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\title
%% The optionnal argument is the short version for headings.
[Bounds for the regularity of product of edge ideals]
%% The mandatory argument is for the title page, summaries, headings
%% if optionnal void.
{Bounds for the regularity of product of edge ideals}
\author[A.~Banerjee]{\firstname{Arindam} \lastname{Banerjee}}
\address{
Department of Mathematics\\
Indian Institute of Technology Kharagpur\\
721302\\
India}
\email{123.arindam@gmail.com}
\author[P.~Das]{\firstname{Priya} \lastname{Das}}
\address{Department of Mathematics\\
National Institute of Technology\\
Calicut\\
Kerala 673601\\
India}
\email{priya.math88@gmail.com}
\author[S ~Selvaraja]{\firstname{S} \lastname{Selvaraja}}
\address{Chennai Mathematical Institute\\
H1 SIPCOT IT Park\\
Siruseri\\
Kelambakkam\\
Chennai\\
India 603103}
\email{selva.y2s@gmail.com, sselvaraja@cmi.ac.in}
\thanks{The first author is partially supported by DST, Govt of India under
the DST-INSPIRE (DST/Inspire/04/2017/000752) Faculty Scheme. The third author is partially supported by DST, Govt of India under
the DST-INSPIRE (DST/Inspire/04/2019/001353) Faculty Scheme.}
\equalenv{author_thm}{cedram_thm}
\keywords{Castelnuovo--Mumford regularity, product of edge ideals, linear resolution}
\subjclass{13D02, 05E45, 05C70}
\datereceived{2021-08-01}
\daterevised{2022-02-15}
\dateaccepted{2022-02-21}
\begin{document}
\newtheorem{setup}{Set-up}
\newtheorem{dis}{Discussion}
\newtheorem{obs}{Observation}
\begin{abstract}
Let $I$ and $J$ be edge ideals in a polynomial ring $R = \mathbb{K}[x_1,\ldots,x_n]$ with $I \subseteq J$.
In this paper, we obtain a general upper and lower bound for the Castelnuovo--Mumford regularity
of $IJ$ in terms of certain invariants associated with $I$ and $J$. % associated to graphs in terms of the
%induced matching number and co-chordal cover number of graphs.
Using these results, we explicitly compute the regularity of $IJ$ for several classes of edge ideals.
In particular, we compute the regularity of $IJ$ when $J$ has a linear resolution.
Finally, we compute the precise expression for the regularity
of $J_1 J_2\cdots J_d$, $d \in \{3,4\}$,
where $J_1,\ldots,J_d$ are edge ideals,
$J_1 \subseteq J_2 \subseteq \cdots \subseteq J_d$ and
$J_d$ is the edge ideal of a complete graph.
\end{abstract}
\maketitle
\section{Introduction}
Let $M$ be a finitely generated graded module over $R = \K[x_1,\ldots,x_n]$ where
$\K$ is a field.
The Castelnuovo--Mumford regularity (or simply, regularity) of $M$, denoted by $\reg(M)$, is defined
to be the least integer $i$ so that, for every $j$, the $j^{\text{th}}$ syzygy of $M$
is generated in degrees $\leq i+j$.
Regularity is an important invariant
in commutative algebra and algebraic geometry that measures the computational
complexity of ideals, modules, and sheaves.
In this paper, we study bounds on the regularity of products of ideals in a
polynomial ring.
The regularity of products of ideals was studied first by Conca and Herzog \cite{CH03}.
They studied whether for homogeneous ideal $I$ and finitely
generated graded module $M$ over $R$, one has
%\begin{equation*}\label{CH:eq}
$\reg(IM) \leq \reg(I) + \reg(M).$
%\end{equation*}
This question is essentially a generalization of the simple fact that the highest degree of a
generator of the product $IM$ is bounded above by the sum of the highest degree of a generator
of $M$ and the highest degree of a generator of $I$. The answer to this question is negative
in general. There are several examples already known with $M=I$ such that
$\reg(I^2)>2\reg(I)$,
see Sturmfels \cite{Stu00}.
They found some special classes of ideals $I$ and modules $M$ such that $\reg(IM) \leq \reg(I)+\reg(M)$.
