# Very well–covered graphs by Betti splittings

Marilena Crupi, Antonino Ficarra  Marilena Crupi, Department of mathematics
and computer sciences, physics and earth sciences, University of Messina,
Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina, Italy<EMAIL_ADDRESS>Antonino Ficarra, Department of mathematics and computer sciences, physics and
earth sciences, University of Messina, Viale Ferdinando Stagno d’Alcontres 31,
98166 Messina, Italy<EMAIL_ADDRESS>
###### Abstract.

A very well–covered graph is an unmixed graph without isolated vertices such
that the height of its edge ideal is half of the number of vertices. We study
these graphs by means of Betti splittings and mapping cone constructions. We
show that the cover ideals of Cohen–Macaulay very well–covered graphs are
splittable. As a consequence, we compute explicitly the minimal graded free
resolution of the cover ideals of such a class of graphs and prove that these
graphs have homological linear quotients. Finally, we conjecture the same is
true for each power of the cover ideal of a Cohen–Macaulay very well–covered
graph, and settle it in the bipartite case.

###### Key words and phrases:

Complexes, Minimal resolutions, Betti splittings, Homological Shift Ideals,
Vertex decomposability, Well–covered graphs, Very well–covered graphs

###### 2020 Mathematics Subject Classification:

13D02, 13P10, 13F55, 13H10, 05C75

## Introduction

Since the foundations of Algebraic Graph Theory, one of the fundamental
problem has been to classify all Cohen–Macaulay graphs [34]. Let
$S=K[x_{1},\ldots,x_{n}]$ be the standard graded polynomial ring with
coefficients in a field $K$. It is well–known that if $G$ is a simple
undirected graph with $|V(G)|=n$, one can associate to $G$ two squarefree
monomial ideals of $S$, the edge ideal $I(G)$ generated by all monomials
$x_{i}x_{j}$ such that $x_{i}x_{j}\in E(G)$, and the cover ideal $J(G)$ which
is generated by monomials $\prod_{x_{i}\in C}x_{i}$, for all minimal vertex
covers $C$ of $G$. A minimal vertex cover of $G$ is a subset $C$ of $V(G)$
such that each edge has at least one vertex in $C$ and no proper subset of $C$
has the same property. $G$ is called a Cohen–Macaulay graph if $S/I(G)$ is a
Cohen–Macaulay ring. To classify all Cohen–Macaulay graphs is an hopeless
task.

However, the dual problem, namely the problem of classifying all
Cohen–Macaulay cover ideals of a graph, is completely solved. Indeed the cover
ideal $J(G)$ of a graph $G$ is the Alexander dual of the edge ideal of $G$,
_i.e._ , $J(G)=I(G)^{\vee}$, and the fundamental Eagon–Reiner Criterion says
that $I(G)^{\vee}$ is Cohen–Macaulay if and only if $I(G)$ has a linear
resolution [6]. All graphs $G$ such that $I(G)$ has a linear resolution have
been characterized by Ralph Fröberg in [13]. Furthermore, Herzog, Hibi and
Zheng showed that for any graph $G$ such that $I(G)$ has a linear resolution,
then $I(G)$ has linear powers, _i.e._ , for all $k\geq 1$, $I(G)^{k}$ has a
linear resolution [18].

Far beyond these classical problems, a new exciting trend has been initiated
by Herzog, Moradi, Rahimbeigi and Zhu in [19] with the introduction of the
homological shift ideals (see, also, [1, 2, 9, 10, 11, 20]). Firstly, given a
vector ${\bf a}=(a_{1},\dots,a_{n})\in{\mathbb{Z}}_{\geq 0}^{n}$, the monomial
$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ will be denoted by ${\bf x^{a}}$. Let
$I\subset S$ be a monomial ideal and let ${\mathbb{F}}$ be its minimal
multigraded free $S$–resolution. Then, the $i$th free $S$–module in
${\mathbb{F}}$ is $F_{i}=\bigoplus_{k=1}^{\beta_{i}(I)}S(-{\bf a}_{ik})$ with
each ${\bf a}_{ik}$ an integer vector in ${\mathbb{Z}}^{n}$ with non–negative
entries. The monomial ideal generated by the monomials ${\bf x}^{{\bf
a}_{ik}}$ ($k=1,\ldots,\beta_{i}(I)$), denoted by $\textup{HS}_{i}(I)$, is
said the _$i$ th homological shift ideal_ of $I$. Note that
$\textup{HS}_{0}(I)=I$. A basic goal of this theory is to find those
combinatorial and homological properties satisfied by all
$\textup{HS}_{i}(I)$, $i=0,\ldots,\operatorname{pd}(I)$. In such a case, we
will speak about a homological shift property of $I$, or if the context is
clear, we will simply speak of homological property. In [19], it is
conjectured that for any graph $G$ whose edge ideal $I(G)$ has a linear
resolution or linear quotients, then $\textup{HS}_{i}(I(G))$ satisfy the same
property for all $i$, _i.e._ , $I(G)$ has homological linear resolution or
homological linear quotients. The conjecture holds true for $i=1$ [19, Theorem
4.7].

Inspired by the problems above, in this article we consider the dual problem
of Herzog et all in [19]. We want to classify the graphs $G$ for which
$I(G)^{\vee}$ has a homological linear resolution and those with homological
linear quotients. Of course, first one needs to know all graphs $G$ such that
$I(G)^{\vee}$ has a linear resolution. One can observe that for $I(G)^{\vee}$
to have a linear resolution, it is necessary that it is equigenerated, which
is equivalent to requiring that $G$ is _unmixed_ in the sense that all the
minimal vertex covers of $G$ have the same cardinality. On the other hand, if
$G$ is unmixed and without isolated vertices, then
$2\textup{height}(I(G))\geq|V(G)|$ [14]. Therefore, to study the question
above it is natural to restrict ourself to the “best possible class” of
graphs: _all unmixed graphs $G$ without isolated vertices and such that
$2\textup{height}(I(G))=|V(G)|$_. In such a case, $G$ is called a _very
well–covered graph_. Such a class of graphs has been studied by many authors
(see, for instance, [5, 8, 25, 26, 27, 28, 29]).

The outline of this article is as follows. Section 1 contains a quick review
on Betti splittings, homological shift ideals, and some notions from the graph
theory. In particular, the classification of all Cohen–Macaulay very
well–covered graphs obtained by Crupi, Rinaldo and Terai in [5] is fundamental
for the development of the article.

In Section 2, we investigate Betti splittings of the cover ideals of
Cohen–Macaulay very well–covered graphs (Proposition 2.3). The Betti splitting
technique introduced in [12] by Francisco, Ha and Van Tuyl is a very useful
tool throughout the article.

In Section 3, if $G$ is a Cohen–Macaulay very well–covered graph, we use the
Betti splitting, discussed in Section 2, to construct explicitly a minimal
free resolution of $I(G)^{\vee}$. A minimal free resolution of $I(G)^{\vee}$
has also been constructed in [25] by different tools. The main advantage of
the Betti splitting technique is that one does not need to verify if the
defined complex is exact or minimal, but only to determine the suitable
comparison maps [12, Proposition 2.1]. The features of our resolution are
required in Section 4 to achieve our main goal. In the last part of Section 3,
we draw some consequences of Theorem 3.5. Firstly, we get a formula for the
Betti numbers of $I(G)^{\vee}$ which is independent on the characteristic of
the base field $K$ (Corollary 3.6). Then, some formulas for the projective
dimension of the cover ideal of a Cohen–Macaulay very well–covered graph $G$
are determined. In particular, in Corollary 3.7, we recover a result due to
Mahmoudi et all [29, Lemma 3.4] and, furthermore, we provide a new proof of it
by Betti splitting (Remark 3.8). Finally, we show that $I(G)^{\vee}$ has the
alternating sum property [35, Definition 4.1] and also we determine a formula
for the multiplicity of $S/I(G)^{\vee}$ (Proposition 3.10).

Section 4 contains our main result which states that the cover ideal of a
Cohen–Macaulay very well–covered graph has homological linear quotients
(Theorem 4.1). Then, we infer that a very well–covered graph $G$ is
Cohen–Macaulay if and only if $I(G)^{\vee}$ has homological linear resolution
which is equivalent to $I(G)^{\vee}$ having homological linear quotients
(Theorem 4.2). An interesting consequence about the cover ideal of the whisker
graph $G^{*}$ of a graph $G$ is drawn in Corollary 4.3.

