# Monomial ideals with regular quotients and some edge rings

Dancheng Lu, Hao Zhou∗ School of Mathematical Sciences, Soochow University,
215006 Suzhou, P.R.China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce and study monomial ideals with regular quotients, which can be
seen as an extension of monomial ideals with linear quotients. Based on these
investigations, we are able to calculate the Betti numbers of toric ideals
belonging to various classes of edge rings.

###### Key words and phrases:

Betti numbers, toric ring, regular quotients, bipartite graphs, initial ideals

###### 2010 Mathematics Subject Classification:

Primary 13D45, 13D05; Secondary 13D02.

*Corresponding author.

## Introduction

Let $\mathbb{K}$ be a field, $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ the polynomial
ring in $n$ variables. The concept of (monomial) ideals with linear quotients
was introduced by Herzog and Takayama [11]. This notion is useful in many
ways, specifically, it is an effective method for proving that an ideal has a
linear resolution and for constructing a minimal graded free resolution for
some ideals, see e.g. [12] and [2]. In this paper, we embark on a study of
monomial ideals with regular quotients, which can be viewed as a natural
extension of monomial ideals possessing linear quotients. Let $I\subset R$ be
a monomial ideal. We say $I$ has regular quotients with respect to the order
$f_{1},\ldots,f_{m}$ of minimal generators of $I$ if the quotient ideal
$(f_{1},\ldots,f_{i-1}):f_{i}$ is generated by a regular sequence on $R$ for
all $i=2,\ldots,m.$ (notice that this property depends on the order of the
generators).

Let $I$ be a monomial ideal with regular quotients. In the most general
scenario, we provide upper bounds for the regularity and projective dimension
of $I$. Furthermore, we explore the conditions under which these upper bounds
are attained, see Corollary 2.2. To study the Betti numbers of such ideals, we
specifically examine the ideals where each quotient is generated by a regular
sequence of the same degree. Let $I=(f_{1},\ldots,f_{m})$ be a monomial ideal
such that the quotient $L_{k}:=(f_{1},\ldots,f_{k-1}):f_{k}$ is generated by a
regular sequence of length $r_{k}$ in which every element has degree $a_{k}$
for $k=2,\ldots,m$. Denote $\deg(f_{k})$ by $d_{k}$ for $k=1,\ldots,m$. By the
construction of mapping cones, we obtain the following upper bound of
$\beta_{i,j}(I)$:

$\beta_{i,j}(I)\leq\beta_{i,j}(R/L_{m}[-d_{m}])+\cdots+\beta_{i,j}(R/L_{2}[-d_{2}])+\beta_{i,j}(R[-d_{1}]).$

Note that the right-hand side represents a number that is determined by the
values of $d_{i}$, $r_{i}$ and $a_{i}$. If $a_{2}=\cdots=a_{m}=1$, i.e. $I$
has linear quotients, the upper bound given above is actually the true value
of $\beta_{i,j}(I)$, see [7, Lemma 2.1 and Proposition 2.2] or [18, Corollary
2.7]. In this paper, we show in Remark 2.9 that this is also the case if
$r_{2}=\cdots=r_{m}=1$. In general, we give a sufficient condition in terms of
a relation among $d_{i}$’s and $a_{i}$’s for which the upper upper is
achieved, see Theorem 2.7 and Corollary 2.10.

Let $G=(V,E)$ be a simple graph with vertex set $V=\\{x_{1},\ldots,x_{n}\\}$
and edge set $E=\\{e_{1},\ldots,e_{r}\\}$. The edge ring $\mathbb{K}[G]$ is
the toric ring $\mathbb{K}[x_{e}\colon\;e\in
E(G)]\subset\mathbb{K}[x_{1},\ldots,x_{n}]$. Here, $x_{e}=\prod_{x_{i}\in
e}x_{i}$ for all $e\in E(G)$. Let $\mathbb{K}[E]$ be the polynomial ring
$\mathbb{K}[e_{1},\ldots,e_{r}]$. Then there is exactly one ring homomorphism

$\phi:\mathbb{K}[E]\rightarrow\mathbb{K}[V]\quad{\mbox{ such that
}}~{}e_{i}\mapsto x_{e_{i}}~{}{i=1,\ldots,r}$

The kernel of the homomorphism map $\phi$ is called the toric ideal or the
defining ideal of $\mathbb{K}[G]$ or $G$, which is denoted by $I_{G}$. It
follows that $\mathbb{K}[G]\cong\mathbb{K}[E]/I_{G}$. In general, calculating
the Betti numbers or other invariants of a toric ideal for a specific type of
graphs is challenging. For recent advancements in this area, please refer to
[1], [4], [6] and [8], etc.

In this paper, we introduce two classes of simple graphs, for which the toric
ideals have an initial ideal with regular quotients. The first class of graphs
is a class of bipartite graphs, which we denote by $B_{\underline{\ell},h}$,
where $h\geq 2$ and
$\underline{\ell}=(\ell_{1},\ldots,\ell_{h})\in\mathbb{Z}_{\geq 1}^{h}.$ We
calculate the regularity and projective dimension of the edge ring of this
class of graphs, and in the case when $\ell_{2}=\cdots=\ell_{h}$, we also
compute the Betti numbers of the edge ring of these graphs. As a byproduct, we
demonstrate the existence of a bipartite graph $G$ such that
$\mathrm{reg}(I_{G})=r$ and $\mathrm{pdim}(I_{G})=p$, where $p,r$ are any
given integers with $p\geq 0$ and $r\geq 2$. We refer to [13] for the
background of this result. The second class of graphs is obtained by attaching
two odd cycles to the graph $B_{\underline{\ell},h}$ when
$\ell_{1}=\cdots=\ell_{h}$. For such graphs, we also compute the Betti numbers
of their edge rings. In all of these computations, we rely on a sufficient
condition that ensures the toric ideal and its initial ideal share the same
Betti numbers. We present this condition in Theorem 3.2, which is inspired by
the proof of [4, Corollary 3.6].

The paper is organized as follows. Section 1 provides a brief overview of
toric ideals of graphs and some homological invariants. In Section 2, we
derive a formula for the graded Betti numbers of a monomial ideal, assuming
that its successive colon ideals are generated by a regular sequence. Section
3 presents a sufficient condition for the graded Betti numbers of the toric
ideal of a simple graph to be equal to the graded Betti numbers of its initial
ideal. In Sections 4 and 5, we introduce two classes of graphs respectively
and compute the graded Betti numbers and other invariants of their toric
ideals, utilizing the results obtained in the preceding sections.

##  1\. Preliminaries

In this section, we provide a brief review of the notation and fundamental
facts that will be utilized later on.

###  1.1. Toric ideals of graphs

Let $G$ be a simple graph, i.e., a finite graph without loops and multiple
edges. Recall that a walk of $G$ of length $q$ is a subgraph $W$ of $G$ such
that
$E(W)=\\{\\{v_{0},v_{1}\\},\\{v_{1},v_{2}\\},\ldots,\\{v_{q-1},v_{q}\\}\\}$,
where $v_{0},v_{1},\ldots,v_{q}$ are vertices of $G$. An walk $W$ of $G$ is
even if $q$ is even, and it is closed if $v_{0}=v_{q}$. A cycle with edge set
$\\{\\{v_{0},v_{1}\\},\\{v_{1},v_{2}\\},\\{v_{q-1},v_{q}=v_{0}\\}\\}$ is a
special even closed walk where $v_{1},\ldots,v_{q}$ are pairwise distinct.

The generators of the toric ideal of $I_{G}$ are binomials which is tightly
related to even closed walks in $G$. Given an even closed walk $W$ of $G$ with

$E(W)=\\{\\{v_{0},v_{1}\\},\\{v_{1},v_{2}\\},\ldots,\\{v_{2q-2},v_{2q-1}\\},\\{v_{2q-1},v_{0}\\}\\}$

We associate $W$ with the binomial defined by

$f_{W}:=\prod\limits_{j=1}^{q}e_{2j-1}-\prod\limits_{j=1}^{q}e_{2j},$

where $e_{j}=\\{v_{j-1},v_{j}\\}$ for $1\leq j\leq 2q-1$ and
$e_{2q}=\\{v_{2q-1},v_{0}\\}$. A binomial $f=u-v\in I_{G}$ is called a
primitive binomial if there is no binomial $g=u^{\prime}-v^{\prime}\in I_{G}$
such that $u^{\prime}|u$ and $v^{\prime}|v$. An even closed walk $W$ of $G$ is
a primitive even closed walk if its associated binomial $f_{W}$ is a primitive
binomial in $I_{G}$.

In this paper, we use $\mathcal{PC}(G)$ for the set of primitive even closed
walks of $G$. Then $\mathcal{PC}(G)$ is a finite set, and the set

$\\{f_{W}\colon\;W\in\mathcal{PC}(G)\\}$

is the universal Gr$\ddot{\mathrm{o}}$bner base of $I_{G}$ by e.g. [3,
Proposition 5.19]. In particular, it is a Gr$\ddot{\mathrm{o}}$bner base of
$I_{G}$ with respect to any monomial order. The set of primitive even walks of
a graph $G$ was described in [16] explicitly.

######  Proposition 1.1.

([16, Lemma 3.2]) Let $G$ be a simple connected graph. Then a primitive even
closed walk $\Gamma$ of $G$ is one of the following:

  1. (1)

$\Gamma$ is an even cycle of $G$;

  2. (2)

$\Gamma$=$(C_{1},C_{2})$, where $C_{1}$ and $C_{2}$ are odd cycles of $G$
having exactly one common vertex;

  3. (3)

$\Gamma$=$(C_{1},\Gamma_{1},C_{2},\Gamma_{2})$, where $C_{1}$ and $C_{2}$ are
odd cycles of $G$ having no common vertex and where $\Gamma_{1}$ and
$\Gamma_{2}$ are walks of $G$ both of which combine a vertex $v_{1}$ of
$C_{1}$ and a vertex $v_{2}$ of $C_{2}$.

