipartite graph, one can write $V(H)=L_{1}\sqcup L_{2}$ such
that $L_{2}\cap V(K_{m})=\emptyset$. Take $V_{1}=L_{1}\cup V(K_{m})$ and
$V_{2}=L_{2}$ as partition of $V(H)\cup V(K_{m})$. From Remark 1.1 it follows
that $\phi(\mathcal{I}_{H}J_{K_{m}})=J_{H}J_{K_{m}}$. As desired, the
statement follows from [1, Theorem 3.1]. ∎

##  4\. Regularity of powers of binomial edge ideals

In this section, we obtain precise expressions for the regularity of powers of
the binomial edge ideal of $d$-sequence unicyclic graphs. The regularity of
powers of the binomial edge ideals of cycle graphs is computed in [16]. We
recall the statement below for the sake of completeness.

######  Remark 4.1.

[16, Theorem 3.6] Let $G=C_{n}$ be a cycle. Then, for all $s\geq 1$,
$\operatorname{reg}(S/J_{G}^{s})=2s+n-4$.

Next, we compute the regularity of powers of binomial edge ideals of connected
unicyclic graphs whose edge-binomial forms a $d$-sequence.

######  Theorem 4.1.

Let $H\in\mathcal{T}_{m}$ be a tree. Let $G$ be a unicyclic graph obtained by
adding an edge between a pendant vertex of $H$ and the center of $H$. Let
$f_{1},\ldots,f_{n}$ be a $d-$sequence edge-binomials of $G$ with $f_{0}=0\in
S$. Then, for any $i=0,1,\ldots,n-1$, we have

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1,\text{
for all }s\geq 1.$

In particular,
$\operatorname{reg}{S}/{J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$, for all
$s\geq 1$.

###### Proof.

We consider a sequence $f_{1},\ldots,f_{n}$ of edge-binomials of $G$ as the
same sequence as in the Theorem 2.1(a). The proof is by induction on $s$. For
$s=1$, the assertion follows from Lemma 3.1(a). We can assume that the
assertion holds for $s-1$. The statement for $s$ is proved by descending
induction on $i$. For $i=n-1$, the statement is verified independently in
Lemma 4.1. Assume that the assertion holds for $i+1$. To prove the statement
for $s$ and $i$, consider the following short exact sequence

$\begin{split}0\longrightarrow\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}:f_{i+1}}(-2)&\longrightarrow\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}}\\\
&\longrightarrow\frac{S}{(f_{1},\ldots,f_{i+1})+J_{G}^{s}}\longrightarrow
0.\end{split}$ (4.1)

By the induction hypothesis on $i$, it follows that
$\operatorname{reg}{S}/{(f_{1},\ldots,f_{i+1})+J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$.
From Lemma 1.3 it follows that
$((f_{1},\ldots,f_{i})+J_{G}^{s}):f_{i+1}=((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}^{s-1}$.
From Remark 1.2(a) one has $((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}^{s-1}=I+J$,
where $I=(f_{1},\ldots,f_{i})+J_{G}^{s-1}$ and $J=J_{K_{n}}$ for some $n<m$
($J_{K_{n}}=\langle f_{kl}\mid k,l\in N_{H_{i}}(\alpha(f_{i+1}))\text{ or
}k,l\in N_{H_{i}}(\beta(f_{i+1}))$). Consider the following two cases:

Case 1: If $((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}^{s-1}=I$. From the induction
hypothesis on $s$, it follows that
$\operatorname{reg}{S}/{I}=2(s-1)+\sum_{j=1}^{m+1}{s_{(j)}}-1$. Applying Lemma
1.1(b) to the short exact sequence (4.1) yields

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m+1}{s_{(j)}}-1.$

Case 2: If $((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}^{s-1}=I+J$. Consider the
short exact sequence

$0\longrightarrow\frac{S}{I\cap
J}\longrightarrow\frac{S}{I}\oplus\frac{S}{J}\longrightarrow\frac{S}{I+J}\longrightarrow
0.$ (4.2)

Since $J$ is a complete graph, from Remark 1.6 it follows that
$\operatorname{reg}{S}/{J}=1$. We claim that $I\cap J=J*\langle
x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$, where $\mathcal{P}$ is the
induced subgraph of $G$ with vertex set $G\setminus\\{k_{0}\\}$ and $k_{0}$
denotes the center of $H$.