Our experiments and the results obtained have lead us to conjecture that given
a Cohen–Macaulay very well–covered graph $G$, all the powers of $I(G)^{\vee}$
have homological linear quotients (Conjecture 4.4). At present we are able to
prove our conjecture only for the subclass of Cohen–Macaulay bipartite graphs.
For this goal, we carefully study the powers of the Hibi ideals. In 1987, Hibi
introduced a fundamental class of ideals associated to a finite poset [22].
More precisely, let $(P,\succeq)$ be a finite poset, a poset ideal
$\mathcal{I}$ of $P$ is a subset of $P$ such that for any
$\alpha\in\mathcal{I}$ and any $\beta\in P$ with $\beta\preceq\alpha$, then
$\beta\in\mathcal{I}$. If $\mathcal{J}(P)$ is the set of all poset ideals of
$P$, ordered by inclusion, then $\mathcal{J}(P)$ is a distributive lattice.
Indeed, Birkhoff’s fundamental theorem [33, Theorem 3.4.1] shows that any
distributive lattice arises in such a way. To any
$\mathcal{I}\in\mathcal{J}(P)$ one can associate the squarefree monomial
$u_{\mathcal{I}}=(\prod_{p\in\mathcal{I}}x_{p})(\prod_{p\in
P\setminus\mathcal{I}}y_{p})$ in the polynomial ring
$K[\\{x_{p},y_{p}\\}_{p\in P}]$. Then the Hibi ideal $H_{P}$ associated to
$(P,\succeq)$ is the squarefree monomial ideal generated by all
$u_{\mathcal{I}}$, $\mathcal{I}\in\mathcal{J}(P)$. The importance of such a
class lies in the fact that the class of Hibi ideals coincides with the class
of the cover ideals of Cohen–Macaulay bipartite graphs [17] (see, also, [16,
Lemma 9.1.9 and Theorem 9.1.13]). Therefore, one can focus on the homological
shifts of powers of Hibi ideals. In Construction 4.6, we associate to any
poset $(P,\succeq)$ and any integer $\ell\geq 1$ a new poset
$(P(\ell),\succeq_{\ell})$ and then we show that the $\ell$th power of $H_{P}$
is equal to $H_{P(\ell)}$ up to polarization (Theorem 4.9).

Finally, observing that polarization commutes with homological shifts (Lemma
1.8) and preserves also the linear quotients property (Lemma 4.10), we state
that all the powers of an Hibi ideal have homological linear quotients
(Corollary 4.11).

All the examples in the article have been verified using Macaulay2 [15] and
the package HomologicalShiftIdeals [10].

##  1\. Preliminaries

In this section for the reader’s convenience we collect some notions and
results we need for the development of the article.

###  1.1. A glimpse to mapping cones and Betti splittings.

Let $S=K[x_{1},\dots,x_{n}]$ be a polynomial ring with coefficients in a field
$K$. If $I$ is a monomial ideal of $S$, it is customary to ask how one can
determine its minimal free resolution.

In order to avoid any ambiguity in what follows, we assume that the trivial
ideals $(0)$ and $S$ are monomial ideals since $(0)=(\varnothing)$ and
$S=(1)$.

Let $G(I)$ be the unique minimal generating set of $I$. In [12], the authors
pointed out that to compute a minimal graded free resolution of $I$, one can
“split” the ideal $I$ into “smaller” ideals $P$, $Q$, _i.e._ , $I=P+Q$ with
$G(I)$ disjoint union of $G(P)$ and $G(Q)$. Hence, to get the minimal free
resolution of $I$, one can use the minimal free resolutions of $P$ and $Q$
together with that of $P\cap Q$. Indeed, let us consider the short exact
sequence

$0\rightarrow P\cap Q\xrightarrow{\ \psi_{-1}}P\oplus Q\xrightarrow{\ \varphi\
}I=P+Q\rightarrow 0,$ (1)

where $\psi_{-1}(w)=(w,-w)$ and $\varphi((w,z))=w+z$. Let $\mathbb{A}$ be the
minimal free resolution of $P\cap Q$ and let $\mathbb{B}$ be the minimal free
resolution of $P\oplus Q$. Note that $\mathbb{B}$ is the direct sum of the
minimal free resolutions of $P$ and $Q$. Since all $S$–modules $A_{i}$ are
free, thus projective, there exists a complex homomorphism
$\psi:\mathbb{A}\rightarrow\mathbb{B}$ lifting the map $\psi_{-1}$, that means
a sequence of maps $\psi_{i}:A_{i}\rightarrow B_{i}$ $(i\geq 0)$, called the
comparison maps, making the following diagram

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(2)

commutative. It is well known that $\psi$ gives rise to an acyclic complex
$C(\psi)$ whose $0$th homology module is
$H_{0}(C(\psi))=\textup{coker}(\psi_{-1})=(P\oplus
Q)/\textup{Im}(\psi_{-1})\cong(P\oplus Q)/(P\cap Q)\cong P+Q=I$, _i.e._ ,
$C(\psi)$ is a free resolution of $I$ (see, for instance, [7, Appendix
A3.12]).

The complex $C(\psi)$ is defined as follows:

  1. (i)

let $C_{0}=B_{0}$, and $C_{i}=A_{i-1}\oplus B_{i}$, for $i>0$;

  2. (ii)

let $d_{0}=\varphi\circ d_{0}^{B}$, $d_{1}=(0,\psi_{0}+d_{1}^{B})$, and
$d_{i}=(-d_{i-1}^{A},\psi_{i-1}+d_{i}^{B})$, for $i>1$.

This procedure, known as the mapping cone, may be visualized as follows:

$\textstyle{\displaystyle\mathbb{A}[-1]:\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\oplus$$\scriptstyle{\psi_{2}}$$\scriptstyle{d_{2}^{A}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\oplus$$\scriptstyle{\psi_{1}}$$\scriptstyle{d_{1}^{A}}$$\textstyle{A_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\oplus$$\scriptstyle{\psi_{0}}$$\scriptstyle{d_{0}^{A}}$$\textstyle{P\cap
Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathbb{B}:\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{3}^{B}}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}^{B}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}^{B}}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}^{B}\
\ }$$\textstyle{P\oplus
Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Here $\mathbb{A}[-1]$ is the complex $\mathbb{A}$ homologically shifted by
$-1$.

Unfortunately, the free resolution $C(\psi)$ is not always minimal. Below, we
describe a special situation in which this happens.

For our purpose, we need to quote the next notions from [12].

######  Definition 1.1.

Let $I$, $P$, $Q$ be monomial ideals of $S$ such that $G(I)$ is the disjoint
union of $G(P)$ and $G(Q)$. We say that $I=P+Q$ is a Betti splitting if

$\beta_{i,j}(I)=\beta_{i,j}(P)+\beta_{i,j}(Q)+\beta_{i-1,j}(P\cap Q),\ \ \
\textup{for all}\ i,j.$ (3)

######  Definition 1.2.

Let $I$ be a monomial ideal of $S$. Let $P$ be the ideal generated by all
elements of $G(P)$ divisible by $x_{i}$ and let $Q$ be the ideal generated by
all other elements of $G(I)$. We call $I=P+Q$ an $x_{i}$–partition of $I$.
If $I=P+Q$ is also a Betti splitting, we call $I=P+Q$ an $x_{i}$–splitting.

The following two results will be needed later.

######  Proposition 1.3.

([12, Corollary 2.7]). Let $I=P+Q$ be a $x_{i}$–partition of $I$ and $P$ be
the ideal generated by all elements of $G(I)$ divisible by $x_{i}$. If the
minimal graded free resolution of $P$ is linear, then $I=P+Q$ is a Betti
splitting.

######  Proposition 1.4.

([3, Proposition 3.1]). Let $d$ be a positive integer, $I$ a monomial ideal
with a $d$–linear resolution, $P,Q\neq 0$ monomial ideals such that $I=P+Q$,
$G(I)=G(P)\cup G(Q)$ and $G(P)\cap G(Q)=\varnothing$. Then the following facts
are equivalent:

  1. (i)

$I=P+Q$ is a Betti splitting of $I$;

  2. (ii)

$P$ and $Q$ have $d$–linear resolutions.

If this is the case, then $P\cap Q$ has a $(d+1)$–linear resolution.

We close the subsection with the next result which shows that understanding
when a monomial ideal has a Betti spitting is equivalent to understanding when
the mapping cone (i)–(ii) gives a minimal free resolution.

######  Theorem 1.5.

([12, Proposition 2.1]). Let $I,P,Q$ be monomial ideals of $S$ such that
$I=P+Q$ and $G(I)$ is the disjoint union of $G(P)$ and $G(Q)$. Furthermore,
consider the short exact sequence (1).

Then the following conditions are equivalent:

  1. (a)

$I=P+Q$ is a Betti splitting.

  2. (b)

For all $i$ and $j$, the map

$\operatorname{Tor}_{i}^{S}(K,P\cap
Q)_{j}\rightarrow\operatorname{Tor}_{i}^{S}(K,P)_{j}\oplus\operatorname{Tor}_{i}^{S}(K,Q)_{j}$

in the long exact sequence in $\operatorname{Tor}$ induced from (1) is the
zero map.

  3. (c)

Applying the mapping cone to (1) gives a minimal free resolution of $I$.

###  1.2. A glimpse to linear quotients, linear powers and homological shift
ideals.

Let $I$ be a monomial ideal of $S=K[x_{1},\dots,x_{n}]$, $I$ has linear
quotients if for some order $u_{1}>\dots>u_{m}$ of its minimal generating set
$G(I)$, all colon ideals $(u_{1},\dots,u_{\ell-1}):u_{\ell}$,
$\ell=2,\dots,m$, are generated by a subset of the set of variables
$\\{x_{1},\dots,x_{n}\\}$. In such a case $u_{1}>u_{2}>\dots>u_{m}$ is called
an admissible order of $I$. By [16, Proposition 1.2.2] each colon ideal
$(u_{1},\dots,u_{\ell-1}):u_{\ell}$ is generated by the monomials

$\frac{u_{j}}{\textup{gcd}(u_{j},u_{\ell})}=\frac{\textup{lcm}(u_{j},u_{\ell})}{u_{\ell}},\
\ \ \ j=1,\dots,\ell-1.$

By [16, Lemma 8.2.3], $u_{1}>\\!\cdots\\!>u_{m}$ is an admissible order of $I$
if and only if for all $j<\ell$ there exist an integer $k<\ell$ and an integer
$p$ such that

$\frac{\textup{lcm}(u_{k},u_{\ell})}{u_{\ell}}=x_{p}\ \ \ \text{and}\ \ \
x_{p}\ \ \ \text{divides}\ \ \ \frac{\textup{lcm}(u_{j},u_{\ell})}{u_{\ell}}.$

One knows that if $I$ is generated in the same degree and it has linear
quotients, then $I$ has a linear resolution. This fact is extremely useful in
combinatorial commutative algebra. Indeed, if one proves that an equigenerated
monomial ideal $I$ and all its powers $I^{k}$ ($k\geq 2$) have linear
quotients, then it will follow that $I$ has linear powers, _i.e._ , $I$ and
all its powers have linear resolutions.