###  1.2. Homological invariants

Let $R:=\mathbb{K}[x_{1},\ldots,x_{n}]$ be the polynomial ring in variables
$x_{1},\ldots,x_{n}$, which is standard graded. For a finitely generated
graded $R-$module $M$, there exists the minimal graded free resolution of $M$
that has the form:

$0\rightarrow\underset{j\in\mathbb{Z}}{\bigoplus}R[-j]^{\beta_{p,j}(M)}\rightarrow\cdots\rightarrow\underset{j\in\mathbb{Z}}{\bigoplus}R[-j]^{\beta_{1,j}(M)}\rightarrow\underset{j\in\mathbb{Z}}{\bigoplus}R[-j]^{\beta_{0,j}(M)}\rightarrow
M\rightarrow 0.$

Here, $R[-j]$ is the cyclic free $R$-module generated in degree $j$ and
$\beta_{i,j}(M)={\rm{dim}}_{\mathbb{K}}\mathrm{Tor}_{i}^{R}(M,\mathbb{K})_{j}$
is called the $(i,j)$-th graded Betti number of $M$. A Betti number
$\beta_{i,i+j}(M)\neq 0$ is called an extremal Betti number of $M$ if
$\beta_{k,k+\ell}(M)=0$ for any pair $(k,\ell)\neq(i,j)$ with $k\geq i$ and
$\ell\geq j$. Many homological invariants of $M$ can be defined in terms of
its minimal graded free resolution. The Castelnuovo-Mumford regularity of $M$
is

$\mbox{reg}\,(M):={\mbox{max}}\,\\{j-i\colon\;\beta_{i,\,j}(M)\neq 0\\}.$

The projective dimension of $M$ is the length of the minimal graded free
resolution:

$\mbox{pdim}\,(M):=\mbox{max}\,\\{i\colon\;\ \beta_{i,\,j}(M)\neq 0\\}.$

The Hilbert series of $M$ is defined to the power series:

$HS(M,t):=\underset{i\in\mathbb{Z}}{\sum}[\mbox{dim}_{\mathbb{K}}(M_{i})]t^{i}.$

It can be computed from the (minimal) graded free resolution of $M$, see [17,
Theorem 16.2].

$HS(M,t)=\frac{\sum_{i,j}(-1)^{i}\beta_{i,\,j}(M)t^{j}}{(1-t)^{n}}.$

Moreover, there exists a unique polynomial $h_{M}(t)\in\mathbb{Z}[t]$ with
$h_{M}(1)\neq 0$, which is called the $h$-polynomial of $M$, such that
$HS(M,t)$ can be reduced to the following form by e.g. [19, Theorem 5.1.4]:

$HS(M,t)=\frac{h_{M}(t)}{(1-t)^{\mbox{dim}(M)}}.$

Here, $\mbox{dim}(M)$ denotes the Krull dimension of $M$. Fix a monomial order
$\prec$ on the ring $R$. If $I$ is a graded ideal, then the initial ideal of
$I$, denoted by ${\rm in}_{\prec}(I)$, is the monomial ideal generated by the
initial terms of the polynomials belonging to $I$. We need the next result in
following sections, see [15, Corollary 2.5].

######  Lemma 1.2.

Let $G$ be a simple graph and let $H$ be an induced subgraph of $G$. Then we
have $\beta_{i,j}(I_{H})\leq\beta_{i,j}(I_{G})$ for all $i,j\geq 0$.

##  2\. monomial ideals which have regular quotients

In this section, we present a generalization of linear quotients by
introducing monomial ideals with regular quotients. We also delve into their
homological aspects, including projective dimension and regularity, and
provide computations of their Betti numbers in certain cases.

Let $R$ denote the standard graded polynomial ring
$\mathbb{K}[x_{1},\dots,x_{n}].$ For a monomial $\alpha$, the support of
$\alpha$, denoted by $\mbox{supp}(\alpha)$, is the set of variables $x$ such
that $x$ divides $\alpha$. It is known that a sequence of monomials
$\alpha_{1},\ldots,\alpha_{r}$ in $R$ is regular if and only if they have
disjoint supports, namely,
$\mbox{supp}(\alpha_{i})\cap\mbox{supp}(\alpha_{j})=\emptyset$ for all $1\leq
i\neq j\leq r$. For the explicit definition of a regular sequence we refer to
[10, page 270].

######  Proposition 2.1.

Suppose the monomial ideal $I$ is minimally generated by $f_{1},\ldots,f_{m}$
such that for $k=2,\ldots,m$, $(f_{1},\ldots,f_{k-1}):f_{k}$ is generated by a
regular sequence $\alpha_{k,1},\ldots,\alpha_{k,r_{k}}$ with
$\deg(\alpha_{k,i})=a_{k,i}$, where $i=1,\ldots,r_{k}$. Denote by $d_{k}$ the
degree of $f_{k}$ for $k=1,\ldots,m$. Then

  1. (1)

$HS(I,t)=\frac{1}{(1-t)^{n}}[t^{d_{1}}+\sum_{k=2}^{m}(1-t^{a_{k,1}})\cdots(1-t^{a_{k,r_{k}}})t^{d_{k}}]$;

  2. (2)

$\mathrm{reg}(I)\leq\max\\{d_{k}+a_{k,1}+\cdots+a_{k,r_{k}}-r_{k}\colon\;2\leq
k\leq m\\}$;

  3. (3)

$\mathrm{Pdim}(I)\leq\max\\{r_{2},\ldots,r_{m}\\}$.

###### Proof.

Recall the known fact that if $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ is a short exact sequence of finitely generated graded
$R$-modules, then $HS(B,t)=HS(A,t)+HS(C,t)$,
$\mathrm{Pdim}(B)\leq\max\\{\mathrm{Pdim}(A),\mathrm{Pdim}(C)\\}$ and
$\mathrm{reg}(B)\leq\max\\{\mathrm{reg}(A),\mathrm{reg}(C)\\}$.

Put $I_{k}=(f_{1},\ldots,f_{k})$ for $k=1,\ldots,m$ and
$L_{k}=(f_{1},\ldots,f_{k-1}):f_{k}$ for $k=2,\ldots,m$. It follows from the
theory of Koszul homology, see e.g. [10], that for $k=2,\ldots,m$, we have

  1. (1)

$HS(R/L_{k}[-d_{k}],t)=[(1-t^{a_{k,1}})\cdots(1-t^{a_{k,r_{k}}})]/(1-t)^{n}$;

  2. (2)

$\mathrm{reg}(R/L_{k}[-d_{k}])=d_{k}+a_{k,1}+\cdots+a_{k,r_{k}}-r_{k}$;

  3. (3)

$\mathrm{Pdim}(R/L_{k}[-d_{k}])=r_{k}$.

Note that $I_{1}$ is isomorphic to the free module $R[-d_{1}]$. It is also
clear that $r_{2}=1$ and $d_{1}\leq d_{2}+a_{2,1}-1$. Hence, the result
follows by applying the facts stated before to the following short exact
sequences:

$0\rightarrow I_{k-1}\rightarrow I_{k}\rightarrow R/L_{k}[-d_{k}]\rightarrow
0,$

where $k=2,\ldots,m.$ $\square$

The upper bounds presented in Proposition 2.1 can be attained under a certain
condition. We will see an application of this result in Section 4.

######  Corollary 2.2.

Let $I$ be as in Proposition 2.1, and let $Q(I)$ and $P(I)$ denote upper
bounds of $\mathrm{reg}(I)$ and $\mathrm{Pdim}(I)$ given in Proposition 2.1
respectively. Let $D(I)$ be the degree of the polynomial $(1-t)^{n}HS(I,t)$.
Then $\mathrm{reg}(I)=Q(I)$ and $\mathrm{Pdim}(I)=P(I)$ whenever
$P(I)+Q(I)=D(I)$. Moreover, $\beta_{P(I),P(I)+Q(I)}(I)$ is the unique extremal
Betti number of $I$ in this case.

###### Proof.

Let $t(I)=\max\\{j\colon\;\beta_{i,j}(I)\neq 0\mbox{ for some }i\geq 0\\}$. In
general, we have

$\mathrm{Pdim}(I)+\mathrm{reg}(I)\geq t(I)\geq D(I).$

From the assumption $P(I)+Q(I)=D(I)$ it follows that $\mathrm{reg}(I)=Q(I)$
and $\mathrm{Pdim}(I)=P(I)$. Moreover, we have $\beta_{i,t(I)}(I)\neq 0$ for
some $i\geq 0$. It is easy to conclude that $i=P(I)$ and the result follows.
$\square$

Computing the Betti numbers of monomial ideals with regular quotients is
generally challenging. For the remainder of this section, we focus on ideals
where each colon ideal involved is generated in a single degree. The following
example illustrates the abundance of such ideals.

######  Example 2.3.

Let $a_{2}\leq a_{3}\leq\cdots\leq a_{m}$ be an arbitrary increasing sequence
of positive integers. Then there exist (square-free) monomials
$f_{1},\cdots,f_{m}$ such that

  1. (1)

$\deg(f_{1})\leq\cdots\leq\deg(f_{m})$;

  2. (2)

$(f_{1},\cdots,f_{k-1}):f_{k}$ is generated by a regular sequence of a single
degree $a_{k}$ for $k=2,\cdots,m$.

###### Proof.

By induction, there exist monomials $g_{1},\cdots,g_{m-1}$ such that
$\deg(g_{1})\leq\cdots\leq\deg(g_{m-1})$ and $(g_{1},\ldots,g_{k-1}):g_{k}$ is
generated by a regular sequence of a single degree $a_{k}$ for
$k=2,\cdots,m-1$. Now, we take monomials $u_{1}$ and $u_{2}$ with
$\deg(u_{1})=\deg(u_{2})=a_{m}$ and such that
$\mathrm{supp}(u_{1})\cap\mathrm{supp}(u_{2})=\emptyset$,
$\mathrm{supp}(u_{i})\cap\mathrm{supp}(g_{j})=\emptyset$ for $i=1,2$ and
$j=1,\dots,m-1$. Then the monomials $f_{1},\ldots,f_{m-1},f_{m}$ is what we
want, where $f_{1}=u_{1}g_{1},\cdots,f_{m-1}=u_{1}g_{m-1},f_{m}=g_{1}\cdots
g_{m-1}u_{2}$. $\square$

For the sake of convenience, we give the following definitions.

######  Definition 2.4.

Let $u_{1},\ldots,u_{r}$ be a regular sequence of monomials. We call it a
regular sequence of type $r$ and in degree $a$ or just a $(a,r)$-sequence if
$\deg(u_{1})=\cdots=\deg(u_{r})=a$.

######  Definition 2.5.

Let $I$ be a monomial ideal of $R$ minimally generated by monomials
$f_{1},\ldots,f_{s}$, and $a$ a positive integer. We say it has $a$-regular
quotients if $(f_{1},\ldots,f_{i-1}):f_{i}$ is generated by a regular sequence
in degree $a$ for $i=2,\ldots,s$.

The following lemma follows directly from Koszul homology theory.