The proof of the claim is similar to the proof of Claim 4.1 of [1, Theorem
4.1]. From Theorem [1, Theorem 3.1] it follows that
$\operatorname{reg}S/(I\cap J)=2+\sum_{j=1}^{m}{s_{(j)}}$. Then applying Lemma
1.1(c) to the short exact sequence (4.2) yields
$\operatorname{reg}{{S}/{(I+J)}}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}-1$, for all
$s\geq 2$. Again applying Lemma 1.1(b) to exact sequence (4.1) yields

$\operatorname{reg}{\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1,\text{
for all }s\geq 2.$

∎

######  Lemma 4.1.

With the hypothesis as in Theorem 4.1, and for all $s>1$, we have

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1})+f_{n}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1.$

###### Proof.

From Lemma 3.1(a), it follows that
$\operatorname{reg}{S}/{J_{G}}=2+\sum_{j=1}^{m}s_{(j)}-1$. By using Theorem
2.1(a) and Remark 1.5, we obtain that
$\operatorname{reg}{S}/{(f_{1},\ldots,f_{n-1})}=2+\sum_{j=1}^{m}s_{(j)}$.
Therefore, applying Lemma 1.1(a) to the short exact sequence

$0\longrightarrow\frac{S}{(f_{1},\ldots,f_{n-1}):f_{n}}(-2)\longrightarrow\frac{S}{(f_{1},\ldots,f_{n-1})}\longrightarrow\frac{S}{(f_{1},\ldots,f_{n})}\longrightarrow
0,$ (4.3)

yields

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1}):f_{n}}=\sum_{j=1}^{m}s_{(j)}.$

Since $f_{1},\ldots,f_{n}$ form a $d$-sequence, one has
$(f_{1},\ldots,f_{n-1}):f_{n}^{s}=(f_{1},\ldots,f_{n-1}):f_{n}$. Now, look at
the following exact sequence

$\begin{split}0\longrightarrow\frac{S}{(f_{1},\ldots,f_{n-1}):f_{n}^{s}}(-2s)&\longrightarrow\frac{S}{(f_{1},\ldots,f_{n-1})}\\\
&\longrightarrow\frac{S}{(f_{1},\ldots,f_{n-1})+f_{n}^{s}}\longrightarrow
0.\end{split}$ (4.4)

Again, applying Lemma 1.1(c) to the short exact sequence (4.4) one can obtain
the desired result. ∎

######  Remark 4.2.

If $m=2$ in Theorem 4.1 then $G$ is a balloon graph. Therefore, we obtained
[16, Remark 3.15] that $\operatorname{reg}S/J_{G}^{s}=2s+n-4$, for all $s\geq
1$.

######  Theorem 4.2.

Let $H\in\mathcal{T}_{m}$ be a tree. Let $G$ be a unicyclic graph obtained by
adding an edge between the center of $H$ and an internal vertex of $H$. Let
$f_{1},\ldots f_{n}$ be a $d$-sequence edge-binomials of $G$ with $f_{0}=0\in
S$. Then, for any $i=0,1,\ldots,n-1$, we have

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1,\text{
for all }s\geq 2.$

In particular,
$\operatorname{reg}{S}/{J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$, for all
$s\geq 2$.

Take $f_{1},\ldots,f_{n}$ to be a sequence of edge-binomials in the same order
as in Theorem 2.1(c). To complete the proof of Theorem 4.2, we need to prove
the following lemmas.