For a monomial $u=x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}\in S$, the
integral vector ${\bf a}=(a_{1},a_{2},\dots,a_{n})$ is called the multidegree
of $u$. We also write $u={\bf x^{a}}$, in particular, for ${\bf a}={\bf
0}=(0,0,\dots,0)$, ${\bf x^{0}}=1$; whereas $\deg(u)=a_{1}+a_{2}+\dots+a_{n}$
is the degree of $u$. It is often customary to identify the multidegree ${\bf
a}$ with the monomial ${\bf x^{a}}$.

We quote the next definition from [19].

######  Definition 1.6.

Let $I\subset S$ be a monomial ideal with minimal multigraded free resolution

${\mathbb{F}}\ \ :\ \ 0\rightarrow F_{p}\rightarrow
F_{p-1}\rightarrow\cdots\rightarrow F_{0}\rightarrow I\rightarrow 0,$

where $F_{i}=\bigoplus_{j=1}^{\beta_{i}(I)}S(-{\bf a}_{i,j})$. The vectors
${\bf a}_{i,j}\in{\mathbb{Z}}^{n}$, $i\geq 0$, $j=1,\dots,\beta_{i}(I)$, are
called the $i$th multigraded shifts of $I$. The monomial ideal

$\textup{HS}_{i}(I)\ =\ ({\bf x}^{{\bf a}_{i,j}}\ :\ j=1,\dots,\beta_{i}(I))$

is called the $i$th homological shift ideal of $I$.

Note that $\textup{HS}_{0}(I)=I$ and $\textup{HS}_{i}(I)=(0)$ for $i<0$ or
$i>\operatorname{pd}(I)$.

The main purpose of the theory of homological shift ideals is to understand
what homological and combinatorial properties are enjoyed by all
$\textup{HS}_{i}(I)$, $i=0,\ldots,\operatorname{pd}(I)$. We call any such
property a homological shift property of $I$, or if the context is clear,
simply homological property. Two important homological shift properties are
the following ones: $I$ has a homological linear resolution if
$\textup{HS}_{i}(I)$ has a linear resolution for all $i$; $I$ has homological
linear quotients if $\textup{HS}_{i}(I)$ has linear quotients for all $i$.

Theorem 1.5(b) implies immediately

######  Proposition 1.7.

Let $I=P+Q$ be a Betti splitting of $I\subset S$. Then

$\textup{HS}_{k}(I)=\textup{HS}_{k-1}(P\cap
Q)+\textup{HS}_{k}(P)+\textup{HS}_{k}(Q),$

for all $k\geq 0$.

Let us recall the technique of polarization which is an operation that
transforms a monomial ideal into a squarefree monomial ideal in a larger
polynomial ring.

Let $u=x_{1}^{b_{1}}x_{2}^{b_{2}}\cdots x_{n}^{b_{n}}\in S$ be a monomial.
Then, the polarization of $u$ is the monomial

$u^{\wp}=\prod_{i=1}^{n}\prod_{j=1}^{b_{i}}x_{i,j}=\prod_{\begin{subarray}{c}i=1,\dots,n\\\
b_{i}>0\end{subarray}}x_{i,1}x_{i,2}\cdots x_{i,b_{i}}$

in the polynomial ring $K[x_{i,j}:i=1,\dots,n,j=1,\dots,b_{i}]$.

For a monomial $u=x_{1}^{b_{1}}x_{2}^{b_{2}}\cdots x_{n}^{b_{n}}$ of $S$, we
define the $x_{i}$–degree of $u$ to be the integer
$\deg_{x_{i}}(u)=b_{i}=\max\\{j\geq 0:x_{i}^{j}\ \textup{divides}\ u\\}$. Let
$I$ be a monomial ideal of $S$ and set $a_{i}=\max\\{\deg_{x_{i}}(u):u\in
G(I)\\}$, $i=1,\dots,n$. Let

$R=K[x_{i,j}:i=1,\dots,n,j=1,\dots,a_{i}]$

be the polynomial ring in the variables $x_{i,j}$,
$i=1,\dots,n,j=1,\dots,a_{i}$. The polarization of the monomial ideal $I$ is
defined to be the squarefree monomial ideal $I^{\wp}$ of $R$ with minimal
generating set $G(I^{\wp})=\\{u^{\wp}:u\in G(I)\\}$.

The polarization commutes with the homological shift ideals as next result
illustrates.

######  Lemma 1.8.

([32, Corollary 1.8], [19, Proposition 1.14]). Let $I\subset S$ be a monomial
ideal. Then $\textup{HS}_{i}(I^{\wp})=\textup{HS}_{i}(I)^{\wp}$ for all $i\geq
0$.

###  1.3. A glimpse to very well–covered graphs.

In this article, a graph will always mean a finite undirected graph without
loops or multiple edges. Let $G$ be a graph with vertex set
$V(G)=\\{x_{1},\dots,x_{n}\\}$ and with edge set $E(G)$. Let $x_{i}\in V(G)$
be a vertex of $G$. The (open) neighborhood of $x_{i}$ is the set

$N(x_{i})=\\{x_{j}\in V(G):x_{i}x_{j}\in E(G)\\},$

whereas the closed neighborhood of $x_{i}$ is the set defined as follows

$N[x_{i}]=\\{x_{j}\in V(G):x_{i}x_{j}\in E(G)\\}\cup\\{x_{i}\\}.$

If $W\subseteq V(G)$, we denote by $G\setminus W$ the subgraph of $G$ with the
vertices of $W$ and their incident edges deleted.

A subset $W$ of $V(G)$ is called a _vertex cover_ if every edge of $G$ is
incident with at least one vertex in $W$. A vertex cover $W$ is called a
_minimal vertex cover_ if there is no proper subset of $W$ which is a vertex
cover of $G$. The set of all minimal vertex cover of $G$ is denote by
$\mathcal{C}(G)$.

Attached to $G$ [34] there are the following squarefree monomial ideals of the
polynomial ring $S=K[x_{1},\ldots,x_{n}]$:

$I(G)=(x_{i}x_{j}\ :\ x_{i}x_{j}\in E(G))$

called the _edge ideal_ of $G$, and

$I(G)^{\vee}=(x_{i_{1}}\cdots x_{i_{s}}\ :\
\mbox{$W=\\{x_{i_{1}},\ldots,x_{i_{s}}\\}$ is a minimal vertex cover of $G$})$

called the _cover ideal_ of $G$.

A graph $G$ is _unmixed_ or _well–covered_ if all the minimal vertex covers
have the same cardinality. In particular, all the associated primes of $I(G)$
have the same height. A graph $G$ is called _Cohen–Macaulay_ over the field
$K$ if $S/I(G)$ is a Cohen–Macaulay ring. It is clear that a Cohen–Macaulay
graph is unmixed [34, Proposition 7.2.9].

Finally, we recall that a pairing off of all the vertices of a graph $G$ is
called a _perfect matching_. Thus $G$ has a perfect matching if and only if
$G$ has an even number of vertices and there is a set of independent edges
covering all the vertices, where for a set of independent edges we mean a set
of pairwise disjoint edges [34].

The next fundamental criterion due to Eagon and Reiner will be crucial.

######  Criterion 1.9.

([6], [16, Theorem 8.1.9]). Let $I\subset S$ be a squarefree monomial ideal.
Then $I$ is Cohen–Macaulay if and only if its Alexander dual $I^{\vee}$ has a
linear resolution.

Now, we analyze the class of very well–covered graphs (see, for instance, [26]
and the references therein). Let $G$ be an unmixed graph without isolated
vertices and let $I(G)$ be its edge ideal in $S$. In [14, Corollary 3.4], the
authors proved the following inequality:

$2\textup{height}(I(G))\geq|V(G)|.$

######  Definition 1.10.

A graph $G$ is called very well–covered if it is unmixed without isolated
vertices and with $2\textup{height}(I(G))=|V(G)|$.

By [8, Theorem 1.2], very well–covered graphs have always perfect matchings.
Hence, for a very well–covered graph $G$ with $2n$ vertices, we may assume
$(*)$ $V(G)=X\cup Y$, $X\cap Y=\varnothing$, with $X=\\{x_{1},\ldots,x_{n}\\}$
a minimal vertex cover of $G$ and $Y=\\{y_{1},\ldots,y_{n}\\}$ a maximal
independent set of $G$ such that $\\{x_{1}y_{1},\ldots,x_{n}y_{n}\\}\subseteq
E(G)$.