######  Lemma 2.6.

Let $L$ be an ideal generated by an $(a,r)$-sequence
$\alpha_{1},\ldots,\alpha_{r}$. Then

$\beta_{i,j}(R/L)=\left\\{\begin{array}[]{ll}\begin{pmatrix}r\\\
i\end{pmatrix},&\hbox{$j=ai,i\geq 0$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$

Let $I=(f_{1},\ldots,f_{m})$ be a monomial ideal such that
$(f_{1},\ldots,f_{k-1}):f_{k}$ is generated by an $(a,r)$-regular sequence for
$k=2,\ldots,m$. Denote $\deg(f_{k})$ by $d_{k}$ for $k=1,\ldots,m$. By
employing the theory of mapping cone, we can derive an upper bound for
$\beta_{i,j}(I)$.

Let $\phi:A\rightarrow B$ be a homomorphism of complexes. Recall the mapping
cone of $\phi$ is the complex $C(\phi)$ with $C(\phi)_{i}=B_{i}\oplus
A_{i-1}$, and with the chain map $d_{i}:C(\phi)_{i}\rightarrow
C(\phi)_{i-1},d_{i}(b,a)=(\phi_{i-1}(a)+\partial_{i}(b),-\partial_{i-1}(a))$
for all $i$. To apply this construction into the short exact sequences:

$0\rightarrow R/L_{k}[-d_{k}]\rightarrow R/I_{k-1}\rightarrow
R/I_{k}\rightarrow 0,\quad\forall\ k=2,\ldots,m$

where $L_{k}=(f_{1},\ldots,f_{k-1}):f_{k}$ and $I_{k}=(f_{1},\ldots,f_{k})$,
we let $K^{(k)}$ be the Koszul complex for the regular sequence which
generates $L_{k}$, $F^{(k-1)}$ be the graded free resolution of $R/I_{k-1}$,
and $\phi^{(k)}:K^{(k)}[-d_{k}]\rightarrow F^{(k-1)}$ a graded complex
homomorphism lifting $R/L_{k}[-d_{k}]\rightarrow R/I_{k-1}$. Then the mapping
cone $C(\phi^{(k)})$ yields a graded free resolution of $R/I_{k}$. Thus by
iterated mapping cones we obtain step by step a graded free resolution of
$R/I$.

######  Theorem 2.7.

Let $I$ be a monomial ideal generated by monomials $f_{1},\ldots,f_{m}$.
Denote $\deg(f_{i})$ by $d_{i}$ for $i=1,\ldots,m$. Suppose that the colon
ideal $(f_{1},\ldots,f_{i-1}):f_{i}$ is generated by an
$(a_{i},r_{i})$-regular sequence for each $i=2,\ldots,m$. Set $a_{1}=r_{1}=0$.
Let $A_{i}$ denote the set $A_{i}=\\{ia_{p}+d_{p}\colon\;p=1,\ldots,m\\}$ for
$i\geq 0$, and $K^{i}_{j}$ the set $\\{1\leq k\leq m\colon\;a_{k}i+d_{k}=j\\}$
for all $i,j\geq 0$. Then

  1. (1)

$\beta_{i,j}(I)\leq\left\\{\begin{array}[]{ll}\sum\limits_{k\in
K_{j}^{i}}\begin{pmatrix}r_{k}\\\ i\end{pmatrix},&\hbox{$j\in A_{i}$;}\\\
0,&\hbox{$j\notin A_{i}$.}\end{array}\right.$

  2. (2)

The upper bound given in (1) is achieved if the following condition holds:

$\begin{split}\mbox{ for all }1\leq i\leq\max\\{r_{2},\ldots,r_{k}\\}+1\mbox{
and all }k=2,\ldots,m,\\\ \mbox{ one
has\quad}a_{k}i+d_{k}\notin\\{a_{\ell}(i-1)+d_{\ell}\colon\;1\leq\ell\leq
k-1\\}.\quad\end{split}$ ($\clubsuit$)

###### Proof.

(1) Based on the preceding discussion, it can be concluded that

$\begin{split}\beta_{i,j}(R/I_{m})&\leq\beta_{i-1,j}(R/L_{m}[-d_{m}])+\beta_{i,j}(R/I_{m-1})\\\
&\leq\cdots\\\
&\leq\beta_{i-1,j}(R/L_{m}[-d_{m}])+\cdots+\beta_{i-1,j}(R/L_{2}[-d_{2}])+\beta_{i,j}(R/(f_{1})).\end{split}$

Hence,

$\begin{split}\beta_{i,j}(I)\leq\beta_{i,j}(R/L_{m}[-d_{m}])+\cdots+\beta_{i,j}(R/L_{2}[-d_{2}])+\beta_{i,j}(R[-d_{1}]).\end{split}$

The latter is equal to the upper bound given above by Lemma 2.6 and so the
result follows.

(2) We proceed by induction on $m$. If $m=1$, then $I\cong R[-d_{1}]$ and so

$\beta_{i,j}(I)=\left\\{\begin{array}[]{ll}1,&\hbox{$i=0,j=d_{1}$;}\\\
0,&\hbox{otherwsie.}\end{array}\right.$

Note that $A_{i}=\\{d_{1}\\}$ for all $i\geq 0$, it is easy to check that the
desired formula is true when $m=1$.

Now, assume $m\geq 2$. By induction hypothesis, one has

$\beta_{i,j}(I_{m-1})=\left\\{\begin{array}[]{ll}\sum\limits_{k\in\underline{K}^{i}_{j}}\begin{pmatrix}r_{k}\\\
i\end{pmatrix},&\hbox{$j\in B_{i}$;}\\\ 0,&\hbox{$j\notin
B_{i}$.}\end{array}\right.$

Here, $B_{i}=\\{a_{p}i+d_{p}\colon\;p=1,\ldots,m-1\\}$ for all $i\geq 0$ and
$\underline{K}^{i}_{j}=K^{i}_{j}\setminus\\{m\\}$ for all $i,j\geq 0$. On the
other hand, the following short exact sequence:

$0\rightarrow I_{m-1}\rightarrow I_{m}\rightarrow R/L_{m}[-d_{m}]\rightarrow
0$

induces the long exact sequence:

$\cdots\rightarrow\mathrm{Tor}_{i+1}^{R}(R/L_{m}[-d_{m}],\mathbb{K})_{j}\rightarrow\mathrm{Tor}_{i}^{R}(I_{m-1},\mathbb{K})_{j}\rightarrow\mathrm{Tor}_{i}^{R}(I_{m},\mathbb{K})_{j}$
$\rightarrow\mathrm{Tor}_{i}^{R}(R/L_{m}[-d_{m}],\mathbb{K})_{j}\rightarrow\mathrm{Tor}_{i-1}^{R}(I_{m-1},\mathbb{K})_{j}\rightarrow\cdots$
(2.1)

Fix $i\geq 0$. If $j\notin A_{i}$, then $j\notin B_{i}$ and $j\neq
ia_{m}+d_{m}$, and so
$\mathrm{Tor}_{i}^{R}(I_{m-1},\mathbb{K})_{j}=\mathrm{Tor}_{i}^{R}(R/L[-d_{m}],\mathbb{K})_{j}=0$
by induction and by Lemma 2.6 respectively. This implies
$\mathrm{Tor}_{i}^{R}(I_{m},\mathbb{K})_{j}=0$ and so
$\beta_{i,j}(I)=\beta_{i,j}(I_{m})=0$. If $j\in A_{i}$, we consider the
following cases.

Case when $j=ia_{m}+d_{m}$: In this case,
$K^{i}_{j}=\underline{K}^{i}_{j}\cup\\{m\\}$. In view of condition
($\clubsuit$ ‣ 2), we have either $j\notin B_{i-1}$ or
$i>\max\\{r_{1},\ldots,r_{m}\\}+1$, and thus
$\mathrm{Tor}_{i-1}^{R}(I_{m-1},\mathbb{K})_{j}=0$. Note that
$\mathrm{Tor}_{i+1}^{R}(R/L_{m}[-d_{m}],\mathbb{K})_{j}=0$ by Lemma 2.6, it
follows that

$\beta_{i,j}(I)=\beta_{i,j}(I_{m-1})+\begin{pmatrix}r_{m}\\\
i\end{pmatrix}=\sum\limits_{k\in K^{i}_{j}}\begin{pmatrix}r_{k}\\\
i\end{pmatrix}.$

Case when $j\neq ia_{m}+d_{m}$: In this case,
$K^{i}_{j}=\underline{K}^{i}_{j}$. Since $j\in A_{i}$, we have either
$j\neq(i+1)a_{m}+d_{m}$ or $i>r_{m}+1$, according to ($\clubsuit$ ‣ 2). It
follows that $\mathrm{Tor}_{\ell}^{R}(R/L_{m}[-d_{m}],\mathbb{K})_{j}=0$ for
$\ell\in\\{i,i+1\\}$ by Lemma 2.6. Hence,

$\beta_{i,j}(I)=\beta_{i,j}(I_{m-1})=\sum\limits_{k\in\underline{K}^{i}_{j}}\begin{pmatrix}r_{k}\\\
i\end{pmatrix}=\sum\limits_{k\in K^{i}_{j}}\begin{pmatrix}r_{k}\\\
i\end{pmatrix}.$

This completes the proof. $\square$

######  Remark 2.8.

Note that if $j\notin A_{i}$, then $K^{i}_{j}=\emptyset$, and in particular,
$\sum\limits_{k\in K_{j}^{i}}\begin{pmatrix}r_{k}\\\ i\end{pmatrix}=0$. Hence,
the conclusion in Theorem 2.7.(2) may be reformulated as:

$\beta_{i,j}(I)=\sum\limits_{k\in K_{j}^{i}}\begin{pmatrix}r_{k}\\\
i\end{pmatrix}.$

######  Remark 2.9.

If $r_{2}=r_{3}=\cdots=r_{m}=1$, then condition ($\clubsuit$ ‣ 2) can be
removed from Theorem 2.7. Since $\beta_{i,j}(I)=0$ for all $i,j$ with $i\geq
2$, it suffices to prove the formula in Theorem 2.7.(2) holds both when $i=0$
and when $i=1$. We show this again by induction on $m$. The case when $i=0$
follows by noting that

$\beta_{0,j}(I_{m})-\beta_{0,j}(I_{m-1})=\left\\{\begin{array}[]{ll}1,&\hbox{$j=d_{m}$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$

On the other hand, since
$\mathrm{Tor}_{2}^{R}(R/L_{m}[-d_{m}],\mathbb{K})_{j}=0$ for any $j\geq 0$, in
view of the long exact sequence (2), we have

$\begin{split}\beta_{1,j}(I_{m})=\beta_{1,j}(I_{m-1})+\beta_{1,j}(R/L_{m}[-d_{m}])+\beta_{0,j}(I_{m})-\beta_{0,j}(I_{m-1})-\beta_{0,j}(R/L_{m}[-d_{m}]).\end{split}$

From this it follows that
$\beta_{1,j}(I_{m})=\beta_{1,j}(I_{m-1})+\beta_{1,j}(R/L_{m}[-d_{m}])$,
completing the proof.