######  Lemma 4.2.

With the hypothesis as in Theorem 4.2, and for any $i=0,1,\ldots,n-1$, we have

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{2}}=4+\sum_{j=1}^{m}{s_{(j)}}-1.$

###### Proof.

From Lemma 3.1(b) one can obtain the regularity of the module
$S/(f_{1},\ldots,f_{n})$. By Theorem 2.1(c) the binomial edge ideal $\langle
f_{1},\ldots,f_{n-1}\rangle$ is associated to a graph in $\mathcal{T}_{m}$.
Thus, by Remark 1.5 it follows that
$\operatorname{reg}S/(f_{1},\ldots,f_{n-1})=2+\sum_{j=1}^{m}{s_{(j)}}$.
Therefore, by applying Lemma 1.1(a) to the short exact sequence (4.3) yields

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1}):f_{n}}=\sum_{j=1}^{m}{s_{(j)}}.$

Now, the proof is by descending induction on $i$. For $i=n-1$, consider the
short exact sequence (4.4) with $s=2$. Since the sequence $f_{1},\ldots,f_{n}$
is a $d$-sequence, one has

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1}):f_{n}^{2}}(-4)=4+\sum_{j=1}^{m}{s_{(j)}}-1>\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1})}.$

Applying Lemma 1.1(c) to the short exact sequence (4.4) for $s=2$ yields that

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1})+J_{G}^{2}}=\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1})+f_{n}^{2}}=4+\sum_{j=1}^{m}{s_{(j)}}-1.$

Next, assume that the assertion holds for $i+1$. Consider the short exact
sequence (4.1) for $s=2$. From Lemma 1.3, it follows that
$((f_{1},\ldots,f_{i})+J_{G}^{2}):f_{i+1}=((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}$.
Following the same approach as in proof of Theorem 4.2 (Case $2$), we get
$\operatorname{reg}{S}/{((f_{1},\ldots,f_{i})+J_{G}^{2}):f_{i+1}}=4+\sum_{j=1}^{m}{s_{(j)}}-1$.
By the induction hypothesis on $i$ that
$\operatorname{reg}{S}/{(f_{1},\ldots,f_{i+1})+J_{G}^{2}}=4+\sum_{j=1}^{m}{s_{(j)}}-1$.
Applying Lemma 1.1(b) to exact sequence (4.1) for $s=2$, yields

$\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{2}}=4+\sum_{j=1}^{m}{s_{(j)}}-1,$

as desired. ∎

######  Lemma 4.3.

With the hypothesis as in Theorem 4.2, and for all $s>2$, we have

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{n-1})+f_{n}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1.$

###### Proof.

Since $f_{1},\ldots,f_{n}$ form a $d$-sequence, one has
$(f_{1},\ldots,f_{n-1}):f_{n}^{s}=(f_{1},\ldots,f_{n-1}):f_{n}$. From Lemma
4.2, we know the regularity of modules ${S}/{(f_{1},\ldots,f_{n-1}):f_{n}}$
and ${S}/{(f_{1},\ldots,f_{n-1})}$. Then applying Lemma 1.1(c) to the short
exact sequence (4.4) one yields
$\operatorname{reg}{S}/{((f_{1},\ldots,f_{n-1})+f_{n}^{s})}=4+\sum_{j=1}^{m}{s_{(j)}}-1$.
∎

###### Proof of Theorem 4.2.

We use induction on $s$ to prove the statement. For $s=2$, the statement
follows from Lemma 4.2. Assume that the assertion holds for $s-1$. The
statement for $s$ is proved by descending induction on $i$. For $i=n-1$, the
statement is verified separately in Lemma 4.3. Next, assume that the assertion
holds for $i+1$. Consider the short exact sequence (4.1). By the induction
hypothesis on $i$ it follows that
${S}/{((f_{1},\ldots,f_{i+1})+J_{G}^{s})}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$. By
the similar argument as in Theorem 4.1 we get
$((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}^{s-1}=(f_{1},\ldots,f_{i})+J_{G}^{s-1}+J_{K_{n}}$,
for some $n<m$. Set $I=(f_{1},\ldots,f_{i})+J_{G}^{s-1}$ and $J=J_{K_{n}}$.
Consider the following two cases:

Case 1: If $((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}^{s-1}=I$, then from induction
hypothesis it follows that
$\operatorname{reg}{S}/{I}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$. Now, from Lemma
1.1(b) it follows that

$\operatorname{reg}\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1.$

Case 2: If $((f_{1},\ldots,f_{i}):f_{i+1})+J_{G}^{s-1}=I+J$. Consider the
short exact sequence (4.2). The regularity of $I$ is known from the induction
hypothesis. From Remark 1.6 it follows that $\operatorname{reg}{S}/{J}=1$,
since $J$ is a complete graph. We claim that $I\cap J=J*\langle
x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$, where $\mathcal{P}$ denotes the
induced subgraph of $G$ with vertex set $G\setminus\\{k_{0}\\}$ and $k_{0}$ is
the center of $G$.

