# Powers of generalized binomial edge ideals of path graphs

Yi-Huang Shen  CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of
Mathematical Sciences, University of Science and Technology of China, Hefei,
Anhui, 230026, P.R. China<EMAIL_ADDRESS>and  Guangjun Zhu∗ School of
Mathematical Sciences, Soochow University, Suzhou, Jiangsu, 215006, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this article, we study the powers of the generalized binomial edge ideal
$\mathcal{J}_{K_{m},P_{n}}$ of a path graph $P_{n}$. We explicitly compute
their regularities and determine the limit of their depths. We also show that
these ordinary powers coincide with their symbolic powers. Additionally, we
study the Rees algebra and the special fiber ring of
$\mathcal{J}_{K_{m},P_{n}}$ via Sagbi basis theory. In particular, we obtain
exact formulas for the regularity of these blowup algebras.

∗ Corresponding author

2020 Mathematics Subject Classification. Primary 13C15, 13P10; Secondary
05E40, 13F20

Keywords: Regularity, depth, generalized binomial edge ideal, path graph, Rees
algebra, special fiber ring

##  1\. Introduction

Let $m$ and $n$ be two positive integers and $[m]$ be the set
$\\{1,2,\ldots,m\\}$. In addition, let
$S={\mathbb{K}}[\bm{X}]\coloneqq{\mathbb{K}}[x_{i,j}:i\in[m],j\in[n]]$ be the
polynomial ring in $m\times n$ variables over a field ${\mathbb{K}}$. In
[MR3290687], Ene et al. introduced the binomial edge ideal of a pair of
graphs. Specifically, let $G_{1}$ and $G_{2}$ be simple graphs on vertex sets
$[m]$ and $[n]$ respectively. Suppose that $e=\\{i,j\\}\in E(G_{1})$ and
$f=\\{k,l\\}\in E(G_{2})$ are two edges with $i<j$ and $k<l$. Then, one can
associate a $2$-minor

$p_{(e,f)}=[i,j\,|\,k,l]\coloneqq x_{i,k}x_{j,l}-x_{i,l}x_{j,k}$

to the pair $(e,f)$. The _binomial edge ideal of the pair $(G_{1},G_{2})$_ is
defined as follows

$\mathcal{J}_{G_{1},G_{2}}\coloneqq(p_{(e,f)}:e\in E(G_{1}),f\in E(G_{2})).$

This is a generalization of the classical _binomial edge ideals_ in
[MR2669070, MR2782571], if one of $G_{1}$ and $G_{2}$ is the complete graph
$K_{2}$. Meanwhile, the _ideals generated by adjacent minors_ in [MR1627343]
turn out to be the binomial edge ideals of a pair of path graphs.

In the last decade, researchers have tried to understand the connection
between the algebraic properties of $\mathcal{J}_{G_{1},G_{2}}$ and the
combinatorial properties of $G_{1}$ and $G_{2}$. In [MR3290687], Ene et al.
proved that $\mathcal{J}_{G_{1},G_{2}}$ is a radical ideal if and only if
either $G_{1}$ or $G_{2}$ is a complete graph. Furthermore,
$\mathcal{J}_{G_{1},G_{2}}$ is a prime ideal if and only if both $G_{1}$ and
$G_{2}$ are complete. The importance of the complete graph in the pair can be
further demonstrated by [MR3859970, Theorem 3.1] and [MR3040610, Theorems 1
and 10].

If $K_{m}$ is a complete graph with $m$ vertices and $G$ is a simple graph,
then $\mathcal{J}_{K_{m},G}$ is the _generalized binomial edge ideal_
associated with $G$, which was previously introduced by Rauh in [MR3011436]
for the study of conditional independence ideals. Inspired by the progress on
classical binomial edge ideals, researchers turn to the study of the
generalized binomial edge ideals. For example, Chaudhry and Irfan in
[MR4233116] proved that, for a block graph $G$, $\mathcal{J}_{K_{m},G}$ is
Cohen–Macaulay if and only if $\mathcal{J}_{K_{m},G}$ is unmixed if and only
if $G$ is a complete graph.

Blowup algebras are ubiquitous in commutative algebra and algebraic geometry.
The _Rees algebra_ $\mathcal{R}(I)\coloneqq\oplus_{k\geq 0}I^{k}T^{k}$ and the
_special fiber ring_
$\mathcal{F}(I)\coloneqq\mathcal{R}(I)/\mathfrak{m}\mathcal{R}(I)$ of an ideal
$I$ are two important blowup algebras that uniquely encode all powers of $I$.
In the case of binomial edge ideals, their properties have been explored by
several researchers. For example, in [MR4173994], Jayanthan et al. proved that
the Rees algebra of an almost complete intersection binomial edge ideal is
Cohen–Macaulay. In [MR4425287], Ene et al. showed that the Rees algebra of the
binomial edge ideal of a closed graph is Cohen–Macaulay. Later, Kumar in
[MR4405525] proved that the special fiber ring of the binomial edge ideal of a
closed graph is Koszul and normal Cohen–Macaulay.

In this article, we are interested in the Castelnuovo–Mumford regularity
(regularity for short) and the depth of powers of generalized binomial edge
ideals. It is well-known that if $I$ is a homogeneous ideal of a polynomial
ring $R$, then $\operatorname{reg}(R/I^{t})$ is asymptotically linear in $t$.
At the same time, $\operatorname{depth}(R/I^{t})$ is constant for sufficiently
large $t$ (cf. [MR1711319, MR1621961, MR0530808]). It is usually difficult to
determine when these phenomena begin. For this problem, the simplest case is
when $I$ is a quadratic squarefree monomial ideal, i.e., when $I$ can be
recognized as the edge ideal of a suitable graph. In this case, many results
have been achieved for simple classes such as forest graphs, cycle graphs,
bipartite graphs, and so on. In addition, a few results are known for the
binomial edge ideal of a graph. For example, in [JKS], Jayanthan et al. gave
an upper bound on the regularity of powers of almost complete intersection
binomial edge ideals using the quadratic sequence approach. Meanwhile, they
gave the exact formulas for the regularity of powers of binomial edge ideals
of several simple graphs such as cycle graphs, star graphs, and balloon
graphs. Recently, in [MR4544259], we gave explicit formulas for the regularity
of powers of binomial edge ideals which are almost complete intersections. At
the same time, in [MR4563443], Wang and Tang studied the depth of powers of
binomial edge ideals of complete bipartite graphs. Ene et al. in [MR4425287]
studied the regularity and the depth of powers of the binomial edge ideals of
connected closed graphs.

However, nearly nothing is known about the algebraic properties of powers of
generalized binomial edge ideals. In this paper we will start such a study by
considering the generalized binomial edge ideal $\mathcal{J}_{K_{m},P_{n}}$ of
the path graph $P_{n}$.

The article is organized as follows. In Section 2, we briefly review essential
definitions and terminology that we will need later. In Section 3, we show
that taking-initial-ideal commutes with taking-powers, when coming to
$\mathcal{J}_{K_{m},P_{n}}$. Using this fact, we study the regularity and the
depth of the powers of $\mathcal{J}_{K_{m},P_{n}}$ via the Sagbi basis theory.
We also show that the symbolic powers and ordinary powers of
$\mathcal{J}_{K_{m},P_{n}}$ coincide. In Section 4, we study the regularities
of the Rees algebra and the special fiber ring of $\mathcal{J}_{K_{m},P_{n}}$,
by considering the corresponding problems of their initial algebras. The final
computation builds on the combinatorial optimization of different flavors. In
the last section, we give applications of the previous results. Since the
philosophy of combinatorial pure subrings applies here, we can consider the
binomial edge ideal of a pair of graphs. In particular, we will give natural
bounds on the regularities of the powers of $\mathcal{J}_{K_{m},G}$ as well as
the two blowup algebras of this generalized binomial edge ideal, if $G$
contains an induced path $P_{n}$.

##  2\. Preliminaries

In this brief section, we provide a concise overview of some combinatorial
notions that will be employed throughout this paper. For a more comprehensive
treatment from an algebraic perspective, we refer the readers to [MR2724673,
MR3838370, MR3362802].

Let $G$ be a simple graph with the vertex set $V(G)$ and the edge set $E(G)$.
For a vertex $v$ of $G$, the set of all neighborhoods of $v$ is denoted by
$N_{G}(v)=\\{u\in V(G):\\{u,v\\}\in E(G)\\}$. A vertex $v$ is called a _leaf_
of $G$ if $N_{G}(v)$ has cardinality one and $v$ is _isolated_ if
$N_{G}(v)=\emptyset$.

For any subset $A$ of $V(G)$, let $G[A]$ denote the _induced subgraph_ of $G$
on the set $A$, i.e., for $u,v\in A$, $\\{u,v\\}\in E(G[A])$ if and only if
$\\{u,v\\}\in E(G)$. At the same time, we denote the induced subgraph of $G$
on the set $V(G)\setminus A$ by $G\setminus A$.

A subset $M\subset E(G)$ is a _matching_ of $G$ if $e\cap
e^{\prime}=\emptyset$ for all distinct edges $e$ and $e^{\prime}$ in $M$. The
_matching number_ of $G$, denoted by $\operatorname{match}(G)$, is the maximum
size of a matching in $G$. If $G$ is a bipartite graph having vertex
partitions $V_{1}$ and $V_{2}$, a _complete matching from $V_{2}$ to $V_{1}$_
is a matching in which there is one edge incident with every vertex in
$V_{2}$. In other words, every vertex in $V_{2}$ is matched against some
vertex in $V_{1}$. Whence, $\operatorname{match}(G)=|V_{2}|$.

A _walk_ $W$ of length $n$ in a graph $G$ is a sequence of vertices
$(w_{1},\ldots,w_{n},w_{n+1})$, such that $\\{w_{i},w_{i+1}\\}\in E(G)$ for
$1\leq i\leq n$. The walk $W$ is _closed_ if $w_{1}=w_{n+1}$. Furthermore, the
walk $W$ is called a _cycle_ if it is closed and the points
$w_{1},\ldots,w_{n}$ are distinct. At the same time, a _path_ is a walk where
all points are distinct. For simplicity, a path of length $n-1$ is denoted by
$P_{n}$, and a cycle of length $n$ is denoted by $C_{n}$.

##  3\. Powers of generalized binomial edge ideals of paths

In this section, we will study the regularity and the depth of the powers of
the generalized binomial edge ideal $\mathcal{J}_{K_{m},P_{n}}$, where $P_{n}$
is a path graph with $n$ vertices. Using Sagbi basis theory, we can turn to
the corresponding study of their initial ideals. However, to make this
approach work, we need to show first in Theorem 3.5 that taking-initial-ideal
commutes with taking-powers when coming to $\mathcal{J}_{K_{m},P_{n}}$. Since
the proof for this is rather technical, we need to make some preparations.
Throughout this paper, we will stay with the following setting:

######  Setting 3.1.

Let $m,n\geq 2$ be two integers. The polynomial ring
$S\coloneqq{\mathbb{K}}[x_{i,j}:i\in[m],j\in[n]]$ over a field ${\mathbb{K}}$
is endowed with the term order $\tau$, which is the lexicographic order on $S$
induced by the natural order

$x_{1,1}>x_{1,2}>\cdots>x_{1,n}>x_{2,1}>x_{2,2}>\cdots>x_{2,n}>\cdots>x_{m,1}>x_{m,2}>\cdots>x_{m,n}.$

Let $P_{n}$ be the path on the set $[n]$ whose edge set is
$E(P_{n})=\Set{\\{i,i+1\\}:i\in[n-1]}$. Furthermore, let $H$ be the graph on
the set $\Set{x_{i,j}:i\in[m],j\in[n]}$ with

$E(H)=\Set{\\{x_{i,j},x_{i^{\prime},j+1}\\}:1\leq i<i^{\prime}\leq m,1\leq
j\leq n-1}.$

For $k\in[n-1]$, let $H_{k}$ be the induced subgraph of $H$ on the set
$\Set{x_{i,j}:i\in[m],j\in\\{k,k+1\\}}$.

Regarding the graph $H$ in 3.1, we have the following two observations:

######  Remark 3.2.

  1. a

If $G$ is a simple graph on the set $[n]$, the Gröbner basis of the
generalized binomial edge ideal $\mathcal{J}_{K_{m},G}$ with respect to the
term order $\tau$ was computed in [MR3011436, Theorem 2]. In particular, the
initial ideal $\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})$ equals the
ideal

$(x_{i,j}x_{i^{\prime},j+1}:1\leq i<i^{\prime}\leq m,1\leq j\leq n-1)$

in $S$. It is clear that this is the edge ideal $I(H)$ of the graph $H$.