Let us see two special cases of Theorem 2.7, which we are interested in.

######  Corollary 2.10.

Let $I$ be a monomial ideal generated by monomials $f_{1},\ldots,f_{m}$.
Denote $\deg(f_{i})$ by $d_{i}$ for $i=1,\ldots,m$. Suppose the colon ideal
$(f_{1},\ldots,f_{k-1}):f_{k}$ is generated by an $(a_{k},r_{k})$-regular
sequence for each $k=2,\ldots,m$. Moreover, if one of the following conditions
is satisfied:

  1. (1)

$a_{2}=\ldots=a_{m}=a$ and $d_{\ell}-d_{k}\neq a$ for $k\geq 2$ and
$1\leq\ell\leq k-1$;

  2. (2)

$a_{1}\leq a_{2}\leq\cdots\leq a_{m}$ and $d_{1}\leq d_{2}\leq\cdots\leq
d_{m}$;

then the conclusion in Theorem 2.7.(2) holds.

###### Proof.

This is because if one of the conditions mentioned above is satisfied then
condition ($\clubsuit$ ‣ 2) in Theorem 2.7 holds true. $\square$

The inspiration for the following result originates from [7]. It suggests if
$I$ is a monomial ideal with $a$-regular quotients, we may rearrange the order
of the generators of $I$ to make their degrees more closely approximate a
monotonically increasing pattern, while preserving the quotient ideals
unchanged. If $a=1$, we recover [7, Lemma 2.1 and Proposition 2.2].

######  Lemma 2.11.

Let $I$ be a monomial ideal having regular quotients with respect to the order
$f_{1},\ldots,f_{m}$ of minimal generators of $I$ such that all the quotient
ideals are generated in the same degree $a$. Then there is a permutation
$\sigma$ on $[n]$ with $\sigma(1)=1$ such that for any $i\in[n]$, one has

$(f_{\sigma(1)},\ldots,f_{\sigma(i-1)}):f_{\sigma(i)}=(f_{1},\ldots,f_{\sigma(i)-1}):f_{\sigma(i)}.$

Moreover, $\deg(f_{\sigma(i)})\leq\deg(f_{\sigma(i+1)})+a-1$ for
$i=1,\ldots,m-1$.

###### Proof.

By induction, we may assume $\deg(f_{i})\leq\deg(f_{i+1})+(a-1)$ for
$i=1,\ldots,m-2$. If $\deg(f_{m-1})\leq\deg(f_{m})+a-1$, we are done. Suppose
now $\deg(f_{m-1})\geq\deg(f_{m})+a$. Then, there is an integer $1\leq j\leq
m-1$ such that $\deg(f_{j})\leq\deg(f_{m})+a-1$ and
$\deg(f_{p})\geq\deg(f_{m})+a$ for $p=j+1,\ldots,m-1$. We only need to prove
$f_{1},\ldots,f_{j},f_{m},f_{j+1},\ldots,f_{m-1}$ is the order we require. In
the following, we denote $u:v$ as the monomial $u/(u,v)$, where $(u,v)$
represents the greatest common divisor of $u$ and $v$.

Since $\deg(f_{p}:f_{m})\geq a+1$ for $p=j+1,\ldots,m-1$, it follows that
$(f_{1},\ldots,f_{j}):f_{m}=(f_{1},\ldots,f_{m-1}):f_{m}$. It remains to show
that for all $p$ with $j+1\leq p\leq m-1$, one has

$(f_{1},\ldots,f_{j},f_{m},f_{j+1},\ldots,f_{p-1}):f_{p}=(f_{1},\ldots,f_{j},f_{j+1},\ldots,f_{p-1}):f_{p}.$

This is equivalent to showing that
$f_{m}:f_{p}\in(f_{1},\ldots,f_{j},f_{j+1},\ldots,f_{p-1}):f_{p}.$ To do this,
we first note that there is an integer $1\leq k\leq j$ such that
$\deg(f_{k}:f_{m})=a$ and $f_{k}:f_{m}$ divides $f_{p}:f_{m}$. Put
$\alpha=f_{k}:f_{m}.$ There are two cases to consider:

The first case is when $\deg(f_{k}:f_{p})>a$. In this case, there is
$1\leq\ell\leq j$ such that $\deg(f_{\ell}:f_{p})=a$ and $f_{\ell}:f_{p}$
divides $f_{k}:f_{p}$. Put $\beta=f_{\ell}:f_{p}$. Then, for any variable
$x\in\mathrm{supp}(\alpha)$, we have

$\deg_{x}(f_{k})=\deg_{x}(f_{m})+\deg_{x}\alpha,\quad\deg_{x}(f_{p})\geq\deg_{x}(f_{m})+\deg_{x}\alpha.$

This implies that $\deg_{x}(f_{k}:f_{p})=0$, and so
$x\notin\mathrm{supp}(\beta)$. Thus,

$\mathrm{supp}(\alpha)\cap\mathrm{supp}(\beta)=\emptyset.$

Take any variable $y\in\mathrm{supp}(\beta)$. Then, we have
$\deg_{y}(f_{m})\geq\deg_{y}(f_{k})$, since
$\mathrm{supp}(\alpha)\cap\mathrm{supp}(\beta)=\emptyset$ as well as
$\alpha=f_{k}:f_{m}.$ On the other hand, since $\beta$ divides $f_{k}:f_{p}$,
it follows that $\deg_{y}(f_{k})\geq\deg_{y}(f_{p})+\deg_{y}(\beta)$. Hence,
$\deg_{y}(f_{m})\geq\deg_{y}(f_{p})+\deg_{y}(\beta)$, and this implies $\beta$
divides $f_{m}:f_{p}$. In particular, $f_{m}:f_{p}$ belongs to
$(f_{1},\ldots,f_{j},f_{j+1},\ldots,f_{p-1}):f_{p}$, as desired.

The second case is when $\deg(f_{k}:f_{p})=a$. In this case, we put
$\beta=f_{k}:f_{p}$. Now, the remaining proof is similar to the one in the
first case and so we omit it. $\square$

######  Example 2.12.

Consider the monomial ideal

$I=(f_{1}=x_{1}x_{2}x_{3}x_{4}x_{5}x_{6},f_{2}=x_{1}x_{2}x_{3}x_{4}x_{7}x_{8}x_{9},f_{3}=x_{3}x_{4}x_{5}x_{6}x_{7},f_{4}=x_{5}x_{6}x_{7}x_{8}).$

It has $2$-regular quotients with respect to the given order, where the
quotients, one by one, are $(x_{5}x_{6}),(x_{1}x_{2}),(x_{3}x_{4})$. It also
has 2-regular quotients with respect to the order $f_{1},f_{3},f_{4},f_{2}$,
where the quotient ideals, in sequence, are
$(x_{1}x_{2}),(x_{3}x_{4}),(x_{5}x_{6})$, as claimed by Lemma 2.11. Since
every quotient ideal is generated by a single element, we obtain the following
Betti table of $I$ by Remark 2.9:

$\begin{array}[]{c|cc}&0&1\\\ \hline\cr 4&1&0\\\ 5&1&1\\\ 6&1&1\\\ 7&1&0\\\
8&0&1\end{array}.$

It is interesting to see that $I$ has regular quotients with respect to many
other orders. For examples:

With respect to the degree increasing order $f_{4},f_{3},f_{1},f_{2}$, $I$ has
$(1,1,2)$-regular quotients. More precisely,
$f_{4}:f_{3}=(x_{8}),(f_{4},f_{3}):f_{1}=(x_{7})$ and
$(f_{4},f_{3},f_{1}):f_{2}=(x_{5}x_{6})$.

With respect to the order $f_{1},f_{4},f_{3},f_{2}$, the ideal $I$ has regular
quotients too: $(f_{4}):f_{1}=(x_{7}x_{8})$,
$(f_{4},f_{1}):f_{3}=(x_{1}x_{2},x_{8})$, and
$(f_{4},f_{1},f_{3}):f_{2}=(x_{5}x_{6})$. This shows the upper bounds given in
Proposition 2.1 may not be achieved.

With respect to the degree decreasing order $f_{2},f_{1},f_{3},f_{4}$, the
ideal $I$ has regular quotients again: $(f_{2}):f_{1}=(x_{7}x_{8}x_{9})$.
$(f_{2},f_{1}):f_{3}=(x_{1}x_{2})$, $(f_{2},f_{1},f_{3}):f_{4}=(x_{3}x_{4})$.

##  3\. A bridge theorem

In this section, we establish a condition that guarantees the equality of
Betti numbers between a toric ideal of an edge ring and its initial ideal.
This condition will be utilized in subsequent sections.

######  Lemma 3.1.

Let $I$ be a homogeneous ideal of $R$ such that ${\mathrm{in}}_{\prec}(I)$ has
a pure resolution for some monomial order $\prec$. Then
$\beta_{i,j}(I)=\beta_{i,j}(\mathrm{in}_{\prec}(I))$ for all $i,j\geq 0$.

###### Proof.

It was shown in [17, Corollary 22.13] that if ${\rm{in}}_{\prec}(I)$ has a
linear resolution, then $\beta_{i,j}(I)=\beta_{i,j}({\rm{in}}_{\prec}(I))$ for
all $i,j\geq 0$. The same proof applies to the case of pure resolution.
$\square$

######  Theorem 3.2.

Let $G$ be a simple graph and $s$ a positive integer. Assume that
$\mathcal{PC}(G)$ can be arranged as $\\{C_{1},\ldots,C_{r}\\}$ such that the
following conditions are satisfied:

  1. (1)

With respect to some monomial order $\prec$, the monomial ideal
$\mathrm{in}_{\prec}(I_{G})$ has $s$-regular quotients with respect to the
order $u_{1},u_{2},\ldots,u_{r}$; Here $u_{i}=\mathrm{in}_{\prec}(f_{C_{i}})$
for $i=1,\ldots,r$.

  2. (2)

There are pairwise distinct positive integers $d_{1},\ldots,d_{m}$ such that

$\begin{split}\deg(u_{1})=\cdots=&\deg(u_{k_{1}})=d_{1},\deg(u_{k_{1}+1})=\cdots=\deg(u_{k_{2}})=d_{2},\\\
&\cdots,\cdots,\deg(u_{k_{m-1}+1})=\cdots=\deg(u_{k_{m}})=d_{m}\end{split}$

for some $k_{i}$’s with $k_{m}=r$ and such that there exists an induced
subgraph $G_{i}$ of $G$ satisfying

$\mathcal{PC}(G_{i})=\\{C_{\ell}\colon\;1\leq\ell\leq k_{i}\\}$

for each $i=1,2,\ldots,m$;

  3. (3)

$d_{k}-d_{\ell}\neq s$ for any pair $k,\ell$ with $1\leq k<\ell\leq m$.