The proof of the claim is similar to the proof of Claim 4.1 of [1, Theorem
4.1]. It follows from Theorem [1, Theorem 3.1] that
$\operatorname{reg}S/(I\cap J)=2+\sum_{j=1}^{m}{s_{(j)}}$. By applying Lemma
1.1(c) to the short exact sequence (4.2) we obtain that
$\operatorname{reg}{S}/({I+J})=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}-1$, for all
$s\geq 3$, i.e.

$\operatorname{reg}{\frac{S}{(f_{1},\ldots,f_{i})+J_{G}^{s}:f_{i+1}}}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}-1,$

for all $s\geq 3$. Thus by applying Lemma 1.1(b) to the short exact sequence
(4.1) one can yields
$\operatorname{reg}{S}/{((f_{1},\ldots,f_{i})+J_{G}^{s})}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$,
for $s\geq 3$. ∎

######  Remark 4.3.

Suppose $G$ in Theorems 4.1 and 4.2 is isomorphic to a connected graph on
$[n]$, which is obtained by adding an edge between two vertices of a path
$P_{n}$. If the girth of $G$ is at least $4$, then we obtain [29, Theorem
4.4.] that $\operatorname{reg}S/J_{G}^{s}=2s+n-4$ for all $s\geq 2$.

##  5\. Regularity of powers of parity binomial edge ideals

In this section, we obtain the exact bound for the regularity of powers of the
parity binomial edge ideal of $d$-sequence unicyclic graphs. As mentioned
earlier, we consider only unicyclic graphs with odd girth. Kumar characterized
graphs whose parity binomial edge ideals are complete intersections; namely,
$\mathcal{I}_{G}$ is a complete intersection if and only if G is a path or odd
cycle (cf. [21]). First, we state the regularity of powers of parity binomial
edge ideals of cycles.

######  Remark 5.1.

Let $G$ be an odd cycle on $[n]$. Then one has
$\operatorname{reg}S/\mathcal{I}_{G}^{s}=2s+n-2$, for all $s\geq 1$.

###### Proof.

The statement follows from [2, Lemma 4.4]. ∎

Next, we compute the regularity of powers of parity binomial edge ideals of
connected unicyclic graphs whose parity edge-binomials form a $d$-sequence.

######  Theorem 5.1.

Let $H\in\mathcal{T}_{m}$ be a tree. Let $G$ be a unicyclic graph obtained by
adding an edge between the pendant vertex of $H$ and the center of $H$ such
that the girth of $G$ is odd. Let $g_{1},\ldots,g_{n}$ be a $d$-sequence
parity edge-binomials of $G$ with $g_{0}=0\in S$. Then, for any
$i=0,1,\ldots,n-1$, for all $s\geq 1$, we have

$\operatorname{reg}\frac{S}{(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}.$

In particular,
$\operatorname{reg}{S}/{\mathcal{I}_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}$, for
all $s\geq 1$.

###### Proof.

We take parity edge-binomials of $G$ in the same order as edge-binomials of
$G$ taken in Theorem 2.1(a). The proof is by induction on $s$. The statement
holds for $s=1$, by Lemma 3.2. Assume that the statement holds for $s-1$. We
prove the statement for $s$ by descending induction on $i$. For $i=n-1$, the
statement holds, by Lemma 5.1. Assume that the assertion holds for $s$ and
$i+1$. We need to prove the statement for $s$ and $i$. Consider the following
short exact sequence