In particular, they showed that if $I$ is a homogeneous ideal in a
polynomial ring $R$ with $\dim(R/I)\leq 1$, then
$\reg(IM) \leq \reg(I)+ \reg(M)$
for any finitely generated module
$M$ over $R$.
In case $M$ is also a homogeneous ideal, the situation becomes particularly interesting.
For example, Sidman proved that if $\dim(R/(I+J)) \leq 1$,
then the regularity of $IJ$ is bounded above by $\reg(I) + \reg(J)$, \cite{Sid02}.
Also, she proved that if two ideals of $R$, say $I$ and $J$, define schemes whose
intersection is a finite set of points,
then $\reg(IJ)\leq \reg(I)+\reg(J)$.
Chardin, Minh and Trung \cite{CMT07} proved that if
$I$ and $J$ are monomial complete intersections, then $\reg(IJ) \leq \reg(I)+\reg(J)$.
Cimpoea\c{s} \cite{Cim09} proved that for two monomial ideals of Borel type $I,J$, we have $\reg(IJ) \leq \reg(I)+\reg(J)$.
Caviglia \cite{Cav07} and Eisenbud, Huneke and Ulrich \cite{EHU06}
studied the more general problem of the regularity of tensor products and various $\Tor$ modules
of $R/I$ and $R/J$.
In this paper, we study the same problem for the case of edge ideals and seek better bounds by
exploiting the combinatorics of the underlying graph.
Let $G$ be a finite simple graph without isolated vertices on the vertex set $\{x_1,\ldots,x_n\}$ and $I(G)\coloneqq (\{x_ix_j \mid
\{x_i, x_j\} \in E(G)\}) \subset R=\K[x_1,\ldots,x_n]$ the \textit{edge ideal} corresponding to $G$.
In general, computing the regularity of $I(G)$ is NP-hard
\cite[Corollary 23]{russ}.
Several recent papers have related the $\reg(I(G))$ with various
invariants of the graph $G$ (see \cite{BBH17} for
a survey in this direction). A primary inspiration for this paper is
Katzman's and Woodroofe's theorem from \cite{kat} and \cite{russ}. They showed that if $G$ is a graph, then
\begin{equation}\label{ag_reg_chg}
\nu(G) + 1 \leq \reg(I(G)) \leq\cochord(G) + 1,
\end{equation}
where $\nu(G)$ denotes the induced matching number of
$G$ (see Section~\ref{pre} for the definition) and
$\cochord(G)$ denotes the co-chordal cover number of $G$ (see Section~\ref{pre} for the definition).
In this context, a natural question is if
$I$ and $J$ are edge ideals in $R$,
then what is the regularity of $IJ$?
This question give rise to two directions of research. One direction is to obtain
the precise expression for $\reg(IJ)$ for particular classes of edge ideals.
Another direction is to obtain upper
and lower bounds for $\reg(IJ)$ using combinatorial invariants associated to graphs.
Therefore, one may ask for edge ideals
$I$ and $J$,
\begin{enumerate}[label=(Q\arabic*)]
\item can we find lower and upper bounds for the regularity of $IJ$ using combinatorial
invariants associated to the graphs?
\item can we find precise expressions for the regularity of $IJ$
for particular classes of graphs?
\end{enumerate}
This paper revolves around these two questions.
Computing the regularity of products of two edge ideals of graphs seems more challenging
compared to the regularity of edge ideals of graphs. Even in the case of simple classes of graphs, a formula for
the regularity of products of two edge ideals is not known.
So, naturally one restricts the attention to important subclasses.
We are therefore interested in families of edge ideals $I$ and $J$ with $I \subseteq J$.
First, we prove a lower bound for the regularity of the product of more than two edge ideals. More precisely,
let $J_1=I(G_1),\dots, J_d=I(G_d)$ be edge ideals of graphs $G_1,\dots,G_d$ with $J_1 \subseteq \cdots \subseteq J_d$.
%Let $I$ and $J$ be as in Set-up~\ref{setup}.
%(with hypothesis as in Theorem~\ref{stu-lemma}).
Then we prove $2d+\nu_{G_1 \cdots G_d}-1 \leq \reg(J_1 \cdots J_d)$,
where $\nu_{G_1 \cdots G_d}$ denotes the joint induced matching number of $G_i$
(see Section~\ref{pre} for the definition) for all
$1 \leq i \leq d$
(Theorem~\ref{main-lower}).