Then $\beta_{i,j}(I_{G})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G}))$ for all
$i,j\geq 0$.

###### Proof.

Observe that $\mathrm{in}_{\prec}(I_{G_{i}})=(u_{1},\ldots,u_{k_{i}})$ for
$i=1,2,\ldots,m$ and that we may regard $G_{i}$ as an induced subgraph of
$G_{i+1}$ for $i=1,\ldots,m-1$. In the sequel, we use induction on $m$. If
$m=1$, then $\beta_{i,j}(\mathrm{in_{<}}(I_{G}))=0$ when $j\neq si+d_{1}$ by
Theorem 2.7. It follows that $\mathrm{in}_{\prec}(I_{G})$ has a pure
resolution, and so we are done by Lemma 3.1.

Suppose $m>1$. Note that $G=G_{m}$. By induction hypothesis, we have
$\beta_{i,j}(I_{G_{m-1}})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G_{m-1}}))$ for
all $i,j\geq 0$. Additionally, by Theorem 2.7.1 as well as [10, Corollary
3.3.3], it follows that $\beta_{i,j}(I_{G})=0$ for all
$j\notin\\{is+d_{k}\colon\;k=1,\ldots,m\\}$.

Fix $i\geq 0$ and let $j\in\\{is+d_{k}\colon\;k=1,\ldots,m-1\\}$. Since
$d_{1},\ldots,d_{m}$ are pairwise distinct,
$\beta_{i,j}(\mathrm{in}_{\prec}(I_{G_{m-1}}))=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G_{m}}))$
by Corollary 2.10. Based on this and by utilizing Lemma 1.2, it can be
inferred that

$\beta_{i,j}(I_{G})\geq\beta_{i,j}(I_{G_{m-1}})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G_{m-1}}))=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G_{m}}))\geq\beta_{i,j}(I_{G}).$

This implies that

$\beta_{i,j}(I_{G})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G}))\mbox{ for all
}i\geq 0\mbox{ and }j\in\\{is+d_{k}\colon\;k=1,\ldots,m-1\\}.$ (3.1)

It remains to show that
$\beta_{i,j}(I_{G})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G}))$ whenever
$j=is+d_{m}$. To do this, we fix $i\geq 0$ and let $j=is+d_{m}$. By putting
$A:=\\{k\geq 0\colon\;ks+d_{t}=j\mbox{\ for some\ }1\leq t\leq m-1\\}$, we
have the coefficient of $t^{j}$ in $(1-t)^{n}HS(I_{G},t)$ is equal to

$(-1)^{i}\beta_{i,j}(I_{G})+\sum_{k\in A}(-1)^{k}\beta_{k,j}(I_{G}),$

and the coefficient of $t^{j}$ in $(1-t)^{n}HS(\mathrm{in}_{\prec}(I_{G}),t)$
is equal to

$(-1)^{i}\beta_{i,j}(\mathrm{in}_{\prec}(I_{G}))+\sum_{k\in
A}(-1)^{k}\beta_{k,j}(\mathrm{in}_{\prec}(I_{G})).$

From (3.1), it follows that

$\sum_{k\in A}(-1)^{k}\beta_{k,j}(I_{G})=\sum_{k\in
A}(-1)^{k}\beta_{k,j}(\mathrm{in}_{\prec}(I_{G})).$

By combining these factors, we obtain
$\beta_{i,j}(I_{G})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{G}))$, as desired.
$\square$

##  4\. Bipartite graphs $B_{\underline{\ell},h}$ and their homological
invariants

Let $h\geq 2$ be an integer, and $\underline{\ell}=(\ell_{1},\ldots,\ell_{h})$
a vector in $\mathbb{Z}_{>0}^{h}$. In this section, we introduce a special
family of bipartite graphs denoted by $B_{\underline{\ell},h}$. We then
compute the regularity and projective dimension of the edge ring
$\mathbb{K}[B_{\underline{\ell},h}]$ in a general setting. Additionally, we
provide an explicit formula for the Betti numbers of the toric ideal of
$B_{\underline{\ell},h}$ in the case when $\ell_{2}=\cdots=\ell_{h}$.

######  Definition 4.1.

Let $h\geq 2$ be an integer and let
$\underline{\ell}=(\ell_{1},\ldots,\ell_{h})$ be a vector in
$\mathbb{Z}_{>0}^{h}$. We define $B_{\underline{\ell},h}$ to be a bipartite
graph consisting of $h$ paths of lengths $2\ell_{1},\ldots,2\ell_{h}$
respectively, all originating from a shared vertex and terminating at another
common vertex, see Fig. 1. More precisely, $B_{\underline{\ell},h}$ is the
bipartite graph with vertex set

$V_{\underline{\ell},h}=\\{y_{1},y_{2}\\}\cup\\{x_{i,j}\colon\;1\leq i\leq
2\ell_{k}-1,1\leq k,j\leq h\\}$

and with edge set

$\begin{split}E_{\underline{\ell},h}&=\\{\\{y_{1},x_{1,1}\\},\ldots,\\{y_{1},x_{1,h}\\}\\}\cup\\{\\{x_{i,j},x_{i+1,j}~{}|~{}1\leq
i\leq 2\ell_{k}-2,1\leq k,j\leq h\\}\\}\\\
&\quad\cup\\{\\{y_{2},x_{2\ell_{1},1}\\},\ldots,\\{y_{2},x_{2\ell_{h},h}\\}\\}.\end{split}$

We label the edges of $B_{\underline{\ell},h}$ as follows:
$e_{1,j}=\\{y_{1},x_{1,j}\\},e_{i,j}=\\{x_{i-1,j},x_{i,j}\\},e_{2\ell,j}=\\{y_{2},x_{2\ell,j}\\}$,
for $i\in\\{2,\ldots,2\ell_{k}-1\\}$ and $k,j\in\\{1,\ldots,h\\}.$

$e_{1,1}$$e_{1,2}$$e_{1,3}$$e_{1,h-2}$$e_{1,h-1}$$e_{1,h}$$e_{2,1}$$e_{2,2}$$e_{2,3}$$e_{2,h-2}$$e_{2,h-1}$$e_{2,h}$$e_{2\ell_{1}-1,1}$$e_{2\ell_{2}-1,2}$$e_{2\ell_{3}-1,3}$$e_{2\ell_{h-2}-1,h-2}$$e_{2\ell_{h-1}-1,h-1}$$e_{2\ell_{h}-1,h}$$e_{2\ell_{1},1}$$e_{2\ell_{2},2}$$e_{2\ell_{3},3}$$e_{2\ell_{h-2},h-2}$$e_{2\ell_{h-1},h-1}$$e_{2\ell_{h},h}$$y_{1}$$y_{2}$Fig.
1: The graph $B_{\underline{\ell},h}$

Let $\prec$ denote the pure lexicographic monomial order on the polynomial
ring $R:=\mathbb{K}[e_{i,j}\colon\;1\leq i\leq 2\ell_{j},1\leq j\leq h]$,
where the variables are ordered in the following way:

$e_{i_{1},j_{1}}\succ
e_{i_{2},j_{2}}\Longleftrightarrow\left\\{\begin{array}[]{ll}\mbox{ either
}i_{1}\mbox{ is odd and }i_{2}\mbox{ is even}\\\ \mbox{or }i_{1}-i_{2}=0\
(\operatorname{mod}2)\mbox{ and }i_{1}<i_{2}\\\ \mbox{or }i_{1}=i_{2}\mbox{
and }j_{1}<j_{2}.\end{array}\right.$

For all $1\leq i<j\leq h$, we set

$u_{i,j}=e_{1,i}e_{3,i}\cdots e_{2\ell_{i}-1,i}e_{2,j}e_{4,j}\cdots
e_{2\ell_{j},j}$

and

$v_{i,j}=e_{1,j}e_{3,j}\cdots e_{2\ell_{j}-1,j}e_{2,i}e_{4,i}\cdots
e_{2\ell_{i},i}.$

Note that both $u_{i,j}$ and $v_{i,j}$ are monomials of degree
$\ell_{i}+\ell_{j}$, and $v_{i,j}\prec u_{i,j}$ with respect to the order
given above.

######  Lemma 4.2.

The universal Gröbner basis of $I_{B_{\underline{\ell},h}}$ is given by

$\mathcal{G}_{\ell,h}=\\{u_{i,j}-v_{i,j}\colon\;1\leq i<j\leq h\\}.$

In particular, the minimal generating set of
$\mathrm{in}_{\prec}(I_{B_{\underline{\ell},h}})$ is

$\\{u_{i,j}\colon\;1\leq i<j\leq h\\}.$

###### Proof.

According to [16, Lemma 3.2] (or see Proposition 1.1), we have the set of
primitive even closed walks of $B_{\underline{\ell},h}$ is

$\mathcal{PC}(B_{\underline{\ell},h})=\\{W_{i,j}=e_{1,i},e_{2,i},\ldots,e_{2\ell_{i},i},e_{2\ell_{j},j},\ldots,e_{2,j},e_{1,j}\colon\;1\leq
i<j\leq h\\}.$

Note that $f_{W_{i,j}}=u_{i,j}-v_{i,j}$ for all $i,j$, it follows that
$\\{u_{i,j}-v_{i,j}\colon\;1\leq i<j\leq h\\}$ is the universal Gröbner basis
of $I_{B_{\underline{\ell},h}}$ by e.g. [3, Proposition 5.19]. Now, the last
assertion follows since $v_{i,j}\prec u_{i,j}$ for all $i,j$ with respect to
the given order. $\square$

It is clear that

$\begin{split}&u_{h-1,h}\prec u_{h-2,h}\prec u_{h-2,h-1}\prec\cdots\prec
u_{i,h}\prec\cdots\prec u_{i,i+1}\\\ &\prec u_{i-1,h}\prec\cdots\prec
u_{i-1,i}\prec\cdots\prec u_{1,h}\prec\cdots\prec u_{1,2}.\end{split}$ (4.1)

We now show that $\mathrm{in}_{\prec}(I_{B_{\underline{\ell},h}})$ has regular
quotients with respect to the above order. To state the result more simply, we
label the monomials in (4.1) one by one:

$m_{1}=u_{h-1,h},\quad m_{2}=u_{h-2,h},\quad\ldots,\quad
m_{\frac{h(h-1)}{2}}=u_{1,2}.$

######  Proposition 4.3.

The initial ideal $\mathrm{in}_{\prec}(I_{B_{\underline{\ell},h}})$ has
regular quotients with respect to the above order. More precisely, the colon
ideal $(m_{1},\ldots,m_{k-1}):m_{k}$ is generated by a regular sequence of
length $r_{k}$ for $k=2,\ldots,\frac{h(h-1)}{2}$, where

$(r_{2},\ldots,r_{\frac{h(h-1)}{2}})=(\underbrace{{1,1}}_{2},\underbrace{2,2,2}_{3},\ldots,\underbrace{h-2,h-2,\ldots,h-2}_{h-1}).$

###### Proof.

By putting $Y_{1,i}=e_{1,i}e_{3,i}\cdots e_{2\ell_{i}-1,i}$ and
$Y_{2,j}=e_{2,j}e_{4,j}\cdots e_{2\ell_{j},j}$, we may write

$u_{i,j}=Y_{1,i}Y_{2,j}$

for $1\leq i<j\leq h.$ Fix $2\leq k\leq\frac{h(h-1)}{2}$, then there exist
unique $i,j$ with $1\leq i<j\leq h$ such that $m_{k-1}=u_{i,j}$. Let $L_{k}$
denote the colon ideal $(m_{1},\ldots,m_{k-1}):m_{k}$.