$\begin{split}0\longrightarrow\frac{S}{(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s}:g_{i+1}}(-2)&\longrightarrow\frac{S}{(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s}}\\\
&\longrightarrow\frac{S}{(g_{1},\ldots,g_{i+1})+\mathcal{I}_{G}^{s}}\longrightarrow
0.\end{split}$ (5.1)

By the induction hypothesis on $i$, it follows that
$\operatorname{reg}{S}/{(g_{1},\ldots,g_{i+1})+\mathcal{I}_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}$.
From Lemma 1.3 and Remark 1.3 it follows that
$((g_{1},\ldots,g_{i}):g_{i+1})+\mathcal{I}_{G}^{s-1}=(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s-1}+J_{K_{n}}$,
for some $n<m$ (where $J_{K_{n}}=\langle f_{kl}\mid k,l\in
N_{H_{i}}(\alpha(g_{i+1}))\text{ or }k,l\in N_{H_{i}}(\beta(g_{i+1}))$). Set
$I=(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s-1}$ and $J=J_{K_{n}}$. Consider the
following two cases:

Case 1: If $((g_{1},\ldots,g_{i}):g_{i+1})+\mathcal{I}_{G}^{s-1}=I$. From the
induction hypothesis on $s$, it follows that
$\operatorname{reg}{S}/{I}=2(s-1)+\sum_{j=1}^{m+1}{s_{(j)}}$. Applying Lemma
1.1(b) to the short exact sequence (4.1) yields

$\operatorname{reg}\frac{S}{(g_{1},\ldots,g_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m+1}{s_{(j)}}.$

Case 2: For $((g_{1},\ldots,g_{i}):g_{i+1})+\mathcal{I}_{G}^{s-1}=I+J$.
Consider the short exact sequence (4.2). Using the induction hypothesis on
$s$, regularity of $I$ follows. The regularity of $J$ follows from Remark 1.6.
To find the regularity of $I\cap J$ we need to prove the following claim.

######  Claim 5.1.

$I\cap J=J*\langle x_{k_{0}},y_{k_{0}},\mathcal{I}_{\mathcal{P}}\rangle$,
where $\mathcal{P}$ denotes the induced subgraph of $G$ on vertex set
$V(G)\setminus k_{0}$ and $k_{0}$ denote the center of $H$. The proof of the
claim is similar to the proof of a claim $4.1$ in [1, Theorem 4.1]. In the
proof instead of edge-binomials $f_{lk_{0}}$ and $f_{kk_{0}}$ we take parity
edge-binomials $g_{lk_{0}}$ and $g_{kk_{0}}$, rest of the proof is similar.

Now, from Theorem 3.1 it follows that $\operatorname{reg}S/(I\cap
J)=2+\sum_{j=1}^{m}{s_{(j)}}$. Then applying Lemma 1.1(c) to the short exact
sequence (4.2) yields
$\operatorname{reg}{{S}/{(I+J)}}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}$, for all
$s>2$. Again applying Lemma 1.1(b) to exact sequence (4.1) one yields

$\operatorname{reg}{\frac{S}{(g_{1},\ldots,g_{i})+I_{G}^{s}}}=2s+\sum_{j=1}^{m}{s_{(j)}},\text{
for all }s>2.$

For $s=2$, first we compute the regularity of
${S}/{(\mathcal{I}_{G}+J_{K_{n}})}$ for some $n<m$ (where $J_{K_{n}}=\langle
f_{kl}\mid k,l\in N_{L_{1}}(\alpha(g_{i+1}))\text{ or }k,l\in
N_{L_{1}}(\beta(g_{i+1}))$). Assume that
$E(G)=E(\mathcal{T}_{m})\cup\\{k_{0},p_{1_{s(1)+1}}\\}$ be an edge set of $G$.
Observe that induced subgraph on the vertex set
$V(G)\setminus\\{p_{1_{1}},p_{1_{2}}\\}$ is a tree. We partition vertex set of
$V(G)\setminus\\{p_{1_{1}},p_{1_{2}}\\}$ into $V_{1}\sqcup V_{2}$. Without
loss of generality one assume that the vertex $k_{0}\in V_{1}$, then we add
vertices $p_{1_{1}}$ and $p_{1_{2}}$ to $V_{2}$. Let
$e=\\{p_{1_{1}},p_{1_{2}}\\}$ be an edge of $G$, note that
$\phi(\mathcal{I}_{G}+J_{K_{n}})=J_{G\setminus e}+g_{e}$. Consider the
following short exact sequence