It is important to point out that when we assume the condition $(*)$ for a
very well–covered graph, we do not force any restriction on the graph. Indeed,
it is only a relabeling of the vertices.

For the reminder of this article, set
$S=K[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ for a field $K$. For a positive
integer $n$, we denote the set $\\{1,2,\dots,n\\}$ by $[n]$.

######  Theorem 1.11.

([5, Theorem 3.6]). Let $G$ be a graph with $2n$ vertices, which are not
isolated, with $\textup{height}(I(G))=n$. We assume condition $(*)$ and also
we assume that if $x_{i}y_{j}\in E(G)$ then $i\leq j$. Then, the following
conditions are equivalent:

  1. (a)

$G$ is a Cohen–Macaulay very well–covered graph.

  2. (b)

The following conditions hold:

    1. (i)

if $x_{i}y_{j}\in E(G)$ then $x_{i}x_{j}\notin E(G)$,

    2. (ii)

if $z_{i}x_{j},y_{j}x_{k}\in E(G)$ then $z_{i}x_{k}\in E(G)$ for any distinct
$i,j,k$ and $z_{i}\in\\{x_{i},y_{i}\\}$.

For our convenience, we reformulate Theorem 1.11, as follows.

######  Characterization 1.12.

([5], [29, Lemma 3.1]). Let $G$ be a very well–covered graph with $2n$
vertices. Then, the following conditions are equivalent.

  1. (a)

$G$ is Cohen–Macaulay.

  2. (b)

There exists a relabeling of $V(G)=\\{x_{1},\dots,x_{n},y_{1},\dots,y_{n}\\}$
such that

    1. (i)

$X=\\{x_{1},\dots,x_{n}\\}$ is a minimal vertex cover of $G$ and
$Y=\\{y_{1},\dots,y_{n}\\}$ is a maximal independent set of $G$,

    2. (ii)

$x_{i}y_{i}\in E(G)$ for all $i\in[n]$,

    3. (iii)

if $x_{i}y_{j}\in E(G)$ then $i\leq j$,

    4. (iv)

if $x_{i}y_{j}\in E(G)$ then $x_{i}x_{j}\notin E(G)$,

    5. (v)

if $z_{i}x_{j},y_{j}x_{k}\in E(G)$ then $z_{i}x_{k}\in E(G)$ for any distinct
$i,j,k$ and $z_{i}\in\\{x_{i},y_{i}\\}$.

The next example illustrates the previous characterization.

######  Example 1.13.

The following graph $G$ is an example of a Cohen–Macaulay very well–covered
graph with $8$ vertices.

$y_{1}$$y_{2}$$x_{1}$$x_{2}$$x_{3}$$y_{3}$$x_{4}$$y_{4}$$G$

Indeed, for $S=K[x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4}]$, $\dim
S/I(G)=4=\operatorname{depth}S/I(G)$ and moreover, its minimal vertex covers
are: $\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\\{x_{1},y_{2},x_{3},x_{4}\\}$,
$\\{y_{1},x_{2},x_{3},x_{4}\\}$, $\\{x_{1},x_{2},x_{3},y_{4}\\}$,
$\\{x_{1},x_{2},y_{3},y_{4}\\}$.

The next graph $G$ is an example of a not Cohen–Macaulay very well–covered
graph with $8$ vertices.

$y_{1}$$y_{2}$$x_{1}$$x_{2}$$x_{3}$$y_{3}$$x_{4}$$y_{4}$$G$

In fact, $G$ is very well–covered. Its minimal vertex covers are the sets
$\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\\{x_{1},y_{2},x_{3},x_{4}\\}$,
$\\{y_{1},x_{2},x_{3},x_{4}\\}$, $\\{x_{1},x_{2},y_{3},y_{4}\\}$. But $G$ is
not Cohen–Macaulay. Note that $\dim
S/I(G)=4\neq\operatorname{depth}\,S/I(G)=3$, with
$S=K[x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4}]$.

##  2\. Betti splittings of cover ideals of very well–covered graphs

In this section we analyze the Betti splittings of the cover ideals of the
class of Cohen–Macaulay very well–covered graphs.

Our first result shows that if we remove some pairs of vertices of a
Cohen–Macaulay very well–covered graph in a suitable way, then we obtain
“smaller graphs” which are again Cohen–Macaulay very well–covered graphs.

######  Proposition 2.1.

Let $G$ be a Cohen–Macaulay very well–covered graph with $2n$ vertices and
assume the condition $(*)$. Then $G\setminus\\{x_{i},y_{i}:i\in A\\}$ is a
Cohen–Macaulay very well–covered graph, for any subset $A\subseteq[n]$.

###### Proof.

Firstly, note that since $G$ is a Cohen–Macaulay very well–covered graph with
$2n$ vertices and condition $(*)$ holds, then $\textup{height}(I(G))=n$.
Moreover, from [5, Lemma 3.5], we may assume that if $x_{i}y_{j}\in E(G)$ then
$i\leq j$.

If $A=\varnothing$, there is nothing to prove. Now, let $A\neq\varnothing$ and
set $A=\\{i_{1},i_{2},\dots,i_{t}\\}$. We show that

$G_{0}\ =\ G\setminus\\{x_{i},y_{i}:i\in A\\}\ =\
G\setminus\\{x_{i_{1}},y_{i_{1}},x_{i_{2}},y_{i_{2}},\dots,x_{i_{t}},y_{i_{t}}\\}$

is a Cohen–Macaulay very well–covered graph. Note that $G_{0}$ has $2(n-t)$
vertices that are not isolated. Moreover, $G_{0}$ satisfies condition $(*)$
for $X_{0}=X\setminus\\{x_{i_{1}},\dots,x_{i_{t}}\\}$ and
$Y_{0}=Y\setminus\\{y_{i_{1}},\dots,y_{i_{t}}\\}$, and $I(G_{0})$ has height
$n-t$, since $X_{0}$ is a minimal vertex cover of $G_{0}$. It is clear that
$G_{0}$ satisfies the conditions (i)–(ii) of Theorem 1.11(b), because $G$
satisfies such conditions. By Theorem 1.11, we get that $G_{0}$ is also a
Cohen–Macaulay very well–covered graph.

Let $F\subseteq[n]$ be a non empty set, we set ${\bf x}_{F}=\prod_{i\in
F}x_{i}$, ${\bf y}_{F}=\prod_{i\in F}y_{i}$. Otherwise, we set ${\bf
x}_{\varnothing}={\bf y}_{\varnothing}=1$. For a monomial $u\in
S=K[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$, we define support of $u$ the set

$\operatorname{supp}(u)=\\{x_{i}:x_{i}\ \textup{divides}\
u\\}\cup\\{y_{j}:y_{j}\ \textup{divides}\ u\\}.$

From now, when we tell about a Cohen–Macaulay very well–covered graph $G$ with
$2n$ vertices, we tacitly assume that there exists a relabeling of the set of
vertices $V(G)=\\{x_{1},\dots,x_{n},y_{1},\dots,y_{n}\\}$ which satisfy the
conditions (i)–(v) of Characterization 1.12.

######  Lemma 2.2.

Let $G$ be a Cohen–Macaulay very well–covered graph with $2n$ vertices. For
each $u\in G(I(G)^{\vee})$ there exists a unique subset $F$ of $[n]$ such that
$u={\bf x}_{F}{\bf y}_{[n]\setminus F}$.

###### Proof.

Let $u\in G(I(G)^{\vee})$. By definition, $u$ is a squarefree monomial whose
support $C=\operatorname{supp}(u)$ is a minimal vertex cover of $G$. Since $G$
is very well–covered, $u$ has degree $|V(G)|/2=n$. By Characterization
1.12(b)(ii), $x_{n}y_{n}\in E(G)$. Hence $z_{n}\in C$ with
$z_{n}\in\\{y_{n},x_{n}\\}$. Note that $C_{1}=C\setminus\\{z_{n}\\}$ is a
vertex cover of $G_{1}=G\setminus\\{y_{n},x_{n}\\}$. By Proposition 2.1,
$G_{1}$ is again Cohen–Macaulay very well–covered. Since
$|C_{1}|=|C|-1=|V(G)|/2-1=|V(G_{1})|/2$, we have that $C_{1}$ is a minimal
vertex cover of $G_{1}$. Thus the monomial $u_{1}$ whose support is $C_{1}$ is
a minimal generator of $I(G_{1})^{\vee}$. By induction, $u_{1}={\bf
x}_{F_{1}}{\bf y}_{[n-1]\setminus F_{1}}$ for a unique subset $F_{1}$ of
$[n-1]$. If $z_{n}=x_{n}$, let $F=F_{1}\cup\\{n\\}$. Otherwise, if
$z_{n}=y_{n}$, let $F=F_{1}$. In both cases, $u=z_{n}u_{1}={\bf x}_{F}{\bf
y}_{[n]\setminus F}$, as required.

Note that if $G$ is a Cohen–Macaulay very well–covered graph with $2n$
vertices, then

$N[x_{n}]=\\{x_{i_{1}},x_{i_{2}},\dots,x_{i_{t}},x_{n},y_{n}\\},\,\,\,\mbox{with
$i_{r}<n$, $r\in[t]$},$

and

$N[y_{n}]=\\{x_{j_{1}},x_{j_{2}},\dots,x_{j_{p}},x_{n},y_{n}\\},\,\,\mbox{with
$j_{q}<n$, $q\in[p]$}.$

Moreover, from Characterization 1.12, $i_{r}\neq j_{q}$, for all $r\in[t]$ and
$q\in[p]$. We will consider again such sets in the next section (see, Setup
3.1 and Lemma 3.2).