%In \cite{selvi_ha} and \cite{JS21}, the authors extended the above results to all powers i.e.,
%for a graph $G$,
%\[
% 2s+\nu(G)-1 \leq \reg(I(G)^s) \leq 2s+\cochord(G)-1 \text{ for all $s \geq 1$}.
%\]
We prove an upper bound for the regularity of product of two edge
ideals in terms of
co-chordal cover numbers. We prove that if
$G$ is a graph and $H$ is a subgraph of $G$ with $I=I(H)$ and $J=I(G)$, then
$\reg(IJ) \leq \max \{\cochord(G)+3,~\reg(I)\}.$
In particular,
$\reg(IJ) \leq \max \{\cochord(G)+3,~\cochord(H)+1\}$ (Theorem~\ref{main}).
The above bound is inspired by the general upper bound for the regularity of powers
of edge ideals given in \cite[Theorem 3.6]{jayanthan} and \cite[Theorem 4.4]{JS21}.
Theorem~\ref{main} has a number of interesting consequences. For example,
Corollary~\ref{mat-bound} says that if $H$ is any subgraph of $G$,
then $\reg(IJ) \leq \ma(G)+3$ where $\ma(G)$ denotes the matching number of $G$.
On the other hand, Corollary~\ref{ind-eql} says that
if $H$ is an induced subgraph of $G$, then
%\begin{equation*}\label{main:induced}
$\nu(H)+3 \leq \reg(IJ) \leq \cochord(G)+3.$
%\end{equation*}
We then move on to compute the precise expression for the regularity of product of edge ideals.
As a consequence of the techniques that we have developed, we explicitly compute the regularity of $IJ$
when $J$ has a linear resolution (Theorem~\ref{regularity}). %As an immediate
%consequence, we get that $IJ$ has linear resolution for several classes of graphs
%(Corollary~\ref{se-cl}).
Next, we study the regularity of products of more than two edge ideals.
We compute the precise expression for $\reg(J_1\cdots J_d)$ when $J_1 \subseteq \cdots \subseteq J_d$, $d \in \{3,4\}$ and
$J_d$ is the edge ideal of complete graph (Theorem~\ref{prd-ideals}).
We use Theorem~\ref{main} and Theorem~\ref{prd-ideals} to get an upper bound for the regularity of
$J_1 \cdots J_d$ in terms of co-chordal cover numbers (Corollary~\ref{prd-us}).
As an immediate consequence of these results, we give sufficient conditions for product of edge ideals
to have linear resolutions (Corollary~\ref{se-cl}, Corollary~\ref{ls-pr}).
Our paper is organized as follows. In Section~\ref{pre}, we collect the necessary notions,
terminologies and some results that are used subsequently.
In Section~\ref{tech} we prove several
technical lemmas which are needed for the proof of our main results, which appear in
Sections~\ref{2-edgeideals} and~\ref{precise}.
%In Section~\ref{app}, we give an example describing the procedure that we explanied in Section~\ref{tech}.
\section{Preliminaries}\label{pre}
In this section, we set up basic definitions and notation needed for the main results. Let $G$ be a finite simple graph with vertex set $V(G)$
and edge set $E(G)$.
A subgraph $L \subseteq G$ is called \textit{induced} if $\{u,v\}$ is an edge of $L$
if and only if $u$ and $v$ are vertices of $L$ and $\{u,v\}$ is an edge of $G$.
For $\{u_1,\ldots,u_r\} \subseteq V(G)$, let $N_G(u_1,\ldots,u_r) = \{v \in V (G)\mid \{u_i, v\} \in E(G)~
\text{for some $1 \leq i \leq r$}\}$
and $N_G[u_1,\ldots,u_r]= N_G(u_1,\ldots,u_r) \cup \{u_1,\ldots,u_r\}$.
For $U \subseteq V(G)$, we denote by $G \setminus U$
the induced subgraph of $G$ on the vertex set $V(G) \setminus U$.
Let $C_k$ denote the cycle on $k$ vertices.
Let $G$ be a graph. We say $2$ non-adjacent edges
$\{f_1,f_2\}$ form a $2K_2$ in $G$ if $G$ does not have an edge with one
endpoint in $f_1$ and the other in $f_2$. A graph without $2K_2$ is called $2K_2$-free
also called \textit{gap-free} graph.