$0\longrightarrow\frac{S}{J_{G\setminus
e}:g_{e}}(-2)\longrightarrow\frac{S}{J_{G\setminus
e}}\longrightarrow\frac{S}{(J_{G\setminus e}+g_{e})}\longrightarrow 0,$ (5.2)

$G\setminus e$ is a tree that belongs to a class of $\mathcal{T}_{m}$ graph.
From Remark 1.5 it follows that $\operatorname{reg}S/{J_{G\setminus
e}}=2+\sum_{j=1}^{m}{s_{(j)}}$. From a part of the proof of [20, Lemma 3.3]
one has $J_{G\setminus e}:g_{e}=J_{(G\setminus e)_{e}}$ and here
$J_{(G\setminus e)_{e}}=J_{G\setminus e}$. Therefore, applying Lemma 1.1(c) to
(5.2) one yields

$\operatorname{reg}\frac{S}{(J_{G\setminus
e}+g_{e})}=2+\sum_{j=1}^{m}{s_{(j)}}-1.$

From Lemma 1.3 it follows that
$(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{2}:g_{i+1}=(g_{1},\ldots,g_{i}):g_{i+1}+\mathcal{I}_{G}={\mathcal{I}_{G}+J_{K_{n}}}.$
Again applying Lemma 1.1(b) to exact sequence (4.1) for $s=2$ one yields

$\operatorname{reg}{\frac{S}{(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{2}}}=4+\sum_{j=1}^{m}{s_{(j)}}.$

∎

######  Lemma 5.1.

Considering the hypothesis of Theorem 5.1, and for all $s>1$, we have

$\operatorname{reg}\frac{S}{(g_{1},\ldots,g_{n-1})+g_{n}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}.$

###### Proof.

Let $e$ be a corresponds to a parity edge-binomial $g_{n}$. From Lemma 1.2(c)
it follows that $\operatorname{reg}S/(\mathcal{I}_{G\setminus
e}:g_{n})=\operatorname{reg}S/J_{(G\setminus
e)_{e}}=2+\sum_{j=1}^{m}{s_{(j)}}-1$. By using Remark 1.1 and Remark 1.5, one
can obtain that
$\operatorname{reg}{S}/{(g_{1},\ldots,g_{n-1})}=2+\sum_{j=1}^{m}s_{(j)}$.
Applying Lemma 1.1(c) to the short exact sequence (4.4) one can obtain desired
result. ∎

######  Remark 5.2.

Note that the balloon graphs are a particular case of graphs discussed in
Theorem 5.1. Hence we obtained the result [29, Theorem 5.7] as a particular
case.

######  Theorem 5.2.

Let $H\in\mathcal{T}_{m}$ be a tree. Let $G$ be a unicyclic graph on $[n]$
with odd girth, which is obtained by adding an edge between the center of $H$
and an internal vertex of $H$. Let $g_{1},\ldots g_{n}$ be a $d$-sequence of
parity edge-binomials of $G$ with $g_{0}=0\in S$. Then, for any
$i=0,1,\ldots,n-1$, and for all $s\geq 1$, we have

$\operatorname{reg}\frac{S}{(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1.$

In particular,
$\operatorname{reg}{S}/{\mathcal{I}_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$,
for all $s\geq 1$.