  2. b

The graph $H$ is bipartite with respect to the bipartition $V_{1}\sqcup
V_{2}$, where

$V_{1}=\Set{x_{i,j}\in\bm{X}:i\in[m],\text{$j$ is odd}}\qquad\text{and}\qquad
V_{2}=\Set{x_{i,j}\in\bm{X}:i\in[m],\text{$j$ is even}}.$

When $m\geq 3$, $H$ has $3$ connected components, where $x_{m,1}$ and
$x_{1,n}$ are isolated vertices. If instead $m=2$, then $H$ has $2+(n-1)=n+1$
connected components, where $x_{m,1}$ and $x_{1,n}$ are still isolated
vertices; see Figure 1.

(A) $m=2$ case

(B) $m=4$ case

Figure 1. Graph $H$

The proof of Theorem 3.5 depends on the presentation ideal of the Rees algebra
$\mathcal{R}(I(H))$ of the edge ideal $I(H)$ and the “lifts” of the Sagbi
basis. Suppose that $\mathcal{G}(I(H))=\\{f_{1},\ldots,f_{q}\\}$ is the
minimal monomial generating set of $I(H)$. Then, there exists a canonical
homomorphism from the polynomial ring $B\coloneqq S[T_{1},\dots,T_{q}]$ to the
Rees algebra $\mathcal{R}(I(H))\coloneqq S[f_{1}T,\dots,f_{q}T]\subset S[T]$,
induced by $T_{i}\mapsto f_{i}T$. Let
$\operatorname{deg}(T_{1})=\cdots=\det(T_{q})=\operatorname{deg}(T)=1$, and
$\operatorname{deg}(x_{i,j})=0$ for every $x_{i,j}\in\bm{X}$. Then, this map
is a graded homomorphism of $S$-algebras. Its kernel $J$ will be called the
_presentation ideal_ of $\mathcal{R}(I(H))$ with respect to
$\mathcal{G}(I(H))$. This is a graded ideal and $\mathcal{R}(I(H))$ has the
_presentation_ $\mathcal{R}(I(H))=B/J$.

Similarly, there is a canonical homomorphism from the polynomial ring
$B^{\prime}\coloneqq{\mathbb{K}}[T_{1},\dots,T_{q}]$ to the special fiber ring
$\mathcal{F}(I(H))\cong{\mathbb{K}}[H]$, induced by $T_{i}\mapsto f_{i}$. The
kernel ideal $J^{\prime}$ of this map is called the _presentation ideal_ of
$\mathcal{F}(I)$. It leads to the _presentation_
$\mathcal{F}(I(H))=B^{\prime}/J^{\prime}$.

In addition, let $W=(w_{1},w_{2},\dots,w_{2s+1}=w_{1})$ be an even closed walk
in $H$ and suppose that $e_{j}=\\{w_{j},w_{j+1}\\}$ for each $j$. We will
write $T_{W^{+}}-T_{W^{-}}$ for the binomial $T_{e_{1}}T_{e_{3}}\cdots
T_{e_{2s-1}}-T_{e_{2}}T_{e_{4}}\cdots T_{e_{2s}}$ in $J$. We are mostly
interested in the case where $W$ is a primitive cycle. Recall that a _cycle_
$C$ is a closed walk with distinct vertices. A _chord_ of a cycle $C$ in a
graph $G$ is an edge of $G$ that connects two non-adjacent vertices of $C$. A
cycle without chords is called _primitive_. Binomials from primitive cycles
are important for describing the presentation ideals.

######  Lemma 3.3 ([MR3362802, Proposition 10.1.14, Theorem 10.1.15]).

Let $I$ be the edge ideal of a bipartite graph.

  1. a

Suppose that $\mathcal{R}(I)=B/J$ is the presentation of the Rees algebra
$\mathcal{R}(I)$. Then $J=BJ_{1}+BP$, where $J_{1}$ is the degree $1$ part of
the graded ideal $J$, and

$P\coloneqq\Set{T_{w^{+}}-T_{w^{-}}:\text{$w$ is a primitive cycle in $H$}}.$

  2. b

Suppose that $\mathcal{F}(I)=B^{\prime}/J^{\prime}$ is the presentation of the
special fiber ring $\mathcal{F}(I)$. Then, $J^{\prime}$ is minimally generated
by the set $P$.

Recall that a bipartite graph is _chordal bipartite_ if every cycle of length
at least six has a chord in it. In other words, the length of every primitive
cycle of this bipartite graph is $4$.

######  Lemma 3.4.

Let $H,H_{1},\dots,H_{n-1}$ be as in 3.1. Then, $H$ is a chordal bipartite
graph.

###### Proof.

Let $C=(c_{1},c_{2},\dots,c_{2s+1}=c_{1})$ be a primitive cycle in $H$ with
$s\geq 2$. By abuse of notation, if $k>2s$, we will identify $c_{k}$ with
$c_{k-2s}$, and if $k<1$, we will identify $c_{k}$ with $c_{k+2s}$. We have
the following two cases. Figure 2 is helpful in understanding the arguments.

  1. a

Suppose that $C$ is a cycle in $H_{k}$ for some $k$. Without loss of
generality, we can assume that $k=1$ and $c_{p}=x_{i_{p},j_{p}}$ for each $p$.
Suppose also that $j_{p}=1$ if $p$ is odd, and $j_{p}=2$ if $p$ is even. We
can also assume that $i_{1}=\min\\{i_{1},i_{3},\dots,i_{2s-1}\\}$. Then,
$i_{1}<i_{2}$ and $i_{1}<i_{2s}$. By symmetry, we assume that $i_{2s}<i_{2}$.
Note that $i_{2s-1}<i_{2s}$ at this time. Therefore, if $s>2$, then $C$ has a
chord $\\{x_{i_{2s-1},1},x_{i_{2},2}\\}$, which violates the primitivity of
$C$.

  2. b

Suppose there is no $k$ such that $C$ is a cycle in $H_{k}$. Then, without
loss of generality, we can assume that $c_{1}=x_{i_{1},1}$. For each $k$ with
$c_{k}=x_{i_{k},1}$, we say that $k$ is _marginal_. Note that in this case, we
have $c_{k\pm 1}=x_{i_{k\pm 1},2}$ with $i_{k-1}\neq i_{k+1}$. If
$i_{k-1}<i_{k+1}$, we say that $k$ is _extendable_ if $c_{k+2}=x_{i_{k+2},3}$.
If instead $i_{k-1}>i_{k+1}$, we say that $k$ is _extendable_ if
$c_{k-2}=x_{i_{k-2},3}$.

    1. i

Suppose there is a marginal $k$ that is extendable. Without loss of
generality, we assume that $i_{k-1}<i_{k+1}$. Note that in this case
$i_{k}<i_{k-1}<i_{k+1}<i_{k+2}$. If $s>2$, then $C$ has a chord
$\\{x_{i_{k-1},2},x_{i_{k+2},3}\\}$, which violates the primitivity of $C$.

    2. ii

Suppose that there is no marginal $k$ that is extendable. Then $C$ is a cycle
in $H_{1}$, a contradiction.

In short, $s=2$ and $C$ is a cycle of length $4$. ∎

$v_{2s-1}$$v_{1}$$v_{2}$$v_{2s}$ (A) Case a

$v_{k}$$v_{k+1}$$v_{k-1}$$v_{k+2}$ (B) Case b

Figure 2. Basic patterns

The following result is indispensable for establishing the regularity result
in Theorem 3.11. It also provides a class of nice ideals sought in [MR1477608,
Section 1].

######  Theorem 3.5.

Under the assumptions in 3.1, we have
$(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}=\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}^{t})$
for all $t\geq 1$.

###### Proof.

We define a term order $\tau^{\prime}$ on $S[T]$ as follows: for any monomials
$u$ and $v$ in $S$, and non-negative integers $i$ and $j$, we set

$uT^{i}<_{\tau^{\prime}}vT^{j}\qquad\Leftrightarrow\qquad
i<j\quad\text{or}\quad i=j\quad\text{and}\quad u<_{\tau}v.$

It follows from [MR1390693, Theorem 2.7] that
$(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}=\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}^{t})$
for all $t\geq 1$, if and only if
$\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))=\operatorname{in}_{\tau^{\prime}}(\mathcal{R}(\mathcal{J}_{K_{m},P_{n}}))$.
In other words, the set

$\mathcal{G}\coloneqq\bm{X}\cup\left\\{[i,j\,|\,k,k+1]T:1\leq i<j\leq m\text{
and }k\in[n-1]\right\\}$

forms a _Sagbi basis_ of $\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ with respect
to $\tau^{\prime}$. For brevity, write $\mathcal{G}=\\{g_{1},\dots,g_{p}\\}$,
and let $F_{1},\dots,F_{q}$ be a system of binomial relations of the affine
semigroup ring
${\mathbb{K}}[\operatorname{in}_{\tau^{\prime}}(g):g\in\mathcal{G}]$. It
follows from [MR1390693, Proposition 1.1] that we have to find
$\lambda_{\alpha}^{(j)}\in{\mathbb{K}}$ such that

$F_{j}(g_{1},\dots,g_{p})=\sum_{\bm{\alpha}}\lambda_{\alpha}^{(j)}g^{\bm{\alpha}}\qquad\text{with
$\operatorname{in}_{\tau^{\prime}}(g^{\bm{\alpha}})\leq\operatorname{in}_{\tau^{\prime}}(F_{j}(g_{1},\dots,g_{p}))$}$

for all $\lambda_{\bm{\alpha}}^{(j)}\neq 0$, where, as usual,
$g^{\bm{\alpha}}\coloneqq g_{1}^{\alpha_{1}}\cdots g_{p}^{\alpha_{p}}$ for a
multi-index $\bm{\alpha}=(\alpha_{1},\dots,\alpha_{p})$. Note that to verify
this condition, it suffices to show that all the
$\operatorname{in}_{\tau^{\prime}}(g^{\bm{\alpha}})$ are distinct. We can
check with ease that this is satisfied in the following eqs. 1, 2, 3, 4 and 5.

Note that
${\mathbb{K}}[\operatorname{in}_{\tau^{\prime}}(g):g\in\mathcal{G}]\cong\mathcal{R}(I(H))$
for the graph $H$ introduced in 3.1. It follows from Lemma 3.3 that we have
the following two cases:

  1. A

First, suppose that $F=F_{j}$ is a binomial generator in $J_{1}$. Since the
edge ideal $I(H)$ is quadratic, we have two subcases.

    1. i

Suppose $F=uvT_{e_{2}}-u^{\prime}v^{\prime}T_{e_{1}}$ such that
$e_{1}\coloneqq\\{u,v\\}$, $e_{2}\coloneqq\\{u^{\prime},v^{\prime}\\}$ and
$e_{1}\cap e_{2}=\emptyset$. We can assume that $u=x_{i,k}$, $v=x_{j,k+1}$,
$u^{\prime}=x_{i^{\prime},k^{\prime}}$ and
$v^{\prime}=x_{j^{\prime},k^{\prime}+1}$ with $1\leq i<j\leq m$, $1\leq
i^{\prime}<j^{\prime}\leq m$, and $k,k^{\prime}\in[n-1]$. In this subcase, we
have the simple equality:

$\displaystyle
x_{i,k}x_{j,k+1}[i^{\prime},j^{\prime}\,|\,k^{\prime},k^{\prime}+1]-x_{i^{\prime},k^{\prime}}x_{j^{\prime},k^{\prime}+1}[i,j\,|\,k,k+1]$
$\displaystyle=\,$ $\displaystyle
x_{i,k+1}x_{j,k}[i^{\prime},j^{\prime}\,|\,k^{\prime},k^{\prime}+1]-x_{i^{\prime},k^{\prime}+1}x_{j^{\prime},k^{\prime}}[i,j\,|\,k,k+1].$
(1)

    2. ii

Suppose $uT_{e_{2}}-vT_{e_{1}}$ such that $e_{1}\coloneqq\\{u,r\\}$ and
$e_{2}\coloneqq\\{v,r\\}$. We have three additional subcases.

      1. a

Suppose $r=x_{i_{1},k}$, $u=x_{i_{2},k+1}$, and $v=x_{i_{3},k+1}$, with $1\leq
i_{1}<i_{2}<i_{3}\leq m$ and $k\in[n-1]$. In this subcase, we have the simple
equality:

$\qquad\qquad\qquad
x_{i_{2},k}[i_{1},i_{3}\,|\,k,k+1]-x_{i_{1},k}[i_{2},i_{3}\,|\,k,k+1]=x_{i_{3},k}[i_{1},i_{2}\,|\,k,k+1].$
(2)

      2. b

Suppose $r=x_{i_{1},k+1}$, $u=x_{i_{2},k}$, and $v=x_{i_{3},k}$, where $1\leq
i_{1}<i_{2}<i_{3}\leq m$ and $k\in[n-1]$. This subcase is similar to the
previous one.