If $j-i\geq 3$, then

$L_{k}=(Y_{1,h-1}Y_{2,h},\ldots,Y_{1,i}Y_{2,j}):Y_{1,i}Y_{2,j-1}=(Y_{1,j-2},\ldots,Y_{1,i+1},Y_{2,j}\ldots,Y_{2,h})$

and so it is generated by a regular sequence of length $h-i-1$.

If $j-i=2$, then

$L_{k}=(Y_{1,h-1}Y_{2,h},\ldots,Y_{1,i}Y_{2,j}):Y_{1,i}Y_{2,i+1}=(Y_{2,i+2}\ldots,Y_{2,h})$

and so it is generated by a regular sequence of length $h-i-1$.

If $j-i=1$, then $i\geq 2$, and

$L_{k}=(Y_{1,h-1}Y_{2,h},\ldots,Y_{1,i}Y_{2,j}):Y_{1,i-1}Y_{2,h}=(Y_{1,h-1},Y_{1,h-2},\ldots,Y_{1,i})$

and so it is generated by a regular sequence of length $h-i$.

The facts above show that if $m_{k-1}=u_{i,j}$ then

$r_{k}=\left\\{\begin{array}[]{ll}h-i-1,&\hbox{$j-i\geq 2$ and $i\geq 1$;}\\\
h-i,&\hbox{$j-i=1$ and $i\geq 2$.}\end{array}\right.$

From this the last assertion of this result follows. $\square$

######  Theorem 4.4.

Let $I$ denote the toric ideal $I_{B_{\underline{\ell},h}}$ and $J$ its
initial ideal $\mathrm{in}_{\prec}(I)$. Then both $R/I$ and $R/J$ are Cohen-
Macaulay. Moreover,

  1. (1)

$\mathrm{Pdim}(I)=\mathrm{Pdim}(J)=h-2$;

  2. (2)

$\mathrm{reg}(I)=\mathrm{reg}(J)=\ell_{1}+\cdots+\ell_{h}-h+2$;

  3. (3)

$\deg h_{R/I}(t)=\deg h_{R/J}(t)=\ell_{1}+\cdots+\ell_{h}-h+1$;

  4. (4)

Denote $p=h-2$ and $q=\ell_{1}+\cdots+\ell_{h}$, then
$\beta_{p,q}(I)=\beta_{p,q}(J)=h-1$, and they are the only extreme Betti-
numbers of $I$ and $J$ respectively.

###### Proof.

Let $n$ denote $2(\ell_{1}+\cdots+\ell_{h})$, the number of variables of
polynomial ring $\mathbb{K}[e_{i,j}\colon\;1\leq i\leq 2\ell_{j},1\leq j\leq
h]$, and let $L_{k}$ denote the ideal $(m_{1},\ldots,m_{k-1}):m_{k}$ for
$k=2,\ldots,\frac{h(h-1)}{2}$. If $m_{k-1}=u_{i,j}$, then, by Proposition 4.3,
we have

$\begin{split}(1-t)^{n}&HS(R/L_{k}[-m_{k}],t)\\\
=&\left\\{\begin{array}[]{ll}(1-t^{\ell_{i}})\cdots(1-t^{\ell_{h}})\frac{t^{\ell_{i}}t^{\ell_{j-1}}}{(1-t^{\ell_{i}})(1-t^{\ell_{j-1}})},&\hbox{$j-i\geq
2$ and $i\geq 1$;}\\\ &\hbox{}\\\
(1-t^{\ell_{i-1}})\cdots(1-t^{\ell_{h}})\frac{t^{\ell_{i-1}}t^{\ell_{h}}}{(1-t^{\ell_{i-1}})(1-t^{\ell_{h}})},&\hbox{$j-i=1$
and $i\geq 2$.}\end{array}\right.\end{split}$

and so the initial term of $(1-t)^{n}HS(R/L_{k}[-m_{k}],t)$ is as follows:

$\begin{split}\left\\{\begin{array}[]{ll}(-1)^{h-i-1}t^{\ell_{i}+\cdots+\ell_{h}},&\hbox{$j-i\geq
2$ and $i\geq 1$;}\\\ (-1)^{h-i}t^{\ell_{i-1}+\cdots+\ell_{h}},&\hbox{$j-i=1$
and $i\geq 2$.}\end{array}\right.\end{split}$

Hence, the initial term of $H(J,t)(1-t)^{n}$ is equivalent to

$(-1)^{h-2}(h-1)t^{\ell_{1}+\cdots+\ell_{h}}.$

By adopting the notions presented in Corollary 2.2, we have
$D(J)=\ell_{1}+\cdots+\ell_{h}$. It also follows from Proposition 4.3 that
$Q(J)=\ell_{1}+\cdots+\ell_{h}-h+2$ and $P(J)=h-2$. This implies
$D(J)=P(J)+Q(J)$, and thus $\mathrm{Pdim}(J)=h-2$ and
$\mathrm{reg}(J)=\ell_{1}+\cdots+\ell_{h}-h+2$ by Corollary 2.2.

In view of [19, Corollary 10.1.21], we have

$\dim(R/J)=\dim\mathbb{K}[B_{\underline{\ell},h}]=2(\ell_{1}+\cdots+\ell_{h})-h+1=n-h+1,$

and

$\mathrm{depth}(R/J)=n-\mathrm{Pdim}(R/J)=n-h+1.$

It follows that both $R/I$ and $R/J$ are Cohen-Macaulay. From this, we have
$\mathrm{Pdim}(I)=\mathrm{Pdim}(J)$ and $\mathrm{reg}(I)=\mathrm{reg}(J)$. The
last statements are also clear now. $\square$

######  Corollary 4.5.

Let $I$ and $J$ be the same as in Theorem 4.4. Then the following are
equivalent:

  1. (1)

$R/I$ is a complete intersection;

  2. (2)

$R/J$ is a complete intersection;

  3. (3)

$R/I$ is Gorenstein;

  4. (4)

$R/J$ is Gorenstein;

  5. (5)

$h=2$.

###### Proof.

If $h=2$, then $B_{\underline{\ell},h}$ is an even cycle and so $I$ and $J$
are both generated by a polynomial. This implies both $R/I$ and $R/J$ are
complete intersections. Conversely, if either $R/I$ or $R/J$ is Gorenstein,
then either $\beta_{p}(I)=1$ or $\beta_{p}(J)=1$. It follows that
$h-1=\beta_{p,q}(I)=\beta_{p,q}(J)\leq 1$, and hence $h=2$. $\square$

######  Corollary 4.6.

Given a vector $(p,r)\in\mathbb{Z}^{2}$ with $p\geq 0$ and $r\geq 2$, there
exists a bipartite graph $G$ such that $\mathrm{pdim}(I_{G})=p$ and
$\mathrm{reg}(I_{G})=r$.

###### Proof.

Put $h=p+2$ and $\underline{\ell}=(\underbrace{1,\ldots,1}_{h-1},r-1)$. Then
the bipartite graph $B_{\underline{\ell},h}$ is what we require. $\square$

In the final part of this section, we compute their Betti numbers of
$\mathbb{K}[B_{\underline{\ell},h}]$ explicitly in the case when
$\ell_{2}=\cdots=\ell_{h}$.

######  Theorem 4.7.

Let $B_{\underline{\ell},h}$ be the bipartite graph defined above with
$\underline{\ell}=(\ell_{1},\ldots,\ell_{h})$.

  1. (1)

If $\ell_{1}=\ell_{2}=\cdots=\ell_{h}$, then

$\beta_{i,j}(I_{B_{\underline{\ell},h}})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{B_{\underline{\ell},h}}))=\left\\{\begin{array}[]{ll}(i+1)\binom{h}{i+2},&\hbox{$j=\ell(i+2)$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$

Here, $\ell=\ell_{1}$.

  2. (2)

If $\ell_{1}\neq\ell_{2}=\cdots=\ell_{h}$, then

$\beta_{i,j}(I_{B_{\underline{\ell},h}})=\beta_{i,j}(\mathrm{in}_{\prec}(I_{B_{\underline{\ell},h}}))=\left\\{\begin{array}[]{ll}(i+1)\binom{h-1}{i+2}&\hbox{$j=\ell(i+2)$;}\\\
(h-1)\binom{h-2}{i}&\hbox{$j=\ell(i+1)+q$;}\\\
0&\hbox{otherwise.}\par\end{array}\right.$

Here, $q=\ell_{1}$ and $\ell=\ell_{2}$.

###### Proof.

Let $J$ denote the monomial ideal
$\mathrm{in}_{\prec}(I_{B_{\underline{\ell},h}})$. In both cases, $J$ has
$\ell$-regular quotients with respect to the given order by Proposition 4.3.

(1) By Corollary 2.10 together with Proposition 4.3, we have
$\beta_{i,j}(J)=0$ for $j\neq\ell(i+2)$; and

$\displaystyle\beta_{i,\ell(i+2)}(J)$
$\displaystyle=\sum_{k=1}^{\frac{h(h-1)}{2}}\binom{r_{k}}{i}=\binom{0}{i}+2\binom{1}{i}+3\binom{2}{i}+\ldots+(h-1)\binom{h-2}{i}$
$\displaystyle=\sum_{q=1}^{h-1}q\binom{q-1}{i}=(i+1)\sum_{q=1}^{h-1}\binom{q}{i+1}=(i+1)\binom{h}{i+2}.$

Here, the second-to-last equality follows from the identity
$k\binom{n}{k}=n\binom{n-1}{k-1}$, and the last equality follows from the
identity $\sum_{j=0}^{n}\binom{j}{k}=\binom{n+1}{k+1}$ for $k\geq 0$.