######  Proposition 2.3.

Let $G$ be a Cohen–Macaulay very well–covered graph with $2n$ vertices. Let
$N[x_{n}]=\\{x_{i_{1}},x_{i_{2}},\dots,x_{i_{t}},x_{n},y_{n}\\}$,
$N[y_{n}]=\\{x_{j_{1}},x_{j_{2}},\dots,x_{j_{p}},x_{n},y_{n}\\}$ and define

$\displaystyle G_{1}\ $ $\displaystyle=\
G\setminus\\{x_{i_{1}},y_{i_{1}},x_{i_{2}},y_{i_{2}},\dots,x_{i_{t}},y_{i_{t}},x_{n},y_{n}\\},$
$\displaystyle G_{2}\ $ $\displaystyle=\
G\setminus\\{x_{j_{1}},y_{j_{1}},\\!x_{j_{2}},\\!y_{j_{2}},\dots,x_{j_{p}},y_{j_{p}},x_{n},y_{n}\\}.$

Then

$I(G)^{\vee}\ =\ x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot
y_{n}I(G_{1})^{\vee}+x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot
x_{n}I(G_{2})^{\vee}$

is a Betti splitting.

###### Proof.

Let $J_{1}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot y_{n}I(G_{1})^{\vee}$ and
$J_{2}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}\cdot x_{n}I(G_{2})^{\vee}$. Note
that $y_{n}$ does not divide any minimal monomial generator of $J_{2}$. Thus
$G(J_{1})\cap G(J_{2})=\varnothing$. We claim that $J_{1}$ and $J_{2}$ have
$n$–linear resolutions. Indeed, by Proposition 2.1, $G_{1}$ and $G_{2}$ are
again Cohen–Macaulay very well–covered graphs and our claim follows from
Criterion 1.9. By virtue of Proposition 1.3, to prove that
$I(G)^{\vee}=J_{1}+J_{2}$ is a Betti splitting, it is enough to show that
$I(G)^{\vee}=J_{1}+J_{2}$ is a $y_{n}$–partition of $I(G)^{\vee}$.
Indeed, let $u$ be a minimal generator of $I(G)^{\vee}$. Then
$C=\operatorname{supp}(u)$ is a minimal vertex cover of $G$. By Lemma 2.2,
$u={\bf x}_{F}{\bf y}_{[n]\setminus F}$ for some $F\subseteq[n]$. Thus, either
$y_{n}\in C$ or $x_{n}\in C$. We distinguish two cases.
Case 1. Suppose $y_{n}\in C$. Since $x_{n}\notin C$, but
$x_{i_{1}}x_{n},\dots,x_{i_{t}}x_{n}\in E(G)$ and $C$ is a vertex cover, then
we obtain that $x_{i_{1}},\dots,x_{i_{t}}\in C$. Hence
$x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot y_{n}$ divides $u$. Note that the
support of $v=u/(x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot y_{n})$ is a vertex
cover of $G_{1}$. Furthermore, $\operatorname{supp}(v)$ is a minimal vertex
cover, since $|\operatorname{supp}(v)|=n-(t+1)=|V(G_{1})|/2$ and $G_{1}$ is a
very well–covered graph (Proposition 2.1). Hence $v\in I(G_{1})^{\vee}$ and
$u\in J_{1}$ in such a case.
Case 2. Suppose $x_{n}\in C$. Since $y_{n}\notin C$, but
$x_{j_{1}}y_{n},\dots,x_{j_{p}}y_{n}\in E(G)$ and $C$ is a vertex cover, then
we obtain that $x_{j_{1}},\dots,x_{j_{p}}\in C$. Hence
$x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot x_{n}$ divides $u$. Note that the
support of $w=u/(x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot x_{n})$ is a vertex
cover of $G_{2}$. Furthermore, $\operatorname{supp}(w)$ is a minimal vertex
cover, since $|\operatorname{supp}(w)|=n-(p+1)=|V(G_{2})|/2$ and $G_{2}$ is a
very well–covered graph (Proposition 2.1). Hence, $w\in I(G_{2})^{\vee}$ and
$u\in J_{2}$.

Hence, we have shown that $G(I(G)^{\vee})$ is contained in $G(J_{1})\cup
G(J_{2})$.

For the opposite inclusion, let $v\in G(I(G_{1})^{\vee})$, then
$\operatorname{supp}(v)$ is a minimal vertex cover of $G_{1}$. We claim that
$C=\operatorname{supp}(x_{i_{1}}\cdots x_{i_{t}}\cdot y_{n}v)$ is a minimal
vertex cover of $G$. Indeed, all edges of $G$ incident with a vertex belonging
to $\\{x_{i_{1}},\dots,x_{i_{t}},y_{n}\\}$ are incident with a vertex of $C$.
Since
$N[x_{n}]\setminus\\{x_{n},y_{n}\\}=\\{x_{i_{1}},\dots,x_{i_{t}}\\}\subset C$,
then each edge which is incident with $x_{n}$ is also incident with a vertex
of $C$. Finally, let $e$ be an edge incident with $y_{i_{k}}$ for some
$k\in[t]$. Then $e=x_{j}y_{i_{k}}\in E(G)$. Since $x_{j}y_{i_{k}}$ and
$x_{i_{k}}x_{n}$ are both edges of $G$, by Characterization 1.12(v),
$x_{j}x_{n}\in E(G)$, too. It follows that $x_{j}\in
N[x_{n}]\setminus\\{x_{n},y_{n}\\}$ and $\\{x_{j},y_{i_{k}}\\}\cap
C\neq\varnothing$. Finally, $C$ is a minimal vertex cover of $G$. This shows
that $G(J_{1})\subseteq G(I(G)^{\vee})$. Similarly, one proves that
$G(J_{2})\subseteq G(I(G)^{\vee})$ by exploiting again condition (v) of
Characterization 1.12. The result follows.

######  Example 2.4.

Let $S=K[x_{1},\ldots,x_{6},y_{1},\ldots,y_{6}]$. Consider the graph $G$
depicted below

$y_{1}$$y_{2}$$x_{1}$$x_{2}$$x_{3}$$y_{3}$$x_{4}$$y_{4}$$x_{5}$$y_{5}$$x_{6}$$y_{6}$$G$

By Characterization 1.12, one verifies that $G$ is a Cohen–Macaulay very
well–covered graph with $12$ vertices. We have

$\displaystyle I(G)\ $ $\displaystyle\ =\
(x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6},x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{1}x_{5},$
$\displaystyle\phantom{\
=(..}x_{1}x_{6},x_{2}x_{3},x_{2}x_{4},x_{2}x_{5},x_{2}x_{6},x_{3}y_{4},x_{3}y_{5},x_{3}y_{6},x_{4}y_{5},x_{4}y_{6}),$
$\displaystyle I(G)^{\vee}\ $ $\displaystyle\ =\
(x_{1}x_{2}x_{3}x_{4}x_{5}x_{6},y_{1}x_{2}x_{3}x_{4}x_{5}x_{6},x_{1}y_{2}x_{3}x_{4}x_{5}x_{6},x_{1}x_{2}x_{3}x_{4}y_{5}x_{6},$
$\displaystyle\phantom{\
=(..}x_{1}x_{2}x_{3}x_{4}x_{5}y_{6},x_{1}x_{2}x_{3}x_{4}y_{5}y_{6},x_{1}x_{2}x_{3}y_{4}y_{5}y_{6},x_{1}x_{2}y_{3}y_{4}y_{5}y_{6}).$

Furthermore, $N[x_{6}]=\\{x_{1},x_{2},x_{6},y_{6}\\}$,
$N[y_{6}]=\\{x_{3},x_{4},x_{6},y_{6}\\}$ and consequentaly

$G_{1}=G\setminus\\{x_{1},y_{1},x_{2},y_{2},x_{6},y_{6}\\},\,\,G_{2}=G\setminus\\{x_{3},y_{3},x_{4},y_{4},x_{6},y_{6}\\},$

_i.e._ ,

$y_{3}$$y_{4}$$x_{3}$$x_{4}$$x_{5}$$y_{5}$$G_{1}$

$y_{1}$$y_{2}$$x_{1}$$x_{2}$$x_{5}$$y_{5}$$G_{2}$

It follows that

$\displaystyle I(G_{1})^{\vee}$ $\displaystyle=$
$\displaystyle(x_{3}x_{4}x_{5},x_{3}x_{4}y_{5},x_{3}y_{4}y_{5},y_{3}y_{4}y_{5}),$
$\displaystyle I(G_{2})^{\vee}$ $\displaystyle=$
$\displaystyle(x_{1}x_{2}x_{5},y_{1}x_{2}x_{5},x_{1}y_{2}x_{5},x_{1}x_{2}y_{5}).$

Finally,

$I(G)^{\vee}=x_{1}x_{2}y_{6}I(G_{1})^{\vee}+x_{3}x_{4}x_{6}I(G_{2})^{\vee}$

is a Betti splitting.