A \textit{matching} in a graph $G$ is a subgraph consisting of pairwise disjoint edges.
The \textit{matching number} of $G$, denoted by $\ma(G)$, is the
maximum cardinality among matchings of $G$.
%and the minimum matching number of $G$, denoted by $\MM(G)$, is
%the minimum cardinality among maximal matchings of $G$.
If the subgraph is an induced subgraph, the matching is an \textit{induced matching}.
The largest size of an induced matching in $G$ is called its \textit{induced
matching number} and denoted by $\nu(G)$.
The \textit{complement} of $G$, denoted by $G^c$, is the graph on the same
vertex set as $G$, where $\{u,v\}$ is an edge of $G^c$ if and only $\{u,v\}\notin E(G)$.
A graph $G$ is \textit{chordal} if every induced
cycle in $G$ has length $3$, and is co-chordal if $G^c$ is chordal.
%A graph $G$ is said to be \textit{weakly chordal} if
%neither $G$ nor $G^c$ contain induced cycle of length $5$ or more.
%It is straightforward to show that a chordal graph is weakly chordal.
The \textit{co-chordal cover number}, denoted $\cochord(G)$, is the
minimum number $n$ such that there exist co-chordal subgraphs
$H_1,\ldots, H_n$ of $G$ with $E(G) = \bigcup_{i=1}^n E(H_i)$.
%By \cite[p. 2]{hibi}, \cite[Theorem 1]{russ}, \cite[p. 10]{jayanthan},
%for any graph $G$, we have
%\begin{equation*}\label{inva-ineq}
% \nu(G) \leq \cochord(G) \leq \MM(G) \leq \ma(G).
%\end{equation*}
Consider graphs $G_i$ for $1 \leq i \leq d$ where $G_i$ is a subgraph
of $G_{i+1}$ for all $1 \leq i \leq d-1$.
The largest size of an induced matching in $G_i$ for all $1 \leq i \leq d$ is called the \textit{joint induced
matching number} and denoted by $\nu_{G_1 \cdots G_d}$.
Note that if $G_i$ is an induced subgraph of $G_{i+1}$ for all
$1 \leq i \leq d-1$, then $\nu_{G_1 \cdots G_d}=\nu(G_1)$.
\begin{exam}
\label{ex:2.1}
Let $G$ be the graph as shown in Figure~\ref{fig:graph_G}.
Then $\{ \{x_1,x_2\}$, $\{x_3,x_4\}$, $\{x_5,x_6\}$, $\{x_7,x_8\}\}$ forms a matching of $G$, but not an induced matching.
The set
$\{ \{x_1,x_2\}, \{x_4,x_5\}\}$ forms an induced matching.
Then $\nu(G) \geq 2$.
It is not hard to verify that $\nu(G)=2$.
Let $H$ be a subgraph of $G$ with $E(H)=\{ \{x_1,x_2\}, \{x_3,x_4\}\}$.
Since $H$ is a disjoint union two edges, $\nu(H)=2$. The set $\{\{x_1,x_2\}\}$ forms an
induced matching of $G$ and $H$. Then $\nu_{HG} \geq 1$. Since the set
$\{\{x_1,x_2\},\{x_3,x_4\}\}$ forms an induced matching of $H$ but not in $G$,
$\nu_{HG} =1$.