It follows from Lemma 3.1 that
$\beta_{i,j}(I_{B_{\underline{\ell},h}})=\beta_{i,j}(J)$ for all $i,j\geq 0$.

(2) In view of Proposition 4.3, the sequence of degrees of generators of $J$
in this case is as follows:

$\underbrace{2\ell,\ldots,2\ell}_{\frac{(h-2)(h-1)}{2}},\underbrace{\ell+q,\ldots,\ell+q}_{h-1}.$

This implies by Theorem 3.2 that
$\beta_{i,j}(I_{B_{\underline{\ell},h}})=\beta_{i,j}(J)$ for all $i,j$.

By Corollary 2.10 together with Proposition 4.3, we have $\beta_{i,j}(J)=0$
for $j\notin\\{\ell(i+2),\ell(i+1)+q\\}$; and

$\displaystyle\beta_{i,\ell(i+2)}(J)$
$\displaystyle=\sum_{k=1}^{\frac{(h-1)(h-2)}{2}}\binom{r_{k}}{i}=\binom{0}{i}+2\binom{1}{i}+3\binom{2}{i}+\cdots+(h-2)\binom{h-3}{i}$
$\displaystyle=\sum_{q=1}^{h-2}q\binom{q-1}{i}=(i+1)\sum_{q=1}^{h-2}\binom{q}{i+1}=(i+1)\binom{h-1}{i+2};$

$\beta_{i,\ell
i+\ell+q}(J)=\sum_{k=\frac{(h-1)(h-2)}{2}+1}^{\frac{h(h-1)}{2}}\binom{r_{k}}{i}=(h-1)\binom{h-2}{i}.\qquad\qquad\qquad\qquad\qquad$

$\square$

##  5\. Graphs $B_{\ell,h}^{s,t}$ and their toric ideals

Let $B_{\ell,h}$ denote the graph $B_{\underline{\ell},h}$ defined in the
preceding section if $\underline{\ell}=(\ell,\ldots,\ell)$. In this section,
we construct a new family of graphs by attaching two odd cycles of length
$2s+1$ and $2t+1$ at the vertex $y_{1}$ and $y_{2}$ of $B_{\ell,h}$
respectively, which we denote by $B_{\ell,h}^{s,t}$. The main objective of our
computations is to determine the Betti numbers of the edge ring
$\mathbb{K}[B_{\ell,h}^{s,t}]$. If $\ell=s=t=1$, the graph $B_{\ell,h}^{s,t}$
is exactly the graph $G_{h}$ defined in [4, Definition 3.1] and thus the
findings in this section can be seen as extensions of certain existing results
in [4].

######  Definition 5.1.

Let $\ell,s,t\geq 1$ and $h\geq 2$ be integers. The graph $B_{\ell,h}^{s,t}$
is the graph which has the vertex set:

$V=V_{\ell,h}\cup\\{z_{1},\cdots,z_{2s},w_{1},\cdots,w_{2t}\\}$

and the edge set

$\displaystyle E$
$\displaystyle=E_{\ell,h}\cup\\{\\{y_{1},z_{1}\\},\\{z_{i},z_{i+1}~{}|~{}1\leq
i\leq 2s-1\\},\\{z_{2s},y_{1}\\}\\}$
$\displaystyle\quad\cup\\{\\{y_{2},w_{1}\\},\\{w_{i},w_{i+1}~{}|~{}1\leq i\leq
2t-1\\},\\{w_{2t},y_{2}\\}\\}.$

We label the edge of $B_{\ell,h}^{s,t}$ as follows:
$e_{1,j}=\\{y_{1},x_{1,j}\\},e_{i,j}=\\{x_{i-1,j},x_{i,j}\\},e_{2\ell,j}=\\{y_{2},x_{2\ell,j}\\}$,
for $i\in\\{2,\ldots,2\ell-1\\}$ and $j\in\\{1,\ldots,h\\}$.
$f_{1}=\\{y_{1},z_{1}\\},f_{2}=\\{z_{1},z_{2}\\},\ldots$,
$f_{2s}=\\{z_{2s-1},z_{2s}\\},f_{2s+1}=\\{z_{2s},y_{1}\\}$,
$g_{1}=\\{y_{2},w_{1}\\},g_{2}=\\{w_{1},w_{2}\\},\ldots,g_{2t}=\\{w_{2t-1},w_{2t}\\}$,
$g_{2t+1}=\\{w_{2t},y_{2}\\}$.

$e_{1,1}$$e_{1,2}$$e_{1,3}$$e_{1,h-2}$$e_{1,h-1}$$e_{1,h}$$e_{2,1}$$e_{2,2}$$e_{2,3}$$e_{2,h-2}$$e_{2,h-1}$$e_{2,h}$$e_{2\ell_{1}-1,1}$$e_{2\ell_{2}-1,2}$$e_{2\ell_{3}-1,3}$$e_{2\ell_{h-2}-1,h-2}$$e_{2\ell_{h-1}-1,h-1}$$e_{2\ell_{h}-1,h}$$e_{2\ell_{1},1}$$e_{2\ell_{2},2}$$e_{2\ell_{3},3}$$e_{2\ell_{h-2},h-2}$$e_{2\ell_{h-1},h-1}$$e_{2\ell_{h},h}$$y_{1}$$y_{2}$$f_{1}$$f_{2}$$f_{2s+1}$$f_{2s}$$e_{2\ell,h}$$g_{1}$$g_{2}$$g_{2t+1}$$g_{2t}$Fig.
2: The graph $B_{\ell,h}^{s,t}$

Let $\prec$ denote the graded reverse lexicographic monomial order on the
polynomial ring
$\mathbb{K}[e_{1,1},\cdots,e_{1,h},\cdots,e_{2\ell,1},\cdots,e_{2\ell,h},f_{1},\cdots,f_{2s+1},g_{1},\cdots,g_{2t+1}]$,
where the variables are ordered in the following way:

$\begin{split}&e_{1,1}\succ\cdots\succ e_{1,h}\succ e_{3,1}\succ\cdots\succ
e_{3,h}\succ\cdots\succ e_{2\ell-1,1}\cdots\succ e_{2\ell-1,h}\succ
g_{1}\succ\cdots\succ g_{2t+1}\\\ &\succ f_{1}\succ\cdots\succ f_{2s+1}\succ
e_{2,1}\succ\cdots\succ e_{2,h}\succ e_{4,1}\succ\cdots\succ
e_{4,h}\succ\cdots\succ e_{2\ell,1}\succ\cdots\succ e_{2\ell,h}.\end{split}$

For simplicity, we set

$Y_{1,i}=\prod\limits_{p=1}^{l}e_{2p-1,i},~{}Y_{2,j}=\prod\limits_{p=1}^{l}e_{2p,j},~{}U=\prod\limits_{p=1}^{s}f_{2p}\prod\limits_{p=1}^{t+1}g_{2p-1}~{}\mathrm{and}~{}V=\prod\limits_{p=1}^{s+1}f_{2p-1}\prod\limits_{p=1}^{t}g_{2p}.$

Here, $1\leq i<j\leq h$. Note that
$\mathrm{deg}(Y_{1,i})=\mathrm{deg}(Y_{2,j})=\ell$ and
$\mathrm{deg}(U)=\mathrm{deg}(V)=s+t+1$.

######  Lemma 5.2.

Let $\ell,r,s\geq 1$ and $h\geq 2$ be integers. Then
$\mathcal{G}_{\ell,h}^{s,t}=\mathcal{G}_{1}\cup\mathcal{G}_{2}\cup\mathcal{G}_{3}$
is the universal Gröbner basis of the toric ideal $I_{B_{\ell,h}^{s,t}}$,
where

$\displaystyle{\mathcal{G}_{1}}=$
$\displaystyle~{}\\{Y_{1,j}Y_{2,i}-Y_{1,i}Y_{2,j}\colon\;1\leq i<j\leq h\\};$
$\displaystyle\mathcal{G}_{2}=$
$\displaystyle~{}\\{Y_{1,i}Y_{1,j}U-Y_{2,i}Y_{2,j}V\colon\;1\leq i<j\leq
h\\};$ $\displaystyle\mathcal{G}_{3}=$
$\displaystyle~{}\\{Y_{1,i}^{2}U-Y_{2,i}^{2}V\colon\;1\leq i\leq h\\}.$

In particular, the minimal generating set of
$\mathrm{in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ is

$\\{Y_{1,j}Y_{2,i}\colon\;1\leq i<j\leq
h\\}\cup\\{Y_{1,i}Y_{1,j}U\colon\;1\leq i<j\leq
h\\}\cup\\{Y_{1,i}^{2}U\colon\;1\leq i\leq h\\}.$

###### Proof.

According to [16, Lemma 3.2], the set of primitive even closed walk
$\mathcal{PC}(B_{\ell,h}^{s,t})$ is the disjoint union $W_{1}\cup W_{2}\cup
W_{3}$, where

$\displaystyle W_{1}$
$\displaystyle=\\{e_{1,i},e_{2,i},\cdots,e_{2\ell,i},e_{2\ell,j},\cdots,e_{2,j},e_{1,j}\colon\;1\leq
i<j\leq h\\}$ $\displaystyle W_{2}$
$\displaystyle=\\{f_{1},\cdots,f_{2s+1},e_{1,i},e_{2,i},\cdots,e_{2\ell,i},g_{1},\cdots,g_{2t+1},e_{2\ell,j},\cdots,e_{2,j},e_{1,j}\colon\;1\leq
i<j\leq h\\}$ $\displaystyle W_{3}$
$\displaystyle=\\{f_{1},\cdots,f_{2s+1},e_{1,i},e_{2,i},\cdots,e_{2\ell,i},g_{1},\cdots,g_{2t+1},e_{2\ell,i},\cdots,e_{2,i},e_{1,i}\colon\;1\leq
i\leq h\\}$

This implies $\mathcal{G}_{\ell,h}^{s,t}$ is the universal Gröbner basis of
$I_{B_{\ell,h}^{s,t}}$. In addition, it is straightforward to get the minimal
generators of $\mathrm{in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ from
$\mathcal{G}_{\ell,h}^{s,t}$ under the above monomial order. $\square$

We label the generators of $\mathrm{in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ one by
one as follows:

$\begin{split}&m_{1}=Y_{1,h}Y_{2,h-1},\quad m_{2}=Y_{1,h}Y_{2,h-2},\quad
m_{3}=Y_{1,h-1}Y_{2,h-2},\ldots,m_{\frac{h(h-1)}{2}}=Y_{1,2}Y_{2,1},\\\
&m_{\frac{h(h-1)}{2}+1}=Y_{1,h}Y_{1,h-1}U,\quad
m_{\frac{h(h-1)}{2}+2}=Y_{1,h}Y_{1,h-2}U,\ldots,m_{h(h-1)}=Y_{1,2}Y_{1,1}U,\\\
&m_{h(h-1)+1}=Y_{1,h}^{2}U,\quad\ldots,\quad
m_{h^{2}}=Y_{1,1}^{2}U.\end{split}$

Then $\mathrm{in}_{\prec}(I_{B_{\ell,h}^{s,t}})=(m_{1},\ldots,m_{h^{2}})$ and
we will show that $\mathrm{in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ has
$\ell$-regular quotients with respect to this order of generators.