      3. c

Suppose $u=x_{i_{1},k}$, $r=x_{i_{2},k+1}$, and $v=x_{i_{3},k+2}$, with $1\leq
i_{1}<i_{2}<i_{3}\leq m$ and $k\in[n-2]$. In this subcase, we have the simple
equality:

$\displaystyle
x_{i_{3},k+2}[i_{1},i_{2}\,|\,k,k+1]-x_{i_{1},k}[i_{2},i_{3}\,|\,k+1,k+2]$
$\displaystyle=\,$ $\displaystyle
x_{i_{3},k}[i_{1},i_{2}\,|\,k+1,k+2]+x_{i_{2},k+2}[i_{1},i_{3}\,|\,k,k+1]$
$\displaystyle-
x_{i_{2},k}[i_{1},i_{3}\,|\,k+1,k+2]-x_{i_{1},k+2}[i_{2},i_{3}\,|\,k,k+1].$
(3)

  2. B

Second, we assume that $F=T_{C^{+}}-T_{C^{-}}$ for some primitive cycle $C$.
It follows from Lemma 3.4 that we can assume that
$C=(c_{1},c_{2},c_{3},c_{4},c_{5}=c_{1})$ in $H$. Without loss of generality,
we may assume that $c_{1}=x_{i_{1},1}$. Then, we have two subcases.

    1. i

Suppose that $C$ is not a cycle in $H_{1}$. In this subcase, we can assume
that $c_{2}=x_{i_{2},2}$, $c_{3}=x_{i_{3},3}$, and $c_{4}=x_{i_{4},2}$ such
that $1\leq i_{1}<i_{2}<i_{4}<i_{3}\leq m$. Now, we have the simple equality:

$\displaystyle[i_{1},i_{2}\,|\,1,2][i_{4},i_{3}\,|\,2,3]-[i_{1},i_{4}\,|\,1,2][i_{2},i_{3}\,|\,2,3]$
$\displaystyle=\,$
$\displaystyle-[i_{1},i_{2}\,|\,2,3][i_{4},i_{3}\,|\,1,2]+[i_{1},i_{4}\,|\,2,3][i_{2},i_{3}\,|\,1,2]$
$\displaystyle-[i_{2},i_{4}\,|\,1,2][i_{1},i_{3}\,|\,2,3]-[i_{2},i_{4}\,|\,2,3][i_{1},i_{3}\,|\,1,2].$
(4)

    2. ii

Suppose that $C$ is a cycle in $H_{1}$. In this subcase, we can assume that
$c_{2}=x_{i_{2},2}$, $c_{3}=x_{i_{3},1}$, and $c_{4}=x_{i_{4},2}$ such that
$1\leq i_{1}<i_{3}<i_{2}<i_{4}\leq m$. Now, we have the well-known Plücker
relation:

$\qquad[i_{1},i_{2}\,|\,1,2][i_{3},i_{4}\,|\,1,2]-[i_{1},i_{4}\,|\,1,2][i_{3},i_{2}\,|\,1,2]=[i_{1},i_{3}\,|\,1,2][i_{2},i_{4}\,|\,1,2].$
(5)

This completes the proof. ∎

In the previous proof, we introduced a term order $\tau^{\prime}$ on $S[T]$,
which is induced from the term order $\tau$ on $S$. We will adopt it from now
on. In particular, we can talk about the initial algebra
$\operatorname{in}_{\tau^{\prime}}(\mathcal{R}(\mathcal{J}_{K_{m},P_{n}}))$.

######  Corollary 3.6.

  1. a

The Rees algebras $\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ and
$\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ are normal
Cohen–Macaulay domains. In particular, if
$\operatorname{char}({\mathbb{K}})=0$, then
$\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ has rational singularities. If
instead $\operatorname{char}({\mathbb{K}})>0$, then
$\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ is F-rational.

  2. b

Let $\mathcal{G}=\\{g_{1},\ldots,g_{q}\\}$ be the natural generating set of
$\mathcal{J}_{K_{m},P_{n}}$. Then $\mathcal{G}$ is a Sagbi basis of the
${\mathbb{K}}$-subalgebra ${\mathbb{K}}[g_{1},\ldots,g_{q}]$ of $S$ with
respect to the lexicographic order $\tau$ on $S$, i.e.,

$\operatorname{in}_{\tau}({\mathbb{K}}[g_{1},\ldots,g_{q}])={\mathbb{K}}[\operatorname{in}_{\tau}(g_{1}),\ldots,\operatorname{in}_{\tau}(g_{q})].$

  3. c

The special fiber rings $\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ and
$\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ are normal
Cohen–Macaulay domains. In particular,
$\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ has rational singularities if
${\mathbb{K}}$ is of characteristic $0$, and
$\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ is F-rational if ${\mathbb{K}}$ is of
positive characteristic.

  4. d

The analytic spreads of $\mathcal{J}_{K_{m},P_{n}}$ and
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})$ coincide. They are given
by

$\begin{cases}n-1,&\text{if $m=2$,}\\\ mn-3,&\text{if $m\geq 3$.}\end{cases}$

  5. e

The associated graded rings $\operatorname{gr}_{\mathcal{J}_{K_{m},P_{n}}}(S)$
and
$\operatorname{gr}_{\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})}(S)$
are Cohen–Macaulay.

###### Proof.

  1. a

Since $\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})=I(H)$ is the edge
ideal of the bipartite graph $H$ constructed in 3.1, the Rees algebra
$\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ is normal
by [MR3362802, Proposition 10.5.8]. It is then Cohen–Macaulay by [MR304376].
Since
$\operatorname{in}_{\tau^{\prime}}(\mathcal{R}(\mathcal{J}_{K_{m},P_{n}}))=\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$
by Theorem 3.5 and [MR1390693, Theorem 2.7], it follows from [MR1390693,
Corollary 2.3] that $\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ has rational
singularities if $\operatorname{char}({\mathbb{K}})=0$, and
$\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ is F-rational if
$\operatorname{char}({\mathbb{K}})>0$. Furthermore, the normality of
$\operatorname{in}_{\tau}(\mathcal{R}(\mathcal{J}_{K_{m},P_{n}}))$ implies
that $\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ is a normal Cohen–Macaulay
domain again by [MR1390693, Corollary 2.3].

  2. b

This part follows from Lemma 3.3 and the part B of the proof of Theorem 3.5.

  3. c

It follows from the previous part b that the special fiber ring
$\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))=\operatorname{in}_{\tau}(\mathcal{F}(\mathcal{J}_{K_{m},P_{n}}))$
is isomorphic to the edge subring ${\mathbb{K}}[H]$ of the graph $H$
constructed in Setting 3.1. Since $H$ is bipartite, ${\mathbb{K}}[H]$ is
Koszul, normal and Cohen–Macaulay by [MR1657721, Theorem 1] and [MR3362802,
Proposition 10.3.1]. The remaining parts follow again from [MR1390693,
Corollary 2.3].

  4. d

Recall that the _analytic spread_ of a graded ideal $I$ is the (Krull)
dimension of the special fiber ring $\mathcal{F}(I)$. By [MR1390693,
Proposition 2.4], the Hilbert functions of
$\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ and
$\operatorname{in}_{\tau}(\mathcal{F}(\mathcal{J}_{K_{m},P_{n}}))$ coincide.
Therefore, they have the same dimension. Meanwhile,
$\operatorname{in}_{\tau}(\mathcal{F}(\mathcal{J}_{K_{m},P_{n}}))=\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$
by b. Thus, the analytic spreads of $\mathcal{J}_{K_{m},P_{n}}$ and
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})$ coincide. Note that
$\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))={\mathbb{K}}[H]$
and $H$ is a bipartite graph. Therefore, it remains to apply 3.2 and
[MR3362802, Corollary 10.1.21].

  5. e

Since the Rees algebras $\mathcal{R}(\mathcal{J}_{K_{m},P_{n}})$ and
$\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ are
Cohen–Macaulay, this part follows from [MR638557, Proposition 1.1]. ∎

Let $I$ be an ideal in $S$. Recall that $I$ satisfies the _persistence
property_ if and only if the sets of associated primes satisfy
$\operatorname{Ass}(I^{t})\subseteq\operatorname{Ass}(I^{t+1})$ for all $t\geq
1$. In addition, the ideal $I$ has the _strong persistence property_ if and
only if $I^{t+1}:I=I^{t}$ for all $t\geq 1$. It is not difficult to see that
the strong persistence property implies the persistence property. On the other
hand, the converse is also known to be false.

######  Corollary 3.7.

Under the assumptions in 3.1, the quadratic ideals
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})$ and
$\mathcal{J}_{K_{m},P_{n}}$ satisfy the strong persistence property. In
particular, both $\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})$ and
$\mathcal{J}_{K_{m},P_{n}}$ satisfy the persistence property.

###### Proof.

Since $\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})=I(H)$ is the edge
ideal of the bipartite graph $H$ constructed in 3.1,
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})$ satisfies the strong
persistence property by [MR3362802, Lemma 7.7.11]. Furthermore, by Theorem
3.5, we have
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}^{t})=(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}$
for all $t\geq 1$. Thus, $\mathcal{J}_{K_{m},P_{n}}$ also satisfies the strong
persistence property by [MR4425287, Proposition 3.13]. ∎

######  Remark 3.8.

The Sagbi-basis property mentioned in Theorem 3.5 is not a common
characteristic for generalized binomial edge ideals. Even for the complete
graph $K_{3}$, one has
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{3},K_{3}}^{2})\neq(\operatorname{in}_{\tau}(\mathcal{J}_{K_{3},K_{3}}))^{2}$.
In other words, the natural generators of the generalized binomial edge ideal
$\mathcal{J}_{K_{3},K_{3}}$ do not form a Sagbi basis under the given term
order.

As further applications of Theorem 3.5, we study the regularity and the depth
of powers of $\mathcal{J}_{K_{m},P_{n}}$ in Theorem 3.11 and Theorem 3.12
respectively. In particular, we give a closed formula of the regularities. To
achieve it, we apply the following lemma:

######  Lemma 3.9 ([MR4210998, Theorem 4.4(2)]).

Let $G$ be a graph. Then, for all $t\geq 1$ we have

$\operatorname{reg}(I(G)^{t})\leq 2t+\operatorname{cochord}(G)-1.$

In particular, we need to compute the co-chordal cover number of the bipartite
graph $H$ from 3.1. For this purpose, recall that a graph $G$ is _chordal_ if
every induced cycle in $G$ has length $3$. Relatedly, a _perfect elimination
order_ of a graph $G$ is an order $v_{1},\dots,v_{n}$ of its vertices such
that for all $i\in[n]$, the neighbor $N_{G_{i}}(v_{i})$ of $v_{i}$, restricted
to the induced subgraph $G_{i}$ of $G$ on the set $\\{v_{i},\ldots,v_{n}\\}$,
induces a complete subgraph in $G$. It is well-known that a graph is chordal
if and only if it admits a perfect elimination order.

On the other hand, a graph $G$ is _co-chordal_ if its complement graph
$G^{\complement}$ is chordal. The _co-chordal cover number_ of a graph $G$,
denoted $\operatorname{cochord}(G)$, is the minimum number $k$ such that there
exist co-chordal subgraphs $G_{1},\ldots,G_{k}$ of $G$ with
$E(G)=\bigcup_{i=1}^{k}E(G_{i})$.

In the following, we compute the co-chordal cover number of the bipartite
graph $H$. An upper bound on this number is sufficient for our application.

######  Lemma 3.10.

For the graph $H$ introduced in 3.1, one has $\operatorname{cochord}(H)\leq
n-1$.

###### Proof.

It is clear that the induced subgraphs $H_{1},H_{2},\dots,H_{n-1}$ are
pairwise isomorphic. Furthermore, $E(H)=\bigcup_{k=1}^{n-1}E(H_{k})$. It
remains to show that $H_{1}$ is co-chordal. Notice that

$\displaystyle E(H_{1}^{\complement})$
$\displaystyle=\Set{\\{x_{i,1},x_{j,1}\\}:1\leq i<j\leq
m}\cup\Set{\\{x_{i,2},x_{j,2}\\}:1\leq i<j\leq m}$
$\displaystyle\quad\cup\Set{\\{x_{i,1},x_{j,2}\\}:1\leq j\leq i\leq m}.$

It can be directly verified that
$x_{1,1},x_{2,1},\dots,x_{m,1},x_{1,2},x_{2,2},\dots,x_{m,2}$ form a perfect
elimination order. Therefore, $H_{1}^{\complement}$ is a chordal graph. ∎

The proof of the following Theorem 3.11 will also use the comparison in
Theorem 5.2. The argument for the latter is logically irrelevant to what is
presented here. Therefore, we decide to postpone the discussion of Theorem 5.2
to a later section of our paper, in order to maintain the consistency and
coherence of our presentation.