##  3\. The resolution of cover ideals of very well–covered graphs

In this section we construct the minimal free resolution of the cover ideal
$I(G)^{\vee}$ of any Cohen–Macaulay very well–covered graph $G$. Our method
uses induction on half of the number of vertices of $G$ and the mapping cone
described in Subsection 1.1. Indeed, the mapping cone applied to the Betti
splitting of $I(G)^{\vee}$ (Proposition 2.3) yields a minimal free resolution
of $I(G)^{\vee}$, if we know the minimal free resolutions of the cover ideals
of three suitable subgraphs which we can associate to the given graph $G$.

Firstly, let $G_{1}$ and $G_{2}$ be two graphs, we define _intersection graph_
of $G_{1}$ ad $G_{2}$ the graph $G$ with $V(G)=V(G_{1})\cap V(G_{2})$ and
$E(G)=\\{e:e\in E(G_{1})\cap E(G_{2})\\}$. We denote $G$ by $G_{1}\cap G_{2}$.

######  Setup 3.1.

Let $G$ be a Cohen–Macaulay very well–covered graph with $2n$ vertices. Let
$N[x_{n}]=\\{x_{i_{1}},x_{i_{2}},\dots,x_{i_{t}},x_{n},y_{n}\\}$,
$N[y_{n}]=\\{x_{j_{1}},x_{j_{2}},\dots,x_{j_{p}},x_{n},y_{n}\\}$ and define

$\displaystyle G_{1}\ $ $\displaystyle=\
G\setminus\\{x_{i_{1}},y_{i_{1}},x_{i_{2}},y_{i_{2}},\dots,x_{i_{t}},y_{i_{t}},x_{n},y_{n}\\},$
$\displaystyle G_{2}\ $ $\displaystyle=\
G\setminus\\{x_{j_{1}},y_{j_{1}},\\!x_{j_{2}},\\!y_{j_{2}},\dots,x_{j_{p}},y_{j_{p}},x_{n},y_{n}\\}.$

By Proposition 2.3, both $G_{1}$ and $G_{2}$ are Cohen–Macaulay very
well–covered graphs. Furthermore, $|V(G_{1})|=2(n-1-t)$ and
$|V(G_{2})|=2(n-1-p)$.
Set $J=I(G)^{\vee}$, $J_{1}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot
y_{n}I(G_{1})^{\vee}$ and $J_{2}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot
x_{n}I(G_{2})^{\vee}$. Then, Proposition 2.3 implies that $J=J_{1}+J_{2}$ is a
Betti splitting of $J$.
Finally, let us consider the subgraph $G_{3}=G_{1}\cap G_{2}$ of $G$. Since
the structure of $G_{1}$ and $G_{2}$, it is clear that $G_{3}$ is
Cohen–Macaulay very well–covered, too.

For instance, in Example 2.4, $G_{3}=G\setminus\\{x_{i},y_{i}:i\neq 5\\}$.

The following lemmas will be crucial in the sequel.

######  Lemma 3.2.

Assume Setup 3.1. Then

$N[x_{n}]\cap N[y_{n}]\ =\ \\{x_{n},y_{n}\\}.$

###### Proof.

For all $s\in[p]$, $x_{j_{s}}y_{n}\in E(G)$. Thus Characterization 1.12(iv)
implies that $x_{j_{s}}x_{n}\notin E(G)$, and so $x_{j_{s}}\notin N[x_{n}]$,
for all $s\in[p]$. Finally $N[x_{n}]\cap N[y_{n}]=\\{x_{n},y_{n}\\}$.

######  Lemma 3.3.

Assume Setup 3.1. Then

$J_{1}\cap J_{2}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot
x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot x_{n}y_{n}I(G_{3})^{\vee}.$

###### Proof.

Set $J_{1,2}=J_{1}\cap J_{2}$. Since $J=J_{1}+J_{2}$ is a Betti splitting and
$J$ has an $n$–linear resolution, then $J_{1,2}$ has an $(n+1)$–linear
resolution (Proposition 1.4). A generating set for $J_{1,2}$ is the set
$\\{\textup{lcm}(u_{1},u_{2}):u_{1}\in G(J_{1}),u_{2}\in G(J_{2})\\}$. Since
$J_{1,2}$ is equigenerated in degree $n+1$, then

$G(J_{1,2})\ =\ \big{\\{}\textup{lcm}(u_{1},u_{2})\ :\ u_{1}\in G(J_{1}),\
u_{2}\in G(J_{2}),\ \deg(\textup{lcm}(u_{1},u_{2}))=n+1\big{\\}}.$

By Lemma 3.2, $N[x_{n}]\cap N[y_{n}]=\\{x_{n},y_{n}\\}$. Thus, by the
presentation of $J_{1}$ and $J_{2}$, we get that each monomial $w\in
G(J_{1,2})$ is divided by $x_{i_{1}}\cdots x_{i_{t}}\cdot x_{j_{1}}\cdots
x_{j_{p}}\cdot x_{n}y_{n}$. Note that $(X\cup
Y)\setminus\\{x_{i_{1}},y_{i_{1}},\dots,x_{i_{t}},y_{i_{t}},x_{j_{1}},y_{j_{1}},\dots,x_{j_{p}},y_{j_{p}},x_{n},y_{n}\\}$
is the vertex set of the graph $G_{3}$. Moreover the support of
$v_{1}=u_{1}/(x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot x_{j_{1}}x_{j_{2}}\cdots
x_{j_{p}}\cdot y_{n})$ is a minimal vertex cover of the very well–covered
graph $G_{3}$. Since $\textup{lcm}(u_{1},u_{2})=x_{n}u_{1}$, we get that
$\textup{lcm}(u_{1},u_{2})=x_{i_{1}}\cdots x_{i_{t}}\cdot x_{j_{1}}\cdots
x_{j_{p}}\cdot x_{n}y_{n}v_{1}\in G(x_{i_{1}}\cdots x_{i_{t}}\cdot
x_{j_{1}}\cdots x_{j_{p}}\cdot x_{n}y_{n}I(G_{3})^{\vee})$.
Conversely, let $w=x_{i_{1}}\cdots x_{i_{t}}\cdot x_{j_{1}}\cdots
x_{j_{p}}\cdot x_{n}y_{n}v\in G(x_{i_{1}}\cdots x_{i_{t}}\cdot x_{j_{1}}\cdots
x_{j_{p}}\cdot x_{n}y_{n}I(G_{3})^{\vee})$, with $v\in I(G_{3})^{\vee}$. Then
$\operatorname{supp}(x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot v)$ is a minimal
vertex cover of $G_{1}$ and $\operatorname{supp}(x_{i_{1}}x_{i_{2}}\cdots
x_{i_{t}}\cdot v)$ is a minimal vertex cover of $G_{2}$. Thus
$u_{1}=w/x_{n}\in G(J_{1})$ and $u_{2}=w/y_{n}\in G(J_{2})$. Since
$w=\textup{lcm}(u_{1},u_{2})$ and $\deg(w)=n+1$, we have that $w\in
G(J_{1,2})$, showing the other inclusion.

We now turn to the construction of the minimal free resolution. For a subset
$C$ of the set of variables $X\cup
Y=\\{x_{1},\dots,x_{n},y_{1},\dots,y_{n}\\}$, we set

${\bf z}_{C}={\bf x}_{C_{x}}{\bf y}_{C_{y}},$ (4)

with $C_{x}=\\{i:x_{i}\in C\\}$ and $C_{y}=\\{j:y_{j}\in C\\}$.

If $G$ is graph such that $V(G)=\\{x_{1},\dots,x_{n},y_{1},\dots,y_{n}\\}$ and
$C\in\mathcal{C}(G)$, we define the set $\mathcal{C}(G;C)=\\{x_{s}:y_{s}\in C\
\text{and}\ (C\setminus y_{s})\cup x_{s}\in\mathcal{C}(G)\\}$. Recall that
$\mathcal{C}(G)$ is the set of all minimal vertex cover of $G$.

For instance, in Example 1.13, for $C=\\{x_{1},x_{2},y_{3},y_{4}\\}$,
$\mathcal{C}(G;C)=\\{x_{3}\\}$. Indeed, $\\{x_{1},x_{2},y_{4}\\}\cup
x_{3}=\\{x_{1},x_{2},x_{3},y_{4}\\}$ is a minimal vertex cover of the given
graph $G$. One can observe that $x_{4}\notin\mathcal{C}(G;C)$, since
$\\{x_{1},x_{2},y_{3}\\}\cup x_{4}$ is not a vertex cover of $G$. Moreover,
for $C=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathcal{C}(G;C)=\varnothing$.

In what follows, we denote by $\binom{\mathcal{C}(G;C)}{i}$ the set of all
subsets of size $i$ of $\mathcal{C}(G;C)$, $0\leq i\leq|\mathcal{C}(G;C)|$.
With abuse of notation, for $\sigma\in\binom{\mathcal{C}(G;C)}{i}$, we set
${\bf x}_{\sigma}=\prod_{x_{s}\in\sigma}x_{s}$. In particular, ${\bf
x}_{\varnothing}=1$.