\begin{figure}
\begin{tikzpicture}[scale=.64]
%\clip(1.42,1.6) rectangle (13.66,5.6);
\draw [line width=1.pt] (4.,5.)-- (5.,5.);
\draw [line width=1.pt] (4.,5.)-- (3.,4.);
\draw [line width=1.pt] (3.,4.)-- (3.,3.);
\draw [line width=1.pt] (3.,3.)-- (4.,2.);
\draw [line width=1.pt] (4.,2.)-- (5.,2.);
\draw [line width=1.pt] (5.96,2.86)-- (5.,2.);
\draw [line width=1.pt] (6.,4.)-- (5.96,2.86);
\draw [line width=1.pt] (5.,5.)-- (6.,4.);
\begin{scriptsize}
\draw [fill=black] (4.,5.) circle (2.0pt);
\draw[color=black] (4.19,5.38) node {$x_8$};
\draw [fill=black] (5.,5.) circle (2.0pt);
\draw[color=black] (5.19,5.38) node {$x_1$};
\draw [fill=black] (3.,4.) circle (2.0pt);
\draw[color=black] (2.57,4.14) node {$x_7$};
\draw [fill=black] (3.,3.) circle (2.0pt);
\draw[color=black] (2.53,3.06) node {$x_6$};
\draw [fill=black] (4.,2.) circle (2.0pt);
\draw[color=black] (3.95,1.76) node {$x_5$};
\draw [fill=black] (5.,2.) circle (2.0pt);
\draw[color=black] (5.35,2.0) node {$x_4$};
\draw [fill=black] (5.96,2.86) circle (2.0pt);
\draw[color=black] (6.35,3.04) node {$x_3$};
\draw [fill=black] (6.,4.) circle (2.0pt);
\draw[color=black] (6.19,4.38) node {$x_2$};
\end{scriptsize}
\end{tikzpicture}
\caption{Graph $G$ for Example~\ref{ex:2.1}}
\label{fig:graph_G}
\end{figure}
Let $H_1$, $H_2$ and $H_3$ be subgraphs of $G$ with
$E(H_1)=\{\{x_1,x_2\},\{x_2,x_3\},\{x_3,x_4\} \}$,
$E(H_2)=\{\{x_4,x_5\},\{x_5,x_6\},\{x_6,x_7\} \}$ and
$E(H_3)=\{\{x_7,x_8\},\{x_8,x_1\}\}$ respectively.
We can seen that $H_1$, $H_2$ and $H_3$ are
co-chordal subgraphs of $G$ and $E(G)=\bigcup_{i=1}^3E(H_i)$.
Therefore, $\cochord(G) \leq 3$. It is also
not hard to verify that $\cochord(G)=3$.
\end{exam}
Polarization is a process to obtain a squarefree monomial ideal from a given monomial ideal.
\begin{defi}\label{pol_def}
Let $M=x_1^{a_1}\cdots x_n^{a_n}$ be a monomial in $R=\K[x_1,\dots,x_n]$.
Then we define the squarefree monomial $P(M)$ \textup{(}\emph{polarization} of $M$\textup{)} as
$$P(M)=x_{11}\cdots x_{1a_1}x_{21}\cdots x_{2a_2}\cdots x_{n1}\cdots x_{na_n}$$ in the polynomial ring
$R_1=\K[x_{ij} \mid 1\leq i\leq n,1\leq j\leq a_i]$.
If $I=(M_1,\dots,M_q)$ is an ideal in $R$, then the polarization of
$I$, denoted by $\widetilde{I}$, is defined as $\widetilde{I}=(P(M_1),\dots,P(M_q))$.
\end{defi}
Let $M$ be a graded $R = \K[x_1,\ldots,x_n]$ module. For non-negative
integers $i, j$, let $\beta_{i,j}(M)$ denote the $(i,j)$-th graded Betti
number of $M$.
In this paper, we repeatedly use an important property of the
polarization, namely:
\begin{coro} \cite[Corollary 1.6.3(a)]{Herzog'sBook}\label{pol_reg}
Let $I \subseteq R=\K[x_1,\ldots,x_n]$ be a monomial ideal.
If $\widetilde{I} \subseteq \widetilde{R}$ is a polarization of $I$, then for
all $i,j$, we have $\beta_{i,j}(R/I)=\beta_{i,j}(\widetilde{R} / \widetilde {I})$.
In particular, $\reg(R/I)=\reg (\widetilde R/\widetilde I)$.
\end{coro}
\section{Technical lemmas}\label{tech}
In this section we prove several technical results concerning the graph associated with
$\widetilde{(IJ:ab)}$, for any $ab \in I$, where $I$ and $J$ are edge ideals and $I \subseteq J$.
We first fix the set-up that we consider throughout this paper.
\begin{setup}\label{setup}
Let $G$ be a graph and $H$ be a subgraph of $G$. Set $I=I(H)$ and $J=I(G)$.
%Let $I$ and $J$ be the edge ideal of a graph $H$ and $G$ respectively
% with $I \subseteq J$.
\end{setup}
For a monomial ideal $K$, let $\mathcal{G}(K)$ denote the minimal generating set of $K$.