######  Theorem 3.11.

Under the assumptions in 3.1, we have

$\operatorname{reg}\left(\frac{S}{\mathcal{J}^{t}_{K_{m},P_{n}}}\right)=\operatorname{reg}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})}\right)=2(t-1)+(n-1)$

for all $t\geq 1$.

###### Proof.

It follows from [MR4425287, Theorem 3.1], [MR2724673, Theorem 3.3.4], and
Theorem 5.2 that

$\operatorname{reg}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})}\right)\geq\operatorname{reg}\left(\frac{S}{\mathcal{J}^{t}_{K_{m},P_{n}}}\right)\geq\operatorname{reg}\left(\frac{S}{\mathcal{J}^{t}_{K_{2},P_{n}}}\right)=2(t-1)+(n-1).$

Thus, it suffices to prove that
$\operatorname{reg}({S}/{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})})\leq
2(t-1)+(n-1)$. To achieve this goal, we note that
$\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})=(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}=I(H)^{t}$
by Theorem 3.5. Furthermore, it follows from Lemmas 3.9 and 3.10 that
$\operatorname{reg}(S/I(H)^{t})\leq 2(t-1)+(n-1)$. So we are done. ∎

Next, we determine the limit of the depth of powers of
$\mathcal{J}_{K_{m},P_{n}}$.

######  Theorem 3.12.

The following results hold under the assumptions in 3.1:

  1. a

$\operatorname{depth}\left(\frac{S}{\mathcal{J}_{K_{m},P_{n}}}\right)=\operatorname{depth}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})}\right)=n+(m-1);$

  2. b

for each $t\geq 1$, we have

$\operatorname{depth}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})}\right)\geq\operatorname{depth}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t+1}_{K_{m},P_{n}})}\right);$

  3. c

$\lim\limits_{t\to\infty}\operatorname{depth}\left(\frac{S}{\mathcal{J}^{t}_{K_{m},P_{n}}}\right)=\lim\limits_{t\to\infty}\operatorname{depth}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})}\right)=\begin{cases}3,&\text{if
$m\geq 3$,}\\\ n+1,&\text{if $m=2$.}\end{cases}$

###### Proof.

  1. a

Recall that a chordal graph is said to be a _generalized block graph_ if it
satisfies: for any three maximal cliques $F_{i}$, $F_{j}$, and $F_{k}$, if
$F_{i}\cap F_{j}\cap F_{k}\neq\emptyset$, then $F_{i}\cap F_{j}=F_{i}\cap
F_{k}=F_{j}\cap F_{k}$. Since the path graph $P_{n}$ is clearly a generalized
block graph whose clique number is $2$, this part follows from [MR4233116,
Theorem 3.3].

  2. b

By Theorem 3.5, we get that
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}^{t})=(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}=I(H)^{t}$
for all $t\geq 1$, where $H$ is the bipartite graph constructed in 3.1.
Therefore, it suffices to prove that
$\operatorname{depth}\left(S/I(H)^{t}\right)\geq\operatorname{depth}\left(S/I(H)^{t+1}\right)$
for all $t\geq 1$. Since $H$ is a bipartite graph, we have
$I(H)^{t}=I(H)^{(t)}$ for any $t\geq 1$ by [MR3362802, Theorem 14.3.6 and
Corollary 14.3.15], where $I(H)^{(t)}$ is the $t$-th symbolic power of $I(H)$.
In addition, note that $x_{m-1,1}$ and $x_{2,n}$ are two leaves of $H$, we
obtain that
$\operatorname{pd}(S/I(H)^{(t+1)})\geq\operatorname{pd}(S/I(H)^{(t)})$ by
[MR3798623, Theorem 5.2]. The desired statement then follows from the
Auslander–Buchsbaum formula; cf, for example, [MR2724673, Corollary A.4.3].

  3. c

For all $t\geq 1$, by [MR2724673, Theorem 3.3.4], we have

$\operatorname{depth}\left(\frac{S}{\mathcal{J}^{t}_{K_{m},P_{n}}}\right)\geq\operatorname{depth}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})}\right).$

It follows from the previous part b, [MR696134, Proposition 3.3], Theorem 3.5,
and 3.6e that

$\displaystyle\lim\limits_{t\to\infty}\operatorname{depth}\left(\frac{S}{\mathcal{J}^{t}_{K_{m},P_{n}}}\right)$
$\displaystyle\geq\lim\limits_{t\to\infty}\operatorname{depth}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})}\right)=\inf_{t\geq
1}\operatorname{depth}\left(\frac{S}{\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})}\right)$
$\displaystyle=\inf_{t\geq
1}\operatorname{depth}\left(\frac{S}{(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}}\right)=\operatorname{dim}(S)-\ell(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$
(6)

where $\ell(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ is the
analytic spread of $\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})$.

At the same time, it follows from that [MR2163482, Theorem 1.2] that
$\lim\limits_{t\to\infty}\operatorname{depth}\left(S/\mathcal{J}^{t}_{K_{m},P_{n}}\right)$
exists and satisfies

$\lim\limits_{t\to\infty}\operatorname{depth}\left(\frac{S}{\mathcal{J}^{t}_{K_{m},P_{n}}}\right)\leq\operatorname{dim}(S)-\ell(\mathcal{J}_{K_{m},P_{n}}).$
(7)

If we combine Equations 6 and 7 together, it remains to apply 3.6d. ∎

We end this section with the study of the symbolic powers of
$\mathcal{J}_{K_{m},P_{n}}$. Let $I$ be an ideal of $S$, and suppose that
$\operatorname{Ass}(I)$ is the set of associated prime ideals of $I$. For any
integer $t\geq 1$, the _$t$ -th symbolic power_ of $I$ is defined by

$I^{(t)}\coloneqq\bigcap\limits_{\mathfrak{p}\in\operatorname{Ass}(I)}(I^{t}S_{\mathfrak{p}}\cap
S).$

In most cases, symbolic powers are not identical to the ordinary powers.
However, Ene and Herzog proved in [MR4143239] that if $G$ is a closed graph
and $J_{G}$ is its binomial edge ideal, then $J_{G}^{(t)}=J_{G}^{t}$ for all
$t\geq 1$. Recall that $G$ is said to be _closed_ if for all edges $\\{i,j\\}$
and $\\{k,l\\}$ of $G$ with $i<j$ and $k<l$, one has $\\{j,l\\}\in E(G)$ if
$i=k$, and $\\{i,k\\}\in E(G)$ if $j=l$. Path graphs are the simplest closed
graphs. In the remaining of this section, we show that the symbolic powers of
the _generalized_ binomial edge ideal of a path graph coincide with the
ordinary powers.

######  Theorem 3.13.

Let $t$ be a positive integer and $\mathcal{J}^{(t)}_{K_{m},P_{n}}$ be the
$t$-th symbolic power of $\mathcal{J}_{K_{m},P_{n}}$. Then, we have
$\mathcal{J}^{(t)}_{K_{m},P_{n}}=\mathcal{J}^{t}_{K_{m},P_{n}}$.

The proof of this result is involved. We need some preparations.

######  Lemma 3.14.

Let $I=I_{2}(\bm{X})$ be the ideal in $S={\mathbb{K}}[\bm{X}]$, which is
generated by all $2$-minors of the $m\times n$ matrix $\bm{X}$. Then, one has
$\operatorname{in}_{\tau}(I^{(t)})=(\operatorname{in}_{\tau}(I))^{(t)}$ for
every integer $t\geq 1$.

###### Proof.

Consider the graph $\widetilde{G}$ with the edge set

$E(\widetilde{G})=\Set{\\{x_{i,j},x_{i^{\prime},j^{\prime}}\\}:1\leq
i<i^{\prime}\leq m,1\leq j<j^{\prime}\leq n}.$

It is clear that $\operatorname{in}_{\tau}(I)$ is the edge ideal of
$\widetilde{G}$ in $S$. To confirm the expected equality, we first show that
$\widetilde{G}$ is _perfect_ in the sense that the _chromatic number_
$\chi(G_{V^{\prime}})$ equals the _clique number_ $\omega(G_{V^{\prime}})$ for
every subset $V^{\prime}$ of $V(\widetilde{G})$. Recall that a graph is called
a _comparability graph_ if there exists a partial ordering of its vertex set
such that two vertices are adjacent if and only if they are comparable. It is
well-known that every comparability graph is perfect, see [MR3362802,
Corollary 14.5.6]. At the same time, the graph $\widetilde{G}$ here is a
comparability graph, where the poset structure can be taken with respect to
the subscripts of the vertices of $\widetilde{G}$: $x_{i,j}\prec
x_{i^{\prime},j^{\prime}}$ if and only if both $i<i^{\prime}$ and
$j<j^{\prime}$. Thus, $\widetilde{G}$ is a perfect graph.

By looking at the graded components, it follows from [MR3362802, Corollary
13.7.2] that $(\operatorname{in}_{\tau}(I))^{(t)}$ is generated by monomials
of the form $u_{1}u_{2}\dots u_{s}$ such that

$u_{k}=x_{i_{k,1},j_{k,1}}x_{i_{k,2},j_{k,2}}\cdots
x_{i_{k,r_{k}},j_{k,r_{k}}}$

with $1\leq i_{k,1}<i_{k,2}<\cdots<i_{k,r_{k}}\leq m$, $1\leq
j_{k,1}<j_{k,2}<\cdots<j_{k,r_{k}}\leq n$, and $\sum_{k=1}^{s}(r_{k}-1)=t$.

At the same time, it follows from [zbMATH07539232, Theorem 4.3.6] that
$\operatorname{in}_{\tau}(I^{(t)})$ is generated by the monomials
$\operatorname{in}_{\tau}(\Delta)$, where $\Delta$ is a product of minors with
$\gamma_{2}(\Delta)=t$ and no factor of size $<2$. These monomials are
precisely those described in the previous paragraph. Thus, the proof is
completed. ∎

Let $G$ be a simple graph and let $c(G)$ denote the number of connected
components of $G$. A vertex $v$ is called a _cut vertex_ of $G$ if
$c(G)<c(G\setminus v)$. Let $A$ be a subset of $V(G)$. By abuse of notation,
we also let $c(A)$ denote the number of connected components of $G\setminus
A$. If $v$ is a cut vertex of the induced subgraph
$G\setminus(A\setminus\\{v\\})$ for any $v\in A$, then we say that $A$ has the
_cut point property_. Set
$\mathcal{C}(G)\coloneqq\\{\emptyset\\}\cup\Set{A:A\text{\ has the cut point
property}}$.

Now, let $G$ be a simple graph on the vertex set $[n]$. For each subset $A$ of
$[n]$, we introduce the ideal

$P_{A}(K_{m},G)\coloneqq(x_{ij}:(i,j)\in[m]\times
A)+\mathcal{J}_{K_{m},\widetilde{G_{1}}}+\cdots+\mathcal{J}_{K_{m},\widetilde{G_{c(A)}}}$

in $S$, where $G_{1},\ldots,G_{c(A)}$ are the connected components of
$G\setminus A$. It is well-known that

$\mathcal{J}_{K_{m},G}=\bigcap_{A\in\mathcal{C}(G)}P_{A}(K_{m},G)$

is the minimal primary decomposition of the radical ideal
$\mathcal{J}_{K_{m},G}$; see [MR3011436, Theorem 7].

######  Lemma 3.15.

Let $A\subset[n]$. Then
$\operatorname{in}_{\tau}(P_{A}(K_{m},G)^{(t)})=(\operatorname{in}_{\tau}(P_{A}(K_{m},G)))^{(t)}$
for any integer $t\geq 1$.

###### Proof.

We follow the strategy of [MR4143239, Lemma 3.2]. The only essential change is
that we consider instead the symbolic Rees algebra
$\mathcal{R}_{s}(I)\coloneqq\oplus_{k\geq 0}I^{(k)}T^{k}$ of an ideal $I$ in
$S$, which is a graded subalgebra of $S[T]$.