######  Proposition 5.3.

Let $\ell,s,t\geq 1$ and $h\geq 2$ be integers. The initial ideal
$\mathrm{in}_{\prec}(I_{B_{\ell,h}^{s,t}})=(m_{1},\ldots,m_{h^{2}})$ has
$\ell$-regular quotients with respect to this order of generators. More
precisely, the colon ideal $(m_{1},\ldots,m_{k-1}):m_{k}$ is generated by a
regular sequence of length $r_{k}$ in a single degree $\ell$ for
$k=2,\ldots,h^{2}$. Here,

$(r_{2},\ldots,r_{\frac{h(h-1)}{2}})=(\underbrace{{1,1}}_{2},\underbrace{2,2,2}_{3},\ldots,\underbrace{h-2,h-2,\ldots,h-2}_{h-1});$

If $\frac{h(h-1)}{2}+1\leq k\leq h^{2}$, then $h-1\leq r_{k}\leq 2h-2$.
Moreover, for any $p$ with $h-1\leq p\leq 2h-2$, one has

$~{}|\\{\frac{h(h-1)}{2}+1\leq k\leq h^{2}\colon\;r_{k}=p\\}|=2h-p-1.$

###### Proof.

Case 1. For $2\leq k\leq\frac{h(h-1)}{2}$, the result follows in a similar way
as in the proof of Proposition 4.3.

Case 2. If $k\in\\{\frac{h(h-1)}{2}+1,\cdots,h(h-1)\\}$, then
$m_{k}=Y_{1,j}Y_{1,i}U$ for some $1\leq i<j\leq h$. If $j<h$, then

$\displaystyle L_{k}$
$\displaystyle=(Y_{1,h}Y_{2,h-1},\ldots,Y_{1,2}Y_{2,1},Y_{1,h}Y_{1,h-1}U,\ldots,Y_{1,j+1}Y_{1,i}U):Y_{1,j}Y_{1,i}U$
$\displaystyle=(Y_{1,i+1},\ldots,\widehat{Y_{1,j}},\ldots,Y_{1,h},Y_{2,1},\ldots,Y_{2,j-1}).$

Here, $\widehat{Y_{1,j}}$ represents the removal of the element $Y_{1,j}$ from
the set of generators. Thus we get $r_{k}=h-i+j-2$. If $j=h$, then

$\displaystyle L_{k}$
$\displaystyle=(Y_{1,h}Y_{2,h-1},\ldots,Y_{1,2}Y_{2,1},Y_{1,h}Y_{1,h-1}U,\ldots,Y_{1,i+2}Y_{1,i+1}U):Y_{1,h}Y_{1,i}U$
$\displaystyle=(Y_{1,i+1},\ldots,\ldots,Y_{1,h-1},Y_{2,1},\ldots,Y_{2,h-1}).$

Thus we also get $r_{k}=2h-i-2=h+j-i-2$.

Case 3. If $k\in\\{h(h-1)+1,\cdots,h^{2}\\}$, then $m_{k}=Y_{1,i}^{2}U$ for
some $1\leq i\leq h$, and

$\displaystyle L_{k}$
$\displaystyle=(Y_{1,h}Y_{2,h-1},\ldots,Y_{1,2}Y_{1,1}U,Y_{1,h}^{2}U,\ldots,Y_{1,i+1}^{2}U):Y_{1,i}^{2}U$
$\displaystyle=(Y_{1,1},\ldots,\widehat{Y_{1,i}},\ldots,Y_{1,h},Y_{2,1},\ldots,Y_{2,i-1}).$

Here, $\widehat{Y_{1,i}}$ represents the removal of the element $Y_{1,i}$ from
the set of generators. It follows that $r_{k}=h+i-2$.

By the above discussion on the three cases, it follows that ${\rm
in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ has $\ell$-regular quotients with respect
to the given order. Integrating the discussions on case 2 and case 3, we see
that if $\frac{h(h-1)}{2}+1\leq k\leq h^{2}$, then $h-1\leq r_{k}\leq 2h-2$.
Moreover, for any $h-1\leq p\leq 2h-2$, one has

$\displaystyle|\\{\frac{h(h-1)}{2}+1\leq k\leq h^{2}\colon\;r_{k}=p\\}|$
$\displaystyle=1+|\\{(i,j)\colon\;j-i=p-h+2,1\leq i<j\leq h\\}|$
$\displaystyle=2h-p-1.$

This completes the proof. $\square$

######  Theorem 5.4.

Fix integers $\ell,s,t\geq 1$ and $h\geq 2$. Then the graded Betti numbers of
$I_{B_{\ell,h}^{s,t}}$ and ${\rm in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ are equal,
and the explicit graded Betti numbers are:

$\beta_{i,j}(I_{B_{\ell,h}^{s,t}})=\left\\{\begin{array}[]{ll}(i+1)\binom{h}{i+2},&\hbox{$j=\ell(i+2)$;}\\\
\binom{2h}{i+2}-(i+3)\binom{h}{i+2},&\hbox{$j=\ell(i+2)+s+t+1$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$

###### Proof.

Proposition 5.3 shows ${\rm in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ has
$\ell$-regular quotients with respect to the ordered set
$\\{m_{1},\ldots,m_{h^{2}}\\}$. Note that
$\mathrm{deg}(m_{1})=\cdots=\mathrm{deg}(m_{\frac{h(h-1)}{2}})=2\ell$ and
$\mathrm{deg}(m_{\frac{h(h-1)}{2}+1})=\cdots=\mathrm{deg}(m_{h^{2}})=2\ell+s+t+1$.
Since $B_{\ell,h}$ is an induced graph of $B_{\ell,h}^{s,t}$ and since
$\mathrm{in}_{\prec}(I_{B_{\ell,h}})=(m_{1},\ldots,m_{\frac{h(h-1)}{2}})$, it
follows that $\beta_{i,j}(I_{B_{\ell,h}^{s,t}})=\beta_{i,j}({\rm
in}_{\prec}(I_{B_{\ell,h}^{s,t}}))$ for all $i,j\geq 0$ by Theorem 3.2.

We still need to calculate the graded betti number of ${\rm
in}_{\prec}(I_{B_{\ell,h}^{s,t}})$ for all $i,j\geq 0$. By Theorem 2.7, we see
that $\beta_{i,j}({\rm in}_{\prec}(I_{B_{\ell,h}^{s,t}}))=0$ for
$j\notin\\{\ell(i+2),\ell(i+2)+s+t+1\\}$, and that $\beta_{i,j}({\rm
in}_{\prec}(I_{B_{\ell,h}^{s,t}}))=\beta_{i,j}({\rm
in}_{\prec}(I_{B_{\underline{\ell},h}}))=(i+1)\binom{h}{i+2}$ if
$j=\ell(i+2)$.

If $j=\ell(i+2)+s+t+1$, it follows from Proposition 5.3 that

$\begin{split}\beta_{i,j}({\rm
in}_{\prec}(I_{B_{\ell,h}^{s,t}}))&=\sum\limits_{p=h-1}^{2h-2}(2h-1-p)\binom{p}{i}\\\
&=\sum\limits_{p=h-1}^{2h-2}2h\binom{p}{i}-\sum\limits_{p=h-1}^{2h-2}(i+1)\binom{p+1}{i+1}\\\
&=2h\binom{2h-1}{i+1}-2h\binom{h-1}{i+1}-(i+1)\binom{2h}{i+2}+(i+1)\binom{h}{i+2}\\\
&=\binom{2h}{i+2}-(i+3)\binom{h}{i+2}.\end{split}$

Here, we employ the equation $\sum_{j=0}^{n}\binom{j}{i}=\binom{n+1}{i+1}$
multiple times. Therefore, the proof is concluded. $\square$

######  Corollary 5.5.

Fix integers $\ell,s,t\geq 1$ and $h\geq 2$. Then we have

  1. (1)

${\rm{pdim}}(I_{B_{\ell,h}^{s,t}})=2h-2$;

  2. (2)

${\rm{reg}}(I_{B_{\ell,h}^{s,t}})=2\ell h+s+t-2h-1$;

  3. (3)

${\rm{deg}}~{}h_{\mathbb{K}[B_{\ell,h}^{s,t}]}(t)=(2\ell-1)h+s+t+1$.

###### Proof.

The assertions (1) and (2) follow from either Theorem 5.4 or Proposition 5.3
directly. In the same vein, we have $\beta_{2h-2,2\ell
h+s+t+1}(I_{B_{\ell,h}^{s,t}})$ is the unique extremal Betti number of
$I_{B_{\ell,h}^{s,t}}$ and hence $\beta_{2h-1,2\ell
h+s+t+1}(\mathbb{K}[B_{\ell,h}^{s,t}])$ is the unique extremal Betti number of
$\mathbb{K}[B_{\ell,h}^{s,t}]$.

Since the graph $B_{\ell,h}^{s,t}$ has $2\ell h+2s+2t+2$ edges and $2\ell
h+2s+2t-h+2$ vertices, we have $\mathbb{K}[E(B_{\ell,h}^{s,t})]$ is a
polynomial ring in $2\ell h+2s+2t+2$ variables, and
${\rm{dim}}(\mathbb{K}[B_{\ell,h}^{s,t}])=2\ell h+2s+2t-h+2$ by [19, Corollary
10.1.21]. It follows that

${\rm{deg}}~{}h_{\mathbb{K}[B_{\ell,h}^{s,t}]}(t)=2\ell h+s+t+1+2\ell
h+2s+2t-h+2-(2\ell h+2s+2t+2),$

as required. $\square$

Acknowledgment: This project is supported by NSFC (No. 11971338)

Data availability: The data used to support the findings of this study are
included within the article.

Statement: On behalf of all authors, the corresponding author states that
there is no conflict of interest.

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