For a monomial $m \in R = \K[x_1,\ldots,x_n]$, the \emph{support} of $m$ is the set of variables appearing in
$m$ and is denoted by $\supp(m)$, i.e. $\supp(m) = \{x_i \mid x_i \text{ divides } m \}$.
The following result is used repeatedly in this paper.
\begin{lemma} \label{stu-lemma}
Let $I$ and $J$ be as in Set-up~\ref{setup}.
Then the \emph{colon ideal}
$(IJ:ab)$ is a generated by
quadratic monomial ideal for any $ab \in I$.
%Set $I=I(H)$ and $J=I(G)$.
More precisely,
\[
(IJ:ab)=J+ K_1 + K_2,
\]
where
$K_1=(pq \mid p \in N_G(a) \text{ and } q \in N_H(b))$ and
$K_2=(rs \mid r \in N_H(a) \text{ and } s\in N_G(b))$.
\end{lemma}
\begin{proof}
% By Lemma~\ref{lemma-degree}, $(IJ:ab)$ is generated by
% quadratic monomial ideal for any $\{a,b\} \in E(H)$.
Let $m \in \mathcal{G}((IJ:ab))$.
By degree considerations $m$ cannot have degree $1$.
Suppose $\deg(m) \geq 3$.
Then there exist $e \in \mathcal{G}(I)$ and
$f \in \mathcal{G}(J)$ such that $ef\mid mab$. Since $m$ is a minimal monomial generator of
$(IJ:ab)$, there does not
exist $m'$, $m' \neq m$ and $m' \mid m$ such that $ef\mid m'ab$.
If there exists $g \in \mathcal{G}(J)$ such that $g\mid m$, then the minimality
of $m$ and $g \in (IJ:ab)$ imply $g=m$.
This is a contradiction to $\deg(m) \geq 3$.
Therefore, $\deg(m)=2$. We assume that $g \nmid m$ for any $g \in \mathcal{G}(J)$.
Then $e \nmid ab$. Let $e=ax$, where $x\mid m$. Therefore,
$xf\mid mb$.
%Now, if $f\mid m$ then by minimality of $m$, $f=m$. So,
%let
If $f=by$, where $y\mid (\frac{m}{x})$, then $xy\mid m$. Hence,
by minimality of $m$, $m$ is a quadratic monomial.
Similarly, for $e=bx$ we can prove that $m$ is quadratic in a similar manner.
Clearly, $J+K_1+K_2 \subseteq (IJ:ab)$. We need to prove the reverse inclusion.
Let $uv \in \mathcal{G}(IJ:ab)$. If $uv \in J$, then we are done. Suppose $uv \notin J$.
Since $uvab \in IJ$, we have the following cases $ua \in I $ and $vb \in J$ or
$ua \in J $ and $vb \in I$ or $ub \in I $ and $va \in J$ or $ub \in J $ and $va \in I$.
In all cases, one can show that either $uv \in K_1$ or $uv \in K_2$. Therefore,
$(IJ:ab)=J+ K_1 + K_2$.
\end{proof}
Let $I$ and $J$ be as in Set-up~\ref{setup}.
Then
for any $ab \in I$, $\widetilde{(IJ:ab)}$ is a quadratic squarefree monomial ideal by
Lemma~\ref{stu-lemma}. There exists a graph $\PP$ associated to $\widetilde{(IJ:ab)}$.
Suppose $xy$ is a minimal generator of $(IJ:ab)$.
If $x \neq y$, then set $\{[x,y]\}=\{x,y\}$ and $\{[x,y]\}$ is
an edge of $\PP$.
If $x=y$, then set $\{[x,y]\}=\{x,z_{x}\}$, where $z_x$ is a new vertex of $\PP$, and $\{[x,y]\}$ is
an edge of $\PP$.
Observe that
$G$ is a subgraph of $\PP$, i.e.
$V(G) \subseteq V(\PP)$ and $E(G) \subseteq E(\PP)$.
For example, let $I=(x_4x_5,x_5x_6,x_4x_6)$ and
$J=(x_1x_2,x_2x_3,x_3x_4,x_4x_5,x_5x_6,x_1x_6,x_4x_6)$. Then $(IJ:x_4x_5)=J+(x_6^2,x_3x_6) \subset
\K[x_1,\ldots,x_6]$ and $\widetilde{(IJ:x_4x_5)}=J+(x_6z_{x_6},x_3x_6)
\subset \K[x_1,\ldots,x_6,z_{x_6}]$. Let $\PP$ be the graph associated to $\widetilde{(IJ:x_4x_5)}$.