For simplicity, we write $P$ instead of $P_{A}(K_{m},G)$, $c$ instead of
$c(A)$, and $J_{k}$ instead of $\mathcal{J}_{K_{m},\widetilde{G_{k}}}$ for
$1\leq k\leq c$. Since the sets of variables $\\{x_{i,j}:i\in[m],j\in
V(\widetilde{G_{k}})\\}$ as well as the set $\Set{x_{i,j}:i\in[m],j\in A}$ are
pairwise disjoint, we have

$\operatorname{in}_{\tau}(P)=(x_{i,j}:i\in[m],j\in
A)+\operatorname{in}_{\tau}(J_{1})+\cdots+\operatorname{in}_{\tau}(J_{c}).$
(8)

It follows from the same pairwise disjointness and [MR4074049, Theorem 3.4]
that

$\displaystyle\mathcal{R}_{s}(P)$
$\displaystyle=\mathcal{R}_{s}((x_{i,j}:i\in[m],j\in
A))\otimes_{{\mathbb{K}}}(\otimes_{k=1}^{c}\mathcal{R}_{s}(J_{k})),$ (9) and
$\displaystyle\mathcal{R}_{s}(\operatorname{in}_{\tau}(P))$
$\displaystyle=\mathcal{R}_{s}((x_{i,j}:i\in[m],j\in
A))\otimes_{{\mathbb{K}}}(\otimes_{k=1}^{c}\mathcal{R}_{s}(\operatorname{in}_{\tau}(J_{k})))$
$\displaystyle=\mathcal{R}_{s}((x_{i,j}:i\in[m],j\in
A))\otimes_{{\mathbb{K}}}(\otimes_{k=1}^{c}\operatorname{in}_{\tau^{\prime}}(\mathcal{R}_{s}(J_{k}))),$
(10)

where the last equality is essentially due to Lemma 3.14. Since it is clear
that $\text{RHS of
\eqref{eqn:symbolic_Rees_init}}\subseteq\operatorname{in}_{\tau^{\prime}}(\text{RHS
of \eqref{eqn:symbolic_Rees}})$, we have

$\mathcal{R}_{s}(\operatorname{in}_{\tau}(P))\subseteq\operatorname{in}_{\tau^{\prime}}(\mathcal{R}_{s}(P)).$
(11)

On the other hand, it is well-known that $\mathcal{R}_{s}(P)$ and
$\operatorname{in}_{\tau^{\prime}}(\mathcal{R}_{s}(P))$ have the same Hilbert
functions. Since $\mathcal{R}_{s}(J_{k})$ and
$\operatorname{in}_{\tau^{\prime}}(\mathcal{R}_{s}(J_{k}))$ have the same
Hilbert functions for every $k$, it follows from eqs. 9 and 10 that
$\mathcal{R}_{s}(\operatorname{in}_{\tau}(P))$ and
$\operatorname{in}_{\tau^{\prime}}(\mathcal{R}_{s}(P))$ have the same Hilbert
functions. Therefore, we have
$\mathcal{R}_{s}(\operatorname{in}_{\tau}(P))=\operatorname{in}_{\tau^{\prime}}(\mathcal{R}_{s}(P))$
from eq. 11. At the level of graded components, we obtain that
$\operatorname{in}_{\tau}(P^{(t)})=(\operatorname{in}_{\tau}(P))^{(t)}$ for
every $t$. ∎

######  Lemma 3.16.

Let $G$ be a generalized block graph. Then,

$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},G})=\bigcap_{\mathfrak{p}\in\operatorname{Ass}(\mathcal{J}_{K_{m},G})}\operatorname{in}_{\tau}(\mathfrak{p}).$

###### Proof.

Without loss of generality, we assume that $G$ is connected. Let $r$ be the
number of maximal cliques of $G$. When $r=1$, $G$ is a complete graph. Since
$\mathcal{J}_{K_{m},G}$ is a prime ideal in this case, the expected result is
trivial. When $r>1$, we apply the following observation from the proof of
[MR4233116, Theorem 3.3]: There exists a leaf order, say, $F_{1},\dots,F_{r}$,
on the clique complex $\Delta(G)$ of $G$. Let $F_{t_{1}},\dots,F_{t_{q}}$ be
the branches of $F_{r}$. Then, the intersection of any pair of facets from
$F_{t_{1}},\dots,F_{t_{q}},F_{r}$ is the same set of vertices, say, $A$. Now,
$\mathcal{J}_{K_{m},G}=J_{1}\cap J_{2}$, where
$J_{1}\coloneqq\bigcap_{B\in\mathcal{C}(G),A\cap B=\emptyset}P_{B}(G)$ and
$J_{2}\coloneqq\bigcap_{B\in\mathcal{C}(G),A\subseteq B}P_{B}(G)$. Note that
$J_{1}=\mathcal{J}_{K_{m},G^{\prime}}$ where $G^{\prime}$ is obtained from $G$
by replacing the cliques $F_{t_{1}},\dots,F_{t_{q}},F_{r}$ by the clique on
the vertex set $F_{r}\cup(\cup_{k=1}^{q}F_{t_{k}})$. At the same time,
$J_{2}=(x_{i,j}:(i,j)\in[m]\times A)+\mathcal{J}_{K_{m},G^{\prime\prime}}$,
where $G^{\prime\prime}$ is the restriction of $G$ to the vertex set
$V(G)\setminus A$. Obviously, $G^{\prime}$ and $G^{\prime\prime}$ are
generalized block graphs with fewer maximal cliques. Since we have
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},G})=\operatorname{in}_{\tau}(J_{1})\cap\operatorname{in}_{\tau}(J_{2})$
from the proof of [MR4233116, Theorem 3.3(c)] in the $r>1$ case, we are done
by induction on $r$. ∎

######  Lemma 3.17.

Let $t$ be a positive integer. Suppose that $I$ is an ideal in $S$ and
$\sigma$ is a term order such that $\operatorname{in}_{\sigma}(I)$ is a
squarefree monomial ideal. It follows that $I$ is a radical ideal and
$I=\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}\mathfrak{p}$. Suppose that
the following conditions are satisfied:

  1. a

$\operatorname{in}_{\sigma}(\mathfrak{p})$ is a squarefree monomial ideal for
each $\mathfrak{p}\in\operatorname{Min}(I)$,

  2. b

$(\operatorname{in}_{\sigma}(I))^{(t)}=(\operatorname{in}_{\sigma}(I))^{t}$,

  3. c

$\operatorname{in}_{\sigma}(\mathfrak{p}^{(t)})=(\operatorname{in}_{\sigma}(\mathfrak{p}))^{(t)}$
for each $\mathfrak{p}\in\operatorname{Min}(I)$,

  4. d

$\operatorname{in}_{\sigma}(I)=\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}\operatorname{in}_{\sigma}(\mathfrak{p})$.

Then $I^{(t)}=I^{t}$.

###### Proof.

Since $\operatorname{in}_{\sigma}(I)$ is a squarefree monomial ideal, we have

$(\operatorname{in}_{\sigma}(I))^{(t)}=\bigcap_{\mathfrak{P}\in\operatorname{Min}(\operatorname{in}_{\sigma}(I))}\mathfrak{P}^{t}.$

Likewise, we have

$(\operatorname{in}_{\sigma}(\mathfrak{p}))^{(t)}=\bigcap_{\mathfrak{P}\in\operatorname{Min}(\operatorname{in}_{\sigma}(\mathfrak{p}))}\mathfrak{P}^{t}$

for each $\mathfrak{p}\in\operatorname{Min}(I)$. Therefore, we have
$\left(\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}\operatorname{in}_{\sigma}(\mathfrak{p})\right)^{(t)}{=}\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}(\operatorname{in}_{\sigma}(\mathfrak{p}))^{(t)}$.
It follows from the remaining assumptions that

$\displaystyle\operatorname{in}_{\sigma}(I^{t})$
$\displaystyle\supseteq(\operatorname{in}_{\sigma}(I))^{t}=(\operatorname{in}_{\sigma}(I))^{(t)}=\left(\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}\operatorname{in}_{\sigma}(\mathfrak{p})\right)^{(t)}{=}\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}(\operatorname{in}_{\sigma}(\mathfrak{p}))^{(t)}$
$\displaystyle=\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}\operatorname{in}_{\sigma}(\mathfrak{p}^{(t)})\supseteq\operatorname{in}_{\sigma}\left(\bigcap_{\mathfrak{p}\in\operatorname{Min}(I)}\mathfrak{p}^{(t)}\right)=\operatorname{in}_{\sigma}(I^{(t)})\supseteq\operatorname{in}_{\sigma}(I^{t}).$

Therefore, we obtain that
$\operatorname{in}_{\sigma}(I^{(t)})=\operatorname{in}_{\sigma}(I^{t})$. Since
$I^{t}\subseteq I^{(t)}$, we get $I^{t}=I^{(t)}$. ∎

###### Proof of Theorem 3.13..

For each $\mathfrak{p}\in\operatorname{Min}(\mathcal{J}_{K_{m},P_{n}})$, we
know $\operatorname{in}_{\tau}(\mathfrak{p})$ is squarefree from eq. 8.
Furthermore, notice that
$\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})=I(H)$ where $H$ is a
bipartite graph. It follows from [MR3362802, Corollary 13.3.6] that
$(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}=(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{(t)}$.
It remains to apply Lemmas 3.16, 3.15 and 3.17. ∎

######  Remark 3.18.

Lemma 3.17 is a modification of [MR4143239, Lemma 3.1]. Note that we cannot
use [MR4143239, Lemma 3.1] directly to prove Theorem 3.13, since the condition
(ii)(a) of [MR4143239, Lemma 3.1] is not satisfied by
$\mathcal{J}_{K_{m},P_{n}}$ when $m=n=3$. In fact, in this case, there is a
prime ideal $\mathfrak{p}$ associated with $\mathcal{J}_{K_{3},P_{3}}$ such
that $\mathfrak{p}^{2}\neq\mathfrak{p}^{(2)}$. This prime ideal is
$P_{\emptyset}(K_{3},P_{3})=I_{2}(\bm{X})$, where $\bm{X}$ is the $3\times 3$
generic matrix. It follows from [zbMATH07539232, Theorem 4.3.6] that
$I_{2}^{(2)}(\bm{X})=I_{2}^{2}(\bm{X})+I_{3}(\bm{X})$.

##  4\. Blowup algebras

In this section, we will use algebraic properties of the initial algebras and
the Sagbi basis theory to study the regularities of the blowup algebras of the
ideal $\mathcal{J}_{K_{m},P_{n}}$. Our approach will involve a combination of
combinatorial optimization techniques to analyze the related algebraic
invariants.

######  Lemma 4.1.

For the bipartite graph $H$ introduced in 3.1, we have

$\operatorname{match}(H)=(m-1)\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n-1}{2}\right\rfloor,$

where $\lfloor\frac{n}{2}\rfloor$ is the largest integer $\leq\frac{n}{2}$.

###### Proof.

We have mentioned in 3.2 that $x_{m,1}$ and $x_{1,n}$ are two isolated
vertices of $H$. Let $H^{\prime}$ be the induced subgraph
$H\setminus\\{x_{m,1},x_{1,n}\\}$. This is a bipartite graph with a
bipartition $V_{1}^{\prime}\sqcup V_{2}^{\prime}$, where
$V_{i}^{\prime}=V_{i}\cap V(H^{\prime})$ for $i=1,2$. Notice that we have a
complete matching from $V_{2}^{\prime}$ to $V_{1}^{\prime}$, given by

$\\{\\{x_{i,j},x_{i+1,j+1}\\}:i\in[m-1],\text{$j$ is
odd}\\}\cup\\{\\{x_{1,j},x_{m,j+1}\\}:\text{$j$ is even}\\};$ (12)

see also Figure 3. Therefore, we have the desired matching number, by counting
the number of edges in eq. 12. ∎

$x_{4,1}$$x_{1,6}$$\times$$\times$ (A) $(m,n)=(4,6)$

$x_{4,1}$$x_{1,7}$$\times$$\times$ (B) $(m,n)=(4,7)$

Figure 3. A complete matching of $H$

######  Theorem 4.2.

Under the assumptions in 3.1, we have

$\operatorname{reg}(\mathcal{R}(\mathcal{J}_{K_{m},P_{n}}))=\operatorname{reg}(\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})))=(m-1)\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n-1}{2}\right\rfloor.$

###### Proof.

Note that the ideal $\mathcal{J}_{K_{m},P_{n}}$ satisfies the following
conditions:

  1. a

the natural generators of $\mathcal{J}_{K_{m},P_{n}}$ form a Gröbner basis
with respect to the term order $\tau$ by 3.2;

  2. b

for each $t\geq 1$,
$\operatorname{in}_{\tau}(\mathcal{J}^{t}_{K_{m},P_{n}})=(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))^{t}$
by Theorem 3.5;

  3. c

the Rees algebras
$\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ is
Cohen–Macaulay by 3.6.

It follows from [MR4405525, Theorem 3.2] that
$\operatorname{reg}(\mathcal{R}(\mathcal{J}_{K_{m},P_{n}}))=\operatorname{reg}(\mathcal{R}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})))$.