######  Construction 3.4.

Assume Setup 3.1. Let

${\mathbb{F}}\ :\ \ \dots\rightarrow F_{i}\xrightarrow{\ d_{i}\
}F_{i-1}\xrightarrow{\ d_{i-1}}\cdots\xrightarrow{\ d_{2}\
}F_{1}\xrightarrow{\ d_{1}\ }F_{0}\xrightarrow{\ d_{0}\ }J\rightarrow 0$

be the complex

  1. -

whose $i$th free module $F_{i}$ has as a basis the symbols ${\bf f}(C;\sigma)$
having multidegree ${\bf z}_{C}{\bf x}_{\sigma}$, where $C\in\mathcal{C}(G)$
and $\sigma\in\binom{\mathcal{C}(G;C)}{i}$, _i.e._ ,
$\sigma\subseteq\mathcal{C}(G;C)$ is a subset of size $i$;

  2. -

and whose $i$th differential is given by $d_{0}({\bf f}(C;\varnothing))={\bf
z}_{C}$ for $i=0$ and for $i>0$ is defined as follows:

$d_{i}({\bf f}(C;\sigma))\ =\
\sum_{x_{s}\in\sigma}(-1)^{\alpha(\sigma;x_{s})}\big{[}y_{s}{\bf
f}((C\setminus y_{s})\cup x_{s};\sigma\setminus x_{s})-x_{s}{\bf
f}(C;\sigma\setminus x_{s})\big{]},$

where $\alpha(\sigma;x_{s})=|\\{x_{j}\in\sigma:j>s\\}|$.

From now on, we set $J_{1,2}=J_{1}\cap J_{2}$ and denote by
$({\mathbb{F}}_{J_{1,2}},d^{J_{1,2}})$, $({\mathbb{F}}_{J_{1}},d^{J_{1}})$,
$({\mathbb{F}}_{J_{2}},d^{J_{2}})$ the minimal free resolutions of $J_{1,2}$,
$J_{1}$, $J_{2}$, respectively.

######  Theorem 3.5.

The complex ${\mathbb{F}}$ given in Construction 3.4 is the minimal free
resolution of $J=I(G)^{\vee}$.

###### Proof.

We proceed by strong induction on $|V(G)|/2=n$. For $n=1$, $I(G)=(x_{1}y_{1})$
and $J=I(G)^{\vee}=(x_{1},y_{1})$. In this case one readily verifies that the
complex ${\mathbb{F}}$ of Construction 3.4 is the minimal free resolution of
$J=I(G)^{\vee}$. So, let $n>1$. The graphs $G_{1},G_{2},G_{3}$ have vertex
sets whose cardinality is less than $|V(G)|$, thus by induction we can assume
that the minimal free resolutions of
$I(G_{1})^{\vee},I(G_{2})^{\vee},I(G_{3})^{\vee}$ are as given in Construction
3.4. As a consequence, we know explicitly the resolutions
${\mathbb{F}}_{J_{1,2}},{\mathbb{F}}_{J_{1}},{\mathbb{F}}_{J_{2}}$, since each
of these resolutions is equal to one of the three previously mentioned
resolutions up to multiplication by a suitable monomial.

Let $({\mathbb{F}}_{J},d^{J})$ be the resolution obtained by the mapping cone
applied to the Betti splitting $J=J_{1}+J_{2}$. Then ${\mathbb{F}}_{J}$ is the
minimal free resolution of $J$. We show that ${\mathbb{F}}_{J}$ can be
identified with ${\mathbb{F}}$. We achieve this goal in three steps.
Step 1. Let us show that the free modules of ${\mathbb{F}}_{J}$ have the basis
described in Construction 3.4. By the mapping cone (Subsection 1.1, (i)),
$F_{i}^{J}=F_{i-1}^{J_{1,2}}\oplus F_{i}^{J_{1}}\oplus F_{i}^{J_{2}}$.

The bases of $F_{i}^{J_{1}},F_{i}^{J_{2}}$ and $F_{i}^{J_{1,2}}$ are the
following ones, respectively:

$\displaystyle\mathcal{B}_{1}\ $ $\displaystyle=\
\Big{\\{}x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot y_{n}\cdot{\bf
f}_{J_{1}}(C_{1};\sigma_{1})\ :\ C_{1}\in\mathcal{C}(G_{1}),\
\sigma_{1}\in\binom{\mathcal{C}(G_{1};C_{1})}{i}\Big{\\}},$
$\displaystyle\mathcal{B}_{2}\ $ $\displaystyle=\
\Big{\\{}x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot x_{n}\cdot{\bf
f}_{J_{2}}(C_{2};\sigma_{2})\ :\ C_{2}\in\mathcal{C}(G_{2}),\
\sigma_{2}\in\binom{\mathcal{C}(G_{2};C_{2})}{i}\Big{\\}},$
$\displaystyle\mathcal{B}_{3}\ $ $\displaystyle=\ \Big{\\{}x_{i_{1}}\cdots
x_{i_{t}}\cdot x_{j_{1}}\cdots x_{j_{p}}\cdot x_{n}y_{n}\cdot{\bf
f}_{J_{1,2}}(C_{3};\sigma_{3})\ :\ C_{3}\in\mathcal{C}(G_{3}),\
\sigma_{3}\in\binom{\mathcal{C}(G_{3};C_{3})}{i}\Big{\\}}.$

Let ${\bf f}(C;\sigma)$ be a basis element of $F_{i}$ as in Construction 3.4,
_i.e._ , with multidegree ${\bf z}_{C}{\bf x}_{\sigma}$ and with
$C\in\mathcal{C}(G)$, $\sigma\in\binom{\mathcal{C}(G;C)}{i}$. We distinguish
three possible cases.
Case 1.1. Let $x_{n}\in C$. Then $N[y_{n}]\setminus
N[x_{n}]=\\{x_{j_{1}},\dots,x_{j_{p}}\\}\subset C$. Indeed, $C$ is a vertex
cover of $G$, $y_{n}\notin C$ (Lemma 2.2) but $x_{j_{r}}y_{n}\in E(G)$ for
$r\in[p]$. Hence, $C_{2}=C\setminus\\{x_{j_{1}},\dots,x_{j_{p}},x_{n}\\}$ is a
minimal vertex cover of $G_{2}$. Furthermore, for all $x_{s}\in\sigma$,
$(C_{2}\setminus y_{s})\cup x_{s}$ is a minimal vertex cover of $G_{2}$
because $(C\setminus y_{s})\cup x_{s}$ is a minimal vertex cover of $G$. Since
${\bf b}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot x_{n}\cdot{\bf
f}_{J_{2}}(C_{2};\sigma)\in\mathcal{B}_{2}$ has the same multidegree of ${\bf
f}(C;\sigma)$, we can identify ${\bf f}(C;\sigma)$ with ${\bf b}$.
Case 1.2. Let $y_{n}\in C$ and $x_{n}\notin\sigma$. Then
$C_{1}=C\setminus\\{x_{1},\dots,x_{i_{t}},y_{n}\\}$ is a minimal vertex cover
of $G_{1}$. As before, we can identify ${\bf f}(C;\sigma)$ with
$x_{i_{1}}\dots x_{i_{t}}\cdot y_{n}\cdot{\bf
f}_{J_{1}}(C_{1};\sigma)\in\mathcal{B}_{1}$.
Case 1.3. Let $y_{n}\in C$ and $x_{n}\in\sigma$. Then both $C$ and
$(C\setminus y_{n})\cup x_{n}$ are minimal vertex covers of $G$. Hence,
$(N[x_{n}]\cup
N[y_{n}])\setminus\\{x_{n},y_{n}\\}=\\{x_{i_{1}},\dots,x_{i_{t}},x_{j_{1}},\dots,x_{j_{p}}\\}\subset
C$. Setting
$C_{3}=C\setminus\\{x_{i_{1}},\dots,x_{i_{t}},x_{j_{1}},\dots,x_{j_{p}},y_{n}\\}$
and $\sigma_{3}=\sigma\setminus x_{n}$, we obtain that $C_{3}$ and
$(C_{3}\setminus y_{s})\cup x_{s}$ are both minimal vertex covers of $G_{3}$
for all $x_{s}\in\sigma_{3}$. Hence, in this case we can identify ${\bf
f}(C;\sigma)$ with $x_{i_{1}}\cdots x_{i_{t}}\cdot x_{j_{1}}\cdots
x_{j_{p}}\cdot x_{n}y_{n}\cdot{\bf
f}_{J_{1,2}}(C_{3};\sigma_{3})\in\mathcal{B}_{3}$.