Then $V(\PP)=V(G) \cup \{z_{x_6}\}$ and $E(\PP)=E(G) \cup \{\{[x_6,x_6]\}, \{x_3,x_6\}\}$.
The following is a useful result on co-chordal graphs that allow us to assume
certain order on their edges.
\begin{lemma}\cite[Lemma 1 and Theorem 2]{benzaken}\label{main-lemmatech}
Let $G$ be a graph and $E(G)=\{e_1\ldots,,e_t\}$. Then $G$ is a co-chordal graph if and only if
there is an ordering of edges of $G$,
$e_{i_1}< \cdots 3$.
Set $\FA_j=g_1g_2g_3$ and $\FA_i=f_1f_2f_3$, where $g_i,f_i \in J_i$
for all $1 \leq i \leq 3$. Since $s \geq 3$, we have $u \mid g_i$ and $u \nmid f_i$ for all $1 \leq i \leq 3$.
Set $g_1=ua$, $g_2=ub$, $g_3=uc$, $f_1=x_1x_2$, $f_2=x_3x_4$ and $f_3=x_5x_6$ ($x_i$ may be equal to $x_j$, for some $1 \leq i,j \leq 5$). Note that $abc \mid f_1f_2f_3$.
If $ab \mid f_i$ and $c \mid f_j$, for some $1 \leq i,j \leq 3$,
then $uaubf_jf_k \in \mathcal{J}$, where $k \neq i,j$.
If $a \mid f_i$, $b \mid f_j$, $c \mid f_k$ for some
$1 \leq i,j,k \leq 3$, then $uaub f_k (\frac{f_if_j}{ab}) \in \mathcal{J}$.
Therefore $u^2 \in (\mathcal{J}:\FA_i)$. Hence the claim.
Let $m \in \mathcal{G}(\mathcal{J}:\FA_i).$ By degree consideration $m$ cannot have degree 1.
We now claim that $\deg(m)=2$. %, for any $m \in \mathcal{G}(\mathcal{J}:\FA_i)$.
%Let $m$ be a minimal monomial generator of $(\mathcal{J}:\FA_i)$.
Suppose $|\supp(m)| \geq 2$.
Since $J_d$ is an edge ideal of a complete graph, $\deg(m)=2$.
Suppose $|\supp(m)|=1$. Assume that $\deg(m) \geq 3$. Set $m=u^s$ for some
$s \geq 3$. Clearly $n_1 \cdots n_d \mid u^s\FA_i$,
where $n_l \in \mathcal{G}(J_l)$ for all $1 \leq l \leq d$.
Then $n_1 \cdots n_{d-1} \mid u^s\FA_i$. Also,
$u^s \in (n_1 \cdots n_{d-1}:\FA_i)$. By the above claim, $u^2 \in (\mathcal{J}:\FA_i)$.
This is a contradiction to $\deg(m) \geq 3$. Therefore
$\deg(m)=2$.
By the above arguments, one can see that the ideal $((\mathcal{J},\FA_1,\ldots,\FA_{i-1}):\FA_i)$ is
generated by quadratic monomial ideals.
Note that $J_d \subseteq (\mathcal{J}:\FA_i)$.
Let $K_i$ be the graph associated
to $\widetilde{((\mathcal{J},\FA_1,\ldots,\FA_{i-1}):\FA_i)}$.
Since $J_d$ is the edge ideal of complete graph, $K_i$ is the graph obtained from
complete graph by attaching pendant to some vertices. Hence
$K_i$ is a co-chordal graph.
By \cite[Theorem 1]{froberg},
$\reg((\mathcal{J},\FA_1,\ldots,\FA_{i-1}):\FA_i))=2$ for all $1 \leq i \leq t$.