We have seen that $\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})=I(H)$
is the edge ideal of the bipartite graph $H$ constructed in 3.1. It follows
from [MR3884545, Theorem 4.2], or implicitly from [MR3362802, Theorems 7.1.8
and 14.3.55], that
$\operatorname{reg}(\mathcal{R}(I(H)))=\operatorname{match}(H)$. The only
remaining step is to apply Lemma 4.1. ∎

In the following, we will use the $a$-invariant of the Cohen–Macaulay special
fiber ring $\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ to compute its regularity;
see [MR3362802, Theorem 6.4.1]. Recall that if $A$ is a standard graded
${\mathbb{K}}$-algebra, the _$a$ -invariant_ of $A$, denoted by $a(A)$, is the
degree, as a rational function, of the Hilbert series of $A$.

######  Remark 4.3.

Notice that
$\operatorname{in}_{\tau}(\mathcal{F}(\mathcal{J}_{K_{m},P_{n}}))={\mathbb{K}}[H]$
for the graph $H$ introduced in 3.1. When $m=2$ or $3$, the graph $H$ is
acyclic; see also 3.2. It follows from Lemma 3.3 that the presentation ideal
of ${\mathbb{K}}[H]$ is the zero ideal. In other words,
$\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ is a
polynomial ring in these cases. In particular, its regularity is zero. Since
$\operatorname{in}_{\tau}\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ and
$\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ are Cohen–Macaulay and have the same
Hilbert series, the regularity of the special fiber ring
$\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ is also zero.

By the previous observation, we will focus on the case when $m\geq 4$. The
technical computation of the $a$-invariant of
$\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ is given in 4.7. Before starting the
proof, let us briefly introduce its strategy, which consists of two different
combinatorial approaches.

The first approach deals with directed graphs, as in Lemma 4.5.

######  Definition 4.4.

Let $D$ be a directed graph with the vertex set $V(D)$ and the edge set
$E(D)$. For every subset $A\subseteq V(D)$, define
$\delta^{+}(A)\coloneqq\Set{(z,w)\in E(D):z\in A,w\notin A}$ to be the set of
edges leaving the vertex set $A$ and $\delta^{-}(A)$ to be the set of edges
entering the vertex set $A$. The edge set $\delta^{+}(A)$ is a _directed cut_
of $D$ if $\emptyset\neq A\subsetneq V(D)$ and $\delta^{-}(A)=\emptyset$.

######  Lemma 4.5 ([MR3362802, Theorem 11.3.2]).

Let $G$ be a connected bipartite graph with bipartition $V_{1}\sqcup V_{2}$.
If $G$ is regarded as a directed graph with all its arrows leaving the vertex
set $V_{1}$, then the following two numbers are equal:

  1. a

$-a({\mathbb{K}}[G])$, minus the $a$-invariant of the edge subring
${\mathbb{K}}[G]$;

  2. b

the maximum number of edge disjoint directed cuts of $G$.

The above lemma invites us to find edge disjoint directed cuts of $H$. A
natural choice of such direct cuts gives the desired $a$-invariant that we
need. However, proving that this is the value that we are looking for is a
different story.

In the second approach, we start with the vector space
$V\coloneqq\operatorname{Mat}_{m\times n}({\mathbb{Q}})$ over ${\mathbb{Q}}$.
It has a canonical basis $\\{\bm{e}_{i,j}:i\in[m],j\in[n]\\}$, where
$\bm{e}_{i,j}=\left(e_{k,\ell}^{i,j}\right)_{\begin{subarray}{c}1\leq k\leq
m\\\ 1\leq\ell\leq n\end{subarray}}\in V$ with

$e_{k,\ell}^{i,j}=\begin{cases}1,&\text{if $(k,\ell)=(i,j)$,}\\\
0,&\text{otherwise.}\end{cases}$

Let $H^{\prime}$ be the graph obtained from $H$ by removing the two isolated
vertices $x_{m,1}$ and $x_{1,n}$. From this graph, we introduce a set of
vectors

$\mathcal{A}\coloneqq\Set{\bm{e}_{i,j}+\bm{e}_{i^{\prime},j^{\prime}}:\\{x_{i,j},x_{i^{\prime},j^{\prime}}\\}\in
E(H^{\prime})}\subset V.$

The set ${\mathbb{R}}_{+}\mathcal{A}$ is called the _edge cone_ of
$H^{\prime}$. At the same time, the _shift polyhedron_ of the edge cone of
$H^{\prime}$ is

$\mathcal{Q}\coloneqq\operatorname{conv}(\operatorname{Mat}_{m\times
n}({\mathbb{Z}})\cap\operatorname{ri}({\mathbb{R}}_{+}\mathcal{A})),$

where $\operatorname{ri}({\mathbb{R}}_{+}\mathcal{A})$ is the interior of
${\mathbb{R}}_{+}\mathcal{A}$ relative to the affine hull of
${\mathbb{R}}_{+}\mathcal{A}$; see also [MR3362802, Corollary 11.2.4]. We will
use the fact that

$-a({\mathbb{K}}[H^{\prime}])=\min\\{\,|\bm{v}|/2:\bm{v}\in\mathcal{Q}\,\\}$
(13)

by [MR3362802, Theorem 11.3.1], where
$|\bm{v}|\coloneqq\sum_{i=1}^{m}\sum_{j=1}^{n}v_{i,j}$ for
$\bm{v}=(v_{i,j})\in V$.

To investigate the shift polyhedron in detail, let $V^{*}$ be the space of
linear functions on $V$. Each $\bm{F}\in V^{*}$ defines a hyperplane
$H_{\bm{F}}\coloneqq\\{\bm{v}\in V:\bm{F}(\bm{v})=0\\}$ and a half-space
$H_{\bm{F}}^{+}\coloneqq\\{\bm{v}\in V:\bm{F}(\bm{v})\geq 0\\}$. Let
$\Set{\bm{E}_{i,j}:i\in[m],j\in[n]}\subset V^{*}$ be the dual basis with
respect to $\\{\bm{e}_{i,j}:i\in[m],j\in[n]\\}$. For simplicity, we will
represent the elements in $V^{*}$ as $m\times n$ matrices like
$\bm{F}=(f_{i,j})\in V^{*}$. For every such function $\bm{F}\in V^{*}$ and
every vector $\bm{v}=(v_{i,j})\in V$, we have
$\bm{F}(\bm{v})=\sum_{i,j}f_{i,j}(v_{i,j})\in{\mathbb{Q}}$.

We have a quick remark.

######  Remark 4.6.

Fix a matrix $\bm{A}=(a_{i,j})$ in $V^{*}$, where

$a_{i,j}=\begin{cases}0,&\text{if $(i,j)=(1,n)$ or $(m,1)$,}\\\ 1,&\text{if
$j$ is otherwise odd,}\\\ -1,&\text{if $j$ is otherwise even,}\end{cases}$

see Figure 4.

$\displaystyle\begin{bmatrix}1&-1&1&-1&1&-1&0\\\ 1&-1&1&-1&1&-1&1\\\
1&-1&1&-1&1&-1&1\\\ 1&-1&1&-1&1&-1&1\\\ 1&-1&1&-1&1&-1&1\\\
0&-1&1&-1&1&-1&1\end{bmatrix}$

(A) $(m,n)=(6,7)$ case

$\displaystyle\begin{bmatrix}1&-1&1&-1&1&-1&1&0\\\ 1&-1&1&-1&1&-1&1&-1\\\
1&-1&1&-1&1&-1&1&-1\\\ 1&-1&1&-1&1&-1&1&-1\\\ 1&-1&1&-1&1&-1&1&-1\\\
0&-1&1&-1&1&-1&1&-1\end{bmatrix}$

(B) $(m,n)=(6,8)$ case

Figure 4. The matrix $\bm{A}$ in $V^{*}$

It is easy to see that $\mathcal{A}\subseteq H_{\bm{A}}\cap
H_{\bm{E}_{1,n}}\cap H_{\bm{E}_{m,1}}$. Since $H^{\prime}$ is a connected
bipartite graph, we get $\operatorname{dim}({\mathbb{K}}[H^{\prime}])=mn-3$ by
[MR3362802, Corollary 10.1.21]. At the same time, the integral points of
$\mathcal{Q}$ define the canonical module of ${\mathbb{K}}[H^{\prime}]$ by
[MR3362802, Proposition 11.2.1]. Therefore, $H_{\bm{A}}\cap
H_{\bm{E}_{1,n}}\cap H_{\bm{E}_{m,1}}$ is the minimal linear space that
contains $\mathcal{Q}$.

The two approaches described above will give us a lower bound and an upper
bound of $-a({\mathbb{K}}[H^{\prime}])$ respectively. Since they coincide, we
obtain the exact value. Now, we carry out this strategy and start the real
computation.

######  Proposition 4.7.

Suppose that $m\geq 4$ is an integer and $H$ is the bipartite graph introduced
in 3.1. Then we have

$-a({\mathbb{K}}[H])=\begin{cases}m\cdot\frac{n}{2},&\text{if $n$ is even},\\\
m\cdot\frac{n+1}{2}-2,&\text{if $n$ is odd}.\end{cases}$

###### Proof.

For the induced subgraph $H^{\prime}$, we have
${\mathbb{K}}[H]={\mathbb{K}}[H^{\prime}]$. Thus, we will prove

$-a({\mathbb{K}}[H^{\prime}])=\begin{cases}m\cdot\frac{n}{2},&\text{if $n$ is
even},\\\ m\cdot\frac{n+1}{2}-2,&\text{if $n$ is odd}\end{cases}$ (14)

by considering the following two steps.

  1. A

First, we show that LHS $\geq$ RHS in eq. 14. Let $V^{\prime}=V(H^{\prime})$,
$V^{\prime}_{1}=\\{x_{i,j}\in V^{\prime}:i\in[m],\text{$j$ is odd}\\}$ and
$V_{2}^{\prime}=V^{\prime}\setminus V_{1}^{\prime}$. Then $H^{\prime}$ is a
connected bipartite graph with bipartition $V_{1}^{\prime}\sqcup
V_{2}^{\prime}$. By Lemma 4.5, we regard $H^{\prime}$ as a directed graph with
all its arrows leaving the vertex set $V_{1}^{\prime}$. For each $u\in
V_{1}^{\prime}$, the directed cut $\delta^{+}(\\{u\\})$ is the set of edges
leaving the vertex $u$. Since

$E(H^{\prime})=\bigsqcup_{u\in V_{1}^{\prime}}\delta^{+}(\\{u\\})$

is a disjoint union, we immediately have

$-a({\mathbb{K}}[H^{\prime}])\geq|V_{1}^{\prime}|=\begin{cases}m\cdot\frac{n}{2}-1,&\text{if
$n$ is even},\\\ m\cdot\frac{n+1}{2}-2,&\text{if $n$ is odd}\end{cases}$

from Lemma 4.5.

At the same time, when $n$ is even, we consider the two special vertices
$u_{1}=x_{1,n-1}\in V_{1}^{\prime}$ and $u_{2}=x_{2,n}\in V_{2}^{\prime}$.
Notice that

$\delta^{+}(\\{u_{1},u_{2}\\})=\Set{(u_{1},x_{i,n}):3\leq i\leq n}$

and

$\delta^{+}(V^{\prime}\setminus\\{u_{2}\\})=\Set{(u_{1},u_{2})}$

are two directed cuts. Thus,

$\delta^{+}(\\{u_{1}\\})=\delta^{+}(\\{u_{1},u_{2}\\})\sqcup\delta^{+}(V^{\prime}\setminus\\{u_{2}\\}).$

As a result, when $n$ is even, we have additionally

$-a({\mathbb{K}}[H^{\prime}])\geq|V_{1}^{\prime}|+1=m\cdot\frac{n}{2},$

from Lemma 4.5. In short, we have LHS $\geq$ RHS in eq. 14.

  2. B

Second, we prove that LHS $\leq$ RHS in eq. 14. By the formula in eq. 13, it
suffices to find a suitable $\hat{\bm{u}}\in\mathcal{Q}$ such that
$|\hat{\bm{u}}|$ is twice the integer in the RHS of eq. 14.

The candidate vector $\hat{\bm{u}}=(u_{i,j})$ in $V$ is given by

$u_{i,j}=\begin{cases}0,&\text{if $(i,j)=(1,n)$ or $(m,1)$,}\\\ 2,&\text{if
$(i,j)=(1,n-1)$,}\\\ 2,&\text{if $(i,j)=(m,2)$ and $n$ is even,}\\\
m-2,&\text{if $(i,j)=(m,2)$ and $n$ is odd,}\\\
1,&\text{otherwise;}\end{cases}$

see Figure 5. It is easy to verify that $|\hat{\bm{u}}|$ satisfies the
requirement. Therefore, it remains to show that $\hat{\bm{u}}$ belongs to
$\mathcal{Q}$.