###### Proof.

We consider parity edge-binomials of $G$ in the same order as edge-binomials
of $G$ in Theorem 2.1(c). Proof is by induction on $s$. For $s=1$ the
statement is proved in Lemma 3.3. Assume that the statement is true for $s-1$.
The statement for $s$ is proved by descending induction on $i$. For $i=n-1$,
the statement holds, by Lemma 5.2. Assume that the assertion holds for $i+1$.
It remains to show that the statement holds for $s$ and $i$. Consider the
short exact sequence (5.1). The statement holds for
${S}/{((g_{1},\ldots,g_{i+1})+\mathcal{I}_{G}^{s})}$ by the reverse induction
on $i$. By a similar argument as in Theorem 5.1 that we take
$I=(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s-1}$ and $J=J_{K_{n}}$. Consider the
following two cases:

Case 1: If $((g_{1},\ldots,g_{i}):g_{i+1})+\mathcal{I}_{G}^{s-1}=I$, then from
induction hypothesis on $s$ it follows that
$\operatorname{reg}{S}/{I}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}-1$. Now, from Lemma
1.1(b) it follows that

$\operatorname{reg}\frac{S}{(g_{1},\ldots,g_{i})+\mathcal{I}_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1.$

Case 2: If $((g_{1},\ldots,g_{i}):g_{i+1})+\mathcal{I}_{G}^{s-1}=I+J$.
Consider the short exact sequence (4.2). To compute the regularity of $I\cap
J$, first, we claim that $I\cap J=J*\langle
x_{k_{0}},y_{k_{0}},\mathcal{I}_{\mathcal{P}}\rangle$, where $\mathcal{P}$
denotes the induced subgraph of $G$ on vertex set $V(G)\setminus k_{0}$ and
$k_{0}$ denote the center of $H$.

The proof of the claim is similar to the proof of Claim 5.1 of Theorem 5.1. It
follows from Theorem 3.1 that $\operatorname{reg}S/(I\cap
J)=2+\sum_{j=1}^{m}{s_{(j)}}$. By applying Lemma 1.1(c) to the short exact
sequence (4.2) we obtain that
$\operatorname{reg}{S}/({I+J})=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}-1$, for all
$s\geq 2$, i.e.

$\operatorname{reg}{\frac{S}{(g_{1},\ldots,g_{i})+J_{G}^{s}:g_{i+1}}}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}-1,$

for all $s\geq 2$. Thus by applying Lemma 1.1(b) to the short exact sequence
(5.1) one yields

$\operatorname{reg}\frac{S}{((g_{1},\ldots,g_{i})+J_{G}^{s})}=2s++\sum_{j=1}^{m}{s_{(j)}}-1,\text{
for all }s\geq 2.$

∎

######  Lemma 5.2.

Under the assumption in Theorem 5.2, and for all $s>2$, we have

$\operatorname{reg}\frac{S}{(g_{1},\ldots,g_{n-1})+g_{n}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}-1.$

###### Proof.

Consider the short exact sequence (4.3). From Lemma 3.3 one has the regularity
of the module $S/\mathcal{I}_{G}$. The graph associated to parity edge-
binomials $g_{1},\ldots,g_{n-1}$ is a tree. From Remark 1.6 it follows that
$\operatorname{reg}S/(g_{1},\ldots,g_{n-1})=2+\sum_{j=1}^{m}{s_{(j)}}$.
Applying Lemma 1.1(a) to exact sequence (4.3) it follows that
$\operatorname{reg}S/((g_{1},\ldots,g_{n-1}):g_{n})=\sum_{j=1}^{m}{s_{(j)}}$.
Since $g_{1},\ldots,g_{n}$ form a $d$-sequence, one has
$(g_{1},\ldots,g_{n-1}):g_{n}^{s}=(g_{1},\ldots,g_{n-1}):g_{n}$. By applying
Lemma 1.1(c) to the short exact sequence (4.4) yields
$\operatorname{reg}{S}/{((g_{1},\ldots,g_{n-1})+g_{n}^{s})}=2s+\sum_{j=1}^{m}{s_{(j)}}-1$,
for all $s\geq 2$, as desired. ∎

######  Remark 5.3.

Note that the graphs were obtained by adding an edge $e$ between two internal
vertices of a path $P_{n}$ such that the girth of $G$ is odd are a particular
case of graphs discussed in Theorem 5.1. Hence we obtained the result [29,
Theorem 5.9].

### Conclusion

###### Proof of Theorem 0.2.

The statement $(i)$ follows from Theorems 4.1 and 5.1, and Remark 2.2(a), and
the statement $(ii)$ follows from Theorems 4.2 and 5.2, and Remark 2.2(b). ∎

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