Considering similar exact sequences as in \eqref{main-exact-seq1}, we get that the inequality
\begin{equation*}\label{main:eq}
\reg\left(\frac{R}{\mathcal{J}}\right) \leq \max\left\{
\begin{array}{l}
\reg\left(\frac{R}{(\mathcal{J} :\FA_1)}\right)+2(d-1),\ldots,
\reg\left(\frac{R}{(\mathcal{J},\FA_1,\ldots,\FA_{t-1}):\FA_t)}
\right) + 2(d-1),\\
\reg\left(\frac{R}{J_1 \cdots J_{d-1}}\right)
\end{array}\right\}
\end{equation*}
holds.
Therefore
$\reg\left(\frac{R}{\mathcal{J}}\right) \leq \max \left\{
\begin{array}{l}
2d, ~\reg\left(\frac{R}{J_1 \cdots J_{d-1}}\right)
\end{array}\right\}.$
Proceeding as in the proof of Theorem~\ref{regularity} we get the desired conclusion.
\end{proof}
As an immediate consequence of Theorem~\ref{main}, Theorem~\ref{prd-ideals}, we obtain an upper bound for the
regularity of products of edge ideals in terms of co-chordal cover numbers.
\begin{coro}\label{prd-us}
Let $J_1=I(G_1),\dots,J_d=I(G_d)$ be the edge ideal of $G_1,\dots,G_d$ with $J_1 \subseteq
\cdots \subseteq J_d$.
\begin{enumerate}[label=(\arabic*)]
\item\label{551}
If $G_3$ is a complete graph, then
\[
\reg(J_1J_2J_3) \leq \max\{6,~\cochord(G_2)+3,~\cochord(G_1)+1\}.
\]
\item
\label{552}
If $G_i$ is a complete graph for all $i=3,4$, then
\[
\reg(J_1J_2J_3J_4) \leq \max\{8,~\cochord(G_2)+3,~\cochord(G_1)+1\}.
\]
\end{enumerate}
\end{coro}
As a consequence of Theorem~\ref{prd-ideals}, we give sufficient conditions
for products of edge ideals to have linear resolutions.
\begin{coro}\label{ls-pr}
Let $J_i=I(G_i)$ be the edge ideal of $G_i$ for all $1 \leq i \leq d$ and $J_1 \subseteq
\cdots \subseteq J_d$.
\begin{enumerate}[label=(\arabic*)]
\item\label{561} If $G_3$ is a complete graph and $ \max\{\cochord(G_2)+3, \cochord(G_1)+1\} \leq 6$, then $J_1J_2J_3$ has
linear resolution.
\item \label{562} If $G_i$ is a complete graph for all $i=3,4$ and
%$\nu_{G_1G_2} \geq 4$ and
$\max\{\cochord(G_2)+3,~\cochord(G_1)+1\} \leq 8$, then
$J_1J_2J_3J_4$ has linear resolution.
\item \label{563} If $G_4$ is a complete graph and $G_i$ is an induced subgraph of $G_{i+1}$ for all $1 \leq i \leq 3$, then
$J_1J_2J_3J_4$ has linear resolution.
\item \label{564} If $G_i$ is a complete graph for all $i=3,4$ and $J_1J_2$ has a linear resolution, then
$J_1J_2J_3J_4$ has a linear resolution.
\end{enumerate}
\end{coro}
\begin{proof}
For~\ref{561} and~\ref{562}, the assertions follow from Theorem~\ref{main} and Theorem~\ref{prd-ideals}.
Consider~\ref{563}. Since $G_4$ is a complete graph and $G_i$ is an induced subgraph of $G_{i+1}$ for all $1 \leq i \leq 3$,
$G_i$ is a complete graph for all $1 \leq i \leq 3$. Therefore, by Corollary~\ref{prd-us}\ref{551},
$J_1J_2J_3$ has a linear resolution. Hence, by Theorem~\ref{prd-ideals}, $J_1J_2J_3J_4$ has a linear resolution.
Finally, consider~\ref{564}. If $J_1J_2$ has a linear resolution, then by Theorem~\ref{prd-ideals}, $J_1J_2J_3$ has linear
resolution. Therefore, by Theorem~\ref{prd-ideals}, $J_1J_2J_3J_4$ has a linear resolution.
\end{proof}
\longthanks{
The computational commutative algebra
package \textsc{Macaulay2} \cite{M2} was heavily used to compute several
examples.
We would also like to express our sincere gratitude
to the anonymous referee for meticulous reading and suggesting several improvements.
}
%\nocite{*}
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