$\displaystyle\begin{bmatrix}1&1&1&1&1&2&0\\\ 1&1&1&1&1&1&1\\\
1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1\\\ 0&4&1&1&1&1&1\end{bmatrix}$

(A) $(m,n)=(6,7)$ case

$\displaystyle\begin{bmatrix}1&1&1&1&1&1&2&0\\\ 1&1&1&1&1&1&1&1\\\
1&1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1&1\\\ 1&1&1&1&1&1&1&1\\\
0&2&1&1&1&1&1&1\end{bmatrix}$

(B) $(m,n)=(6,8)$ case

Figure 5. The extremal vector $\hat{\bm{u}}$ in $V$

First, we consider the case where $n$ is odd. We do this in two sub-steps.

    1. (a)

In the first sub-step, we show that $\hat{\bm{u}}$ belongs to the polyhedron
${\mathbb{R}}_{+}\mathcal{A}$.

Let $\widetilde{H}$ be a subgraph of $H^{\prime}$. By abuse of notation, the
_degree matrix_ of $\widetilde{H}$ is the $m\times n$ matrix
$\bm{D}=(d_{i,j})$, where

$\qquad\qquad\qquad d_{i,j}=\begin{cases}0,&\text{if $(i,j)=(1,n)$ or
$(m,1)$,}\\\ \text{the degree of the vertex $x_{i,j}$ in
$\widetilde{H}$,}&\text{otherwise.}\end{cases}$

We will construct subgraphs $\widetilde{H}$ of $H^{\prime}$ such that the
degree matrix of $\widetilde{H}$ is given by the $\hat{\bm{u}}$ above; we will
call such subgraphs of _$\hat{\bm{u}}$ -type_.

The first instance $\widetilde{H}_{1}$ of $\hat{\bm{u}}$-type can be
constructed as follows. For simplicity, we will say that edges of the form
$\\{x_{i,j},x_{i^{\prime},j+1}\\}$ belong to the zone $\mathcal{Z}_{j}$ for
each $j\in[n-1]$. For $\widetilde{H}_{1}$, the edges in zone $\mathcal{Z}_{1}$
are

$\Set{\\{x_{1,1},x_{m-1,2}\\},\\{x_{2,1},x_{m,2}\\},\\{x_{3,1},x_{m,2}\\},\dots,\\{x_{m-1,1},x_{m,2}\\}}.$

Note that it contains two long parallel edges. For $2\leq j<n$, where $j$ is
odd, there are only two long parallel edges in the zone $\mathcal{Z}_{j}$:
$\\{x_{1,j},x_{m-1,j+1}\\}$ and $\\{x_{2,j},x_{m,j+1}\\}$. For $2\leq j<n$
with $j$ being even, there are $m-2$ parallel slightly shorter edges in the
zone $\mathcal{Z}_{j}$: $\\{x_{i,j},x_{i+2,j+1}\\}$ with $1\leq i\leq m-2$.
Finally, we supplement the last zone $\mathcal{Z}_{n-1}$ with the extra edge
$\\{x_{1,n-1},x_{2,n}\\}$. Then, we get all the edges for the graph
$\widetilde{H}_{1}$. At this point, the reader is invited to look at the first
graph of Figure 6. Notice that
$\hat{\bm{u}}=\sum_{\\{x_{i,j},x_{i^{\prime},j+1}\\}\in
E(\widetilde{H}_{1})}(\bm{e}_{i,j}+\bm{e}_{i^{\prime},j+1})$. Consequently,
$\hat{\bm{u}}\in{\mathbb{R}}_{+}\mathcal{A}$.

$\times$$\times$ (A) $\widetilde{H}_{1}$

$\times$$\times$ (B) $\widetilde{H}_{2}$

$\times$$\times$ (C) $\widetilde{H}_{3}$

Figure 6. Three subgraphs of $\hat{\bm{u}}$-type in the case $(m,n)=(6,7)$

    2. (b)

Next, we show that
$\hat{\bm{u}}\in\operatorname{ri}({\mathbb{R}}_{+}\mathcal{A})$. For this
purpose, we show that for any $\bm{F}\in V^{*}$ such that $H_{\bm{F}}$ is a
supporting hyperplane of ${\mathbb{R}}_{+}\mathcal{A}$ and $\hat{\bm{u}}\in
H_{\bm{F}}$, we have $\mathcal{Q}\subseteq H_{\bm{F}}$ (and equivalently,
${\mathbb{R}}_{+}\mathcal{A}\subseteq H_{\bm{F}}$). Without loss of
generality, we can assume that $\bm{F}$ is represented by the matrix
$(f_{i,j})$ with $f_{1,n}=f_{m,1}=0$. Whence, it remains to prove that
$\bm{F}$ is a multiple of the matrix $\bm{A}$, which was defined earlier in
4.6.

To prove this, we still use the $\hat{\bm{u}}$-type subgraphs. Let
$\widetilde{H}$ be such a subgraph. Since
$\hat{\bm{u}}=\sum_{\\{x_{i,j},x_{i^{\prime},j+1}\\}\in
E(\widetilde{H})}(\bm{e}_{i,j}+\bm{e}_{i^{\prime},j+1})\in H_{\bm{F}}$, we
have $\sum_{\\{x_{i,j},x_{i^{\prime},j+1}\\}\in
E(\widetilde{H})}(f_{i,j}+f_{i^{\prime},j+1})=0$. On the other hand, if
$\\{x_{i,j},x_{i^{\prime},j+1}\\}\in E(\widetilde{H})$, then
$\bm{e}_{i,j}+\bm{e}_{i^{\prime},j+1}\in\mathcal{A}$. Since $H_{\bm{F}}$ is a
supporting hyperplane, we have $f_{i,j}+f_{i^{\prime},j+1}\geq 0$. Therefore,
we have indeed $f_{i,j}=-f_{i^{\prime},j+1}$.

In addition to the subgraph $\widetilde{H}_{1}$ in the previous part, we will
construct subgraphs $\widetilde{H}_{2}$ and $\widetilde{H}_{3}$ of
$\hat{\bm{u}}$-type, such that the subgraph $\widehat{H}$ of $H^{\prime}$ with
edges $E(\widetilde{H}_{1})\cup E(\widetilde{H}_{2})\cup E(\widetilde{H}_{3})$
is a connected graph. Now, $f_{i,j}=-f_{i^{\prime},j+1}\in{\mathbb{Z}}$
whenever $\\{x_{i,j},x_{i^{\prime},j+1}\\}\in E(\widehat{H})$. Since
$\widehat{H}$ is connected, this implies that $\bm{F}$ is a multiple of
$\bm{A}$.

The graph $\widetilde{H}_{2}$ is constructed from $\widetilde{H}_{1}$ as
follows. When $j$ is odd, the zone $\mathcal{Z}_{j}$ contains two long
parallel edges in $\widetilde{H}_{1}$. We cross them for $\widetilde{H}_{2}$.
When $j$ is even, the zone $\mathcal{Z}_{j}$ contains $m-2$ parallel edges of
slope $-2$ in $\widetilde{H}_{1}$. We make $m-3$ of them as parallel edges of
slope $-1$, and the remaining one to be $\\{x_{1,j},x_{m,j+1}\\}$. At this
point, the reader is invited to look at the second graph of Figure 6.
Meanwhile, note that for each $j$, one has

$\qquad\qquad\qquad\Set{x_{i,j},x_{i^{\prime},j+1}:\\{x_{i,j},x_{i^{\prime},j+1}\\}\in
E(\widetilde{H}_{1})}=\Set{x_{i,j},x_{i^{\prime},j+1}:\\{x_{i,j},x_{i^{\prime},j+1}\\}\in
E(\widetilde{H}_{2})}.$

For later reference, we denote this vertex set as $\widetilde{V}_{j}$. It is
crucial to observe that the edges $(E(\widetilde{H}_{1})\cup
E(\widetilde{H}_{2}))\cap\mathcal{Z}_{j}$ define a connected subgraph over
$\widetilde{V}_{j}$.

The graph $\widetilde{H}_{3}$ is constructed from $\widetilde{H}_{1}$ as
follows. For $1\leq j\leq n-2$ with $j$ being odd, we change the edges
$\\{x_{1,j},x_{m-1,j+1}\\}$ and $\\{x_{m-2,j+1},x_{m,j+2}\\}$ in
$\widetilde{H}_{1}$ to the edges $\\{x_{1,j},x_{m-2,j+1}\\}$ and
$\\{x_{m-1,j+1},x_{m,j+2}\\}$ in $\widetilde{H}_{3}$. For $1\leq j\leq n-2$
with $j$ being even, we change the edges $\\{x_{1,j},x_{3,j+1}\\}$ and
$\\{x_{2,j+1},x_{m,j+2}\\}$ in $\widetilde{H}_{1}$ to the edges
$\\{x_{1,j},x_{2,j+1}\\}$ and $\\{x_{3,j+1},x_{m,j+2}\\}$ in
$\widetilde{H}_{3}$. At this point, the reader is invited to look at the third
graph of Figure 6. Note that for each $j$, the vertex set

$\Set{x_{i,j},x_{i^{\prime},j+1}:\\{x_{i,j},x_{i^{\prime},j+1}\\}\in
E(\widetilde{H}_{3})}$

intersects both $\widetilde{V}_{j}$ and $\widetilde{V}_{j+1}$. This fact makes
the combined subgraph $\widehat{H}$ to be connected.

In summary, we have completed the proof for the case when $n$ is odd. The case
when $n$ is even is analogous, so the details will be omitted. We only give
the construction of the corresponding graphs $\widetilde{H}_{1}$,
$\widetilde{H}_{2}$, and $\widetilde{H}_{3}$ for the case $(m,n)=(6,8)$, see
Figure 7. ∎

$\times$$\times$ (A) $\widetilde{H}_{1}$

$\times$$\times$ (B) $\widetilde{H}_{2}$

$\times$$\times$ (C) $\widetilde{H}_{3}$

Figure 7. Three subgraphs of $\hat{\bm{u}}$-type in the case $(m,n)=(6,8)$

######  Theorem 4.8.

Under the assumptions in 3.1, we suppose additionally that $m\geq 4$. Then, we
have

$\displaystyle\operatorname{reg}(\mathcal{F}(\mathcal{J}_{K_{m},P_{n}}))$
$\displaystyle=\operatorname{reg}(\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})))$
$\displaystyle=\begin{cases}(mn-3)-(m\cdot\frac{n}{2}),&\text{if $n$ is
even},\\\ (mn-3)-(m\cdot\frac{n+1}{2}-2),&\text{if $n$ is odd}.\end{cases}$

###### Proof.

Since
$\operatorname{in}_{\tau}(\mathcal{F}(\mathcal{J}_{K_{m},P_{n}}))=\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$,
the two special fiber rings $\mathcal{F}(\mathcal{J}_{K_{m},P_{n}})$ and
$\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ have the
same Hilbert series by [MR1390693, Proposition 2.4]. At the same time, these
two special fiber rings are Cohen–Macaulay domains by 3.6. Thus,
$\operatorname{reg}(\mathcal{F}(\mathcal{J}_{K_{m},P_{n}}))=\operatorname{reg}(\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}})))$
by [MR3838370, Corollary 2.18].

Note that the algebras
$\mathcal{F}(\operatorname{in}_{\tau}(\mathcal{J}_{K_{m},P_{n}}))$ and
${\mathbb{K}}[H]$ are isomorphic. Since $m\geq 4$, we have
$\operatorname{dim}({\mathbb{K}}[H])=mn-3$ by [MR3362802, Corollary 10.1.21].
The desired result then follows from 4.7 and the equality

$\operatorname{reg}({\mathbb{K}}[H])=\operatorname{dim}({\mathbb{K}}[H])+a({\mathbb{K}}[H])$

in the Cohen–Macaulay case, see [MR3362802, Theorem 6.4.1]. ∎

##  5\. Applications

In this section, we will explore the generalized binomial edge ideal of a
simple graph and its blowup algebras. We will compare them with the
corresponding parts of an induced subgraph. Our analysis will be based on the
regularity results from previous sections, which serve as natural lower bounds
for the comparison.

We will start with the comparison result in a more general setting. Recall
that if $G^{\prime}$ is an induced subgraph of a graph $G$, then for the
graded Betti numbers of the powers of the binomial edge ideals, one has
$\beta_{ij}(S/J^{s}_{G^{\prime}})\leq\beta_{ij}(S/J^{s}_{G})$ for all $i,j\geq
0$ and $s\geq 1$; see [JKS, Proposition 3.3]. This result for the classical
binomial edge ideal can be easily generalized to the binomial edge ideal of a
pair, as follows.

######  Setting 5.1.

For $i=1,2$, let $G_{i}$ be a simple graph on the vertex set $[n_{i}]$, and
let $H_{i}$ be its induced subgraph. Correspondingly, we have
$\mathcal{J}_{G_{1},G_{2}}$, the binomial ideal of the pair $(G_{1},G_{2})$ in
the polynomial ring ${\mathbb{K}}[\bm{X}]$, as well as
$\mathcal{J}_{H_{1},H_{2}}$, the binomial ideal of the pair $(H_{1},H_{2})$ in
the polynomial ring ${\mathbb{K}}[\bm{Y}]$. If we consider $\bm{X}$ to be an
$n_{1}\times n_{2}$ matrix of variables, then we can naturally regard $\bm{Y}$
as a $|V(H_{1})|\times|V(H_{2})|$ submatrix.

######  Theorem 5.2.

Under the assumptions in 5.1, we have

$\beta_{ij}({\mathbb{K}}[\bm{Y}]/\mathcal{J}^{s}_{H_{1},H_{2}})\leq\beta_{ij}({\mathbb{K}}[\bm{X}]/\mathcal{J}^{s}_{G_{1},G_{2}})$

for all $i,j$ and $s\geq 1$. In particular, we have

$\operatorname{reg}(\mathcal{J}^{s}_{H_{1},H_{2}})\leq\operatorname{reg}(\mathcal{J}^{s}_{G_{1},G_{2}})\qquad\text{and}\qquad\operatorname{pd}(\mathcal{J}^{s}_{H_{1},H_{2}})\leq\operatorname{pd}(\mathcal{J}^{s}_{G_{1},G_{2}})$

for all $s\geq 1$.

###### Proof.

First, we show that
$\mathcal{J}^{s}_{H_{1},H_{2}}=\mathcal{J}^{s}_{G_{1},G_{2}}\cap{\mathbb{K}}[\bm{Y}]$
for all $s\geq 1$. Since the natural generators of
$\mathcal{J}^{s}_{H_{1},H_{2}}$ are automatically contained in
$\mathcal{J}^{s}_{G_{1},G_{2}}$, one has
$\mathcal{J}^{s}_{H_{1},H_{2}}\subseteq\mathcal{J}^{s}_{G_{1},G_{2}}\cap{\mathbb{K}}[\bm{Y}]$.
For the converse inclusion, let
$g\in\mathcal{J}^{s}_{G_{1},G_{2}}\cap{\mathbb{K}}[\bm{Y}]$. We can write $g$
as a finite sum

$g=\sum_{\begin{subarray}{c}(e_{i},f_{i})\in E(G_{1})\times E(G_{2}),\\\ 1\leq
i\leq
s\end{subarray}}h_{(e_{1},f_{1}),\ldots,(e_{s},f_{s})}p_{(e_{1},f_{1})}\cdots
p_{(e_{s},f_{s})},$

where $h_{(e_{1},f_{1}),\ldots,(e_{s},f_{s})}\in{\mathbb{K}}[\bm{X}]$. Now,
consider the ${\mathbb{K}}$-algebra homomorphism
$\pi:{\mathbb{K}}[\bm{X}]\to{\mathbb{K}}[\bm{Y}]$ by setting

$\pi(x_{i,j})=\begin{cases}x_{i,j},&\text{if $x_{i,j}$ is a variable in
$\bm{Y}$,}\\\ 0,&\text{otherwise}.\end{cases}$

Thus,

$\pi(p_{(e,f)})=\begin{cases}p_{(e,f)},&\text{if $(e,f)\in E(H_{1})\times
E(H_{2})$,}\\\ 0,&\text{otherwise.}\end{cases}$

Since $g\in{\mathbb{K}}[\bm{Y}]$, we have $\pi(g)=g$. Therefore, we get

$\displaystyle g$ $\displaystyle=\sum_{\begin{subarray}{c}(e_{i},f_{i})\in
E(G_{1})\times E(G_{2}),\\\ 1\leq i\leq
s\end{subarray}}\pi(h_{(e_{1},f_{1}),\ldots,(e_{s},f_{s})})\pi(p_{(e_{1},f_{1})})\cdots\pi(p_{(e_{s},f_{s})})$
$\displaystyle=\sum_{\begin{subarray}{c}(e_{i},f_{i})\in E(H_{1})\times
E(H_{2}),\\\ 1\leq i\leq
s\end{subarray}}\pi(h_{(e_{1},f_{1}),\ldots,(e_{s},f_{s})})p_{(e_{1},f_{1})}\cdots
p_{(e_{s},f_{s})}.$

Thus, $g\in\mathcal{J}^{s}_{H_{1},H_{2}}$. This completes our proof for
$\mathcal{J}^{s}_{H_{1},H_{2}}=\mathcal{J}^{s}_{G_{1},G_{2}}\cap{\mathbb{K}}[\bm{Y}]$.

Consequently, ${\mathbb{K}}[\bm{Y}]/\mathcal{J}^{s}_{H_{1},H_{2}}$ is a
${\mathbb{K}}$-subalgebra of
${\mathbb{K}}[\bm{X}]/\mathcal{J}^{s}_{G_{1},G_{2}}$. Let
$\overline{\pi}:{\mathbb{K}}[\bm{X}]/\mathcal{J}^{s}_{G_{1},G_{2}}\to{\mathbb{K}}[\bm{Y}]/\mathcal{J}^{s}_{H_{1},H_{2}}$
be the homomorphism induced by $\pi$. Since
$\pi(\mathcal{J}^{s}_{G_{1},G_{2}})\subseteq\mathcal{J}^{s}_{H_{1},H_{2}}$,
the map $\overline{\pi}$ is well-defined. Notice that the restriction of
$\overline{\pi}$ to ${\mathbb{K}}[\bm{Y}]/\mathcal{J}^{s}_{H_{1},H_{2}}$ is
the identity map. Thus, $\overline{\pi}$ is surjective, and
${\mathbb{K}}[\bm{Y}]/\mathcal{J}^{s}_{H_{1},H_{2}}$ is an algebra retract of
${\mathbb{K}}[\bm{X}]/\mathcal{J}^{s}_{G_{1},G_{2}}$. Now, the expected
inequalities follow from [MR1789447, Corollary 2.5]. ∎

######  Corollary 5.3.

Let $G$ be a simple graph and $G^{\prime}$ be its induced subgraph. Then we
have
$\operatorname{reg}(\mathcal{J}^{s}_{K_{m},G^{\prime}})\leq\operatorname{reg}(\mathcal{J}^{s}_{K_{m},G})$
for all $s\geq 1$.

Kumar proved in [MR4405525, Theorems 3.5 and 4.6] that if $G^{\prime}$ is an
induced subgraph of a graph $G$, then
$\operatorname{reg}(\mathcal{R}(J_{G^{\prime}}))\leq\operatorname{reg}(\mathcal{R}(J_{G}))$
and
$\operatorname{reg}(\mathcal{F}(J_{G^{\prime}}))\leq\operatorname{reg}(\mathcal{F}(J_{G}))$
for the regularities of the blowup algebras of classical binomial edge ideals.
We can generalize this to the binomial edge ideals of pairs.

######  Theorem 5.4.

Under the assumptions in 5.1, we have

$\operatorname{reg}(\mathcal{R}(\mathcal{J}_{H_{1},H_{2}}))\leq\operatorname{reg}(\mathcal{R}(\mathcal{J}_{G_{1},G_{2}}))$

and

$\operatorname{reg}(\mathcal{F}(\mathcal{J}_{H_{1},H_{2}}))\leq\operatorname{reg}(\mathcal{F}(\mathcal{J}_{G_{1},G_{2}})).$

###### Proof.

Let $\pi:{\mathbb{K}}[\bm{X}]\to{\mathbb{K}}[\bm{Y}]$ be the map defined in
the proof of Theorem 5.2. We have
$\pi(\mathcal{J}_{G_{1},G_{2}}^{s})=\mathcal{J}_{H_{1},H_{2}}^{s}=\mathcal{J}_{G_{1},G_{2}}^{s}\cap{\mathbb{K}}[\bm{Y}]$
for all $s\geq 0$. This fact induces the graded embedding map

$\iota:\mathcal{R}(\mathcal{J}_{H_{1},H_{2}}){\hookrightarrow}\mathcal{R}(\mathcal{J}_{G_{1},G_{2}})$

as well as a graded epimorphism

$\pi^{*}:\mathcal{R}(\mathcal{J}_{G_{1},G_{2}})\twoheadrightarrow\mathcal{R}(\mathcal{J}_{H_{1},H_{2}}).$

Notice that ${\pi}^{*}\circ\iota$ is the identity map on
$\mathcal{R}(\mathcal{J}_{H_{1},H_{2}})$. It follows that
$\mathcal{R}(\mathcal{J}_{H_{1},H_{2}})$ is an algebra retract of
$\mathcal{R}(\mathcal{J}_{G_{1},G_{2}})$.

Meanwhile, notice that

$\displaystyle\mathcal{F}(\mathcal{J}_{G_{1},G_{2}})$
$\displaystyle\cong\mathcal{R}(\mathcal{J}_{G_{1},G_{2}})\otimes_{{\mathbb{K}}[\bm{X}]}{\mathbb{K}}\cong{\mathbb{K}}[p_{(e,f)}:e\in
E(G_{1}),f\in E(G_{2})]\subseteq{\mathbb{K}}[\bm{X}],$ and
$\displaystyle\mathcal{F}(\mathcal{J}_{H_{1},H_{2}})$
$\displaystyle\cong\mathcal{R}(\mathcal{J}_{H_{1},H_{2}})\otimes_{{\mathbb{K}}[\bm{X}]}{\mathbb{K}}\cong\mathcal{R}(\mathcal{J}_{H_{1},H_{2}})\otimes_{{\mathbb{K}}[\bm{Y}]}{\mathbb{K}}$
$\displaystyle\cong{\mathbb{K}}[p_{(e,f)}:e\in E(H_{1}),f\in
E(H_{2})]\subseteq{\mathbb{K}}[\bm{Y}]\subseteq{\mathbb{K}}[\bm{X}].$

Therefore, we have a graded embedding map
$\iota^{\triangle}:\mathcal{F}(\mathcal{J}_{H_{1},H_{2}})\hookrightarrow\mathcal{F}(\mathcal{J}_{G_{1},G_{2}})$.
On the other hand, by tensoring $\pi^{*}$ with ${\mathbb{K}}$, we have an
induced graded epimorphism
$\pi^{\triangle}:\mathcal{F}(\mathcal{J}_{G_{1},G_{2}})\twoheadrightarrow\mathcal{F}(\mathcal{J}_{H_{1},H_{2}})$.
Notice that ${\pi}^{\triangle}\circ\iota^{\triangle}$ is the identity map on
$\mathcal{F}(\mathcal{J}_{H_{1},H_{2}})$. It follows that
$\mathcal{F}(\mathcal{J}_{H_{1},H_{2}})$ is an algebra retract of
$\mathcal{F}(\mathcal{J}_{G_{1},G_{2}})$.

To complete the proof, it remains to apply [MR1789447, Corollary 2.5]. ∎

######  Corollary 5.5.

Let $G$ be a graph and $G^{\prime}$ be its induced subgraph. Then,

$\operatorname{reg}(\mathcal{R}(\mathcal{J}_{K_{m},G^{\prime}}))\leq\operatorname{reg}(\mathcal{R}(\mathcal{J}_{K_{m},G}))$

and

$\operatorname{reg}(\mathcal{F}(\mathcal{J}_{K_{m},G^{\prime}}))\leq\operatorname{reg}(\mathcal{F}(\mathcal{J}_{K_{m},G})).$

######  Corollary 5.6.

Let $G$ be a graph which contains an induced path with $n$ vertices. Then, we
have

$\operatorname{reg}\left(\frac{S}{\mathcal{J}^{t}_{K_{m},G}}\right)\geq
2(t-1)+(n-1)$

for each $t\geq 1$. Furthermore,

$\operatorname{reg}(\mathcal{R}(\mathcal{J}_{K_{m},G}))\geq(m-1)\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n-1}{2}\right\rfloor,$

and

$\operatorname{reg}(\mathcal{R}(\mathcal{J}_{K_{m},G}))\geq\begin{cases}(mn-3)-(m\cdot\frac{n}{2}),&\text{if
$n$ is even},\\\ (mn-3)-(m\cdot\frac{n+1}{2}-2),&\text{if $n$ is
odd}.\end{cases}$

###### Proof.

These results follow from Theorems 5.2, 3.11, 5.4, 4.2 and 4.8. ∎

######  Acknowledgment.

The authors are grateful to the software systems Macaulay2 [M2] and Normaliz
[Normaliz], for serving as excellent sources of inspiration. This work is
supported by the Natural Science Foundation of Jiangsu Province (No.
BK20221353). In addition, the first author is partially supported by the Anhui
Initiative in Quantum Information Technologies (No. AHY150200) and the
“Innovation Program for Quantum Science and Technology” (2021ZD0302902). And
the second author is supported by the Foundation of the Priority Academic
Program Development of Jiangsu Higher Education Institutions.

## References