Conversely, any basis element ${\bf b}\in\mathcal{B}_{i}$ ($i=1,2,3$) can be
identified with a basis element ${\bf f}(C;\sigma)$, as given in Construction
3.4. Thus, we realize that the modules $F_{i}^{J}$ have the required bases as
described in Construction 3.4.
Step 2. Let $\psi_{-1}:u\in J_{1,2}\mapsto(u,-u)\in J_{1}\oplus J_{2}$. In
order to apply the mapping cone, we need to construct the comparison maps
$\psi_{i}$ making the following diagram

$\textstyle{\displaystyle{\mathbb{F}}_{J_{1,2}}:\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{2}^{J_{1,2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{2}}$$\scriptstyle{d_{2}^{J_{1,2}}}$$\textstyle{F_{1}^{J_{1,2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{1}}$$\scriptstyle{d_{1}^{J_{1,2}}}$$\textstyle{F_{0}^{J_{1,2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{0}}$$\scriptstyle{d_{0}^{J_{1,2}}}$$\textstyle{J_{1,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{-1}}$$\textstyle{0}$$\textstyle{{\mathbb{F}}_{J_{1}}\oplus{\mathbb{F}}_{J_{2}}:\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{2}^{J_{1}}\oplus
F_{2}^{J_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}^{J_{1}}\oplus
d_{2}^{J_{2}}}$$\textstyle{F_{1}^{J_{1}}\oplus
F_{1}^{J_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}^{J_{1}}\oplus
d_{1}^{J_{2}}}$$\textstyle{F_{0}^{J_{1}}\oplus
F_{0}^{J_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}^{J_{1}}\oplus
d_{0}^{J_{2}}}$$\textstyle{J_{1}\oplus
J_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

commutative. Using the notation in (4), let ${\bf b}=x_{i_{1}}\cdots
x_{i_{t}}\cdot x_{j_{1}}\cdots x_{j_{p}}\cdot x_{n}y_{n}\cdot{\bf
f}_{J_{1,2}}(C_{3};\sigma_{3})={\bf z}_{N[x_{n}]\cup N[y_{n}]}\cdot{\bf
f}_{J_{1,2}}(C_{3};\sigma_{3})\in\mathcal{B}_{3}$ be a basis element of
$F_{i}^{J_{1,2}}$. We define

$\psi_{i}({\bf b})\ =\ \big{(}{\bf z}_{N[x_{n}]}\cdot{\bf
f}_{J_{1}}(C_{3}\cup\\{x_{j_{1}},\dots,x_{j_{p}}\\};\sigma_{3}),\ -{\bf
z}_{N[y_{n}]}\cdot{\bf
f}_{J_{2}}(C_{3}\cup\\{x_{i_{1}},\dots,x_{i_{t}}\\};\sigma_{3})\big{)}.$

To simplify the notation, we set
$C_{1}=C_{3}\cup\\{x_{j_{1}},\dots,x_{j_{p}}\\}$ and
$C_{2}=C_{3}\cup\\{x_{i_{1}},\dots,x_{i_{t}}\\}$. One can note that $\psi_{i}$
is well defined. Indeed, $C_{1}\in\mathcal{C}(G_{1})$,
$C_{2}\in\mathcal{C}(G_{2})$, $\sigma_{3}\subseteq\mathcal{C}(G_{1};C_{1})$,
and $\sigma_{3}\subseteq\mathcal{C}(G_{2};C_{2})$.

We need to verify that for all $i\geq 0$ and all ${\bf b}\in\mathcal{B}_{3}$,
it is

$\psi_{i-1}\circ d_{i}^{J_{1,2}}({\bf b})\ =\ (d_{i}^{J_{1}}\oplus
d_{i}^{J_{2}})\circ\psi_{i}({\bf b}).$

For $i=0$, we have $\sigma_{3}=\varnothing$, thus ${\bf x}_{\sigma_{3}}=1$ and
both sides of the equation are equal to

$({\bf z}_{N[x_{n}]\cup N[y_{n}]}{\bf z}_{C_{3}},-{\bf z}_{N[x_{n}]\cup
N[y_{n}]}{\bf z}_{C_{3}}).$

Let $i>0$. To further simplify the notation, let us write the generic basis
element $({\bf z}_{N[x_{n}]}{\bf f}_{J_{1}}(C_{1};\sigma_{1}),{\bf
z}_{N[y_{n}]}{\bf f}_{J_{2}}(C_{2};\sigma_{2}))\in F_{i}^{J_{1}}\oplus
F_{i}^{J_{2}}$ as ${\bf z}_{N[x_{n}]}{\bf f}_{J_{1}}(C_{1};\sigma_{1})+{\bf
z}_{N[y_{n}]}{\bf f}_{J_{2}}(C_{2};\sigma_{2})$. By induction we know
explicitly $d_{i}^{J_{1,2}},d_{i}^{J_{1}},d_{i}^{J_{2}}$. Let us compute
$\psi_{i-1}\circ d_{i}^{J_{1,2}}({\bf b})$. We have

$\displaystyle d_{i}^{J_{1,2}}({\bf b})$ $\displaystyle={\bf z}_{N[x_{n}]\cup
N[y_{n}]}\sum_{x_{s}\in\sigma_{3}}(-1)^{\alpha(\sigma_{3};x_{s})}\big{[}y_{s}{\bf
f}_{J_{1,2}}((C_{3}\setminus y_{s})\cup x_{s};\sigma_{3}\setminus
x_{s})-x_{s}{\bf f}_{J_{1,2}}(C_{3};\sigma_{3}\setminus x_{s})\big{]}.$

Hence,

$\displaystyle\psi_{i-1}\circ d_{i}^{J_{1,2}}({\bf b})\ =\
\sum_{x_{s}\in\sigma_{3}}(-1)^{\alpha(\sigma_{3};x_{s})}\big{[}$
$\displaystyle\ y_{s}{\bf z}_{N[x_{n}]}\cdot{\bf f}_{J_{1}}((C_{1}\setminus
y_{s})\cup x_{s};\sigma_{3}\setminus x_{s})$ (5) $\displaystyle-$
$\displaystyle\ y_{s}{\bf z}_{N[y_{n}]}\cdot{\bf f}_{J_{2}}((C_{2}\setminus
y_{s})\cup x_{s};\sigma_{3}\setminus x_{s})$ $\displaystyle-$ $\displaystyle\
x_{s}{\bf z}_{N[x_{n}]}\cdot{\bf f}_{J_{1}}(C_{1};\sigma_{3}\setminus x_{s})$
$\displaystyle+$ $\displaystyle\ x_{s}{\bf z}_{N[y_{n}]}\cdot{\bf
f}_{J_{2}}(C_{2};\sigma_{3}\setminus x_{s})\big{]}.$

Now, let us compute $(d_{i}^{J_{1}}\oplus d_{i}^{J_{2}})\circ\psi_{i}({\bf
b})$. Firstly, we have

$\psi_{i}({\bf b})\ =\ {\bf z}_{N[x_{n}]}\cdot{\bf
f}_{J_{1}}(C_{1};\sigma_{3})-{\bf z}_{N[y_{n}]}\cdot{\bf
f}_{J_{2}}(C_{2};\sigma_{3}),$

then

$\displaystyle(d_{i}^{J_{1}}\oplus d_{i}^{J_{2}})$
$\displaystyle\circ\psi_{i}({\bf b})\ ={\bf z}_{N[x_{n}]}\cdot
d_{i}^{J_{1}}({\bf f}_{J_{1}}(C_{1};\sigma_{3}))-{\bf z}_{N[y_{n}]}\cdot
d_{i}^{J_{2}}({\bf f}_{J_{2}}(C_{2};\sigma_{3}))$ $\displaystyle=$
$\displaystyle\ {\bf
z}_{N[x_{n}]}\sum_{x_{s}\in\sigma_{3}}(-1)^{\alpha(\sigma_{3};x_{s})}\big{[}y_{s}{\bf
f}_{J_{1}}((C_{1}\setminus y_{s})\cup x_{s};\sigma_{3}\setminus
x_{s})-x_{s}{\bf f}_{J_{1}}(C_{1};\sigma_{3}\setminus x_{s})\big{]}$
$\displaystyle-$ $\displaystyle\ {\bf
z}_{N[y_{n}]}\sum_{x_{s}\in\sigma_{3}}(-1)^{\alpha(\sigma_{3};x_{s})}\big{[}y_{s}{\bf
f}_{J_{2}}((C_{2}\setminus y_{s})\cup x_{s};\sigma_{3}\setminus
x_{s})-x_{s}{\bf f}_{J_{2}}(C_{2};\sigma_{3}\setminus x_{s})\big{]}.$

A comparison with equation (5) shows that $\psi_{i-1}\circ
d_{i}^{J_{1,2}}({\bf b})\ =\ (d_{i}^{J_{1}}\oplus
d_{i}^{J_{2}})\circ\psi_{i}({\bf b})$.
Step 3. It remains to prove that the differentials $d_{i}^{J}$ act as
described in Construction 3.4. So, let ${\bf f}(C;\sigma)$,
$C\in\mathcal{C}(G)$ and $\sigma\in\binom{\mathcal{C}(G;C)}{i}$ be a basis
element. Let us compute $d_{i}^{J}({\bf f}(C;\sigma))$. For $i=0$, we easily
see that $d_{0}^{J}({\bf f}(C;\varnothing))={\bf z}_{C}$, as desired.

Let $i>0$. We distinguish three cases.
Case 3.1. Let $x_{n}\in C$. Set
$C_{2}=C\setminus\\{x_{j_{1}},\dots,x_{j_{p}},x_{n}\\}$. Then, by the mapping
cone (Subsection 1.1, (ii)), we have:

$\displaystyle d_{i}^{J}({\bf f}$ $\displaystyle(C;\sigma))\ =\
d_{i}^{J_{2}}(x_{j_{1}}\cdots x_{j_{p}}\cdot x_{n}\cdot{\bf
f}_{J_{2}}(C_{2};\sigma))$ $\displaystyle=\ x_{j_{1}}\cdots x_{j_{p}}\cdot
x_{n}\sum_{x_{s}\in\sigma}(-1)^{\alpha(\sigma;x_{s})}\big{[}y_{s}{\bf
f}_{J_{2}}((C_{2}\setminus y_{s})\cup x_{s};\sigma\setminus x_{s})-x_{s}{\bf
f}_{J_{2}}(C_{2};\sigma\setminus x_{s})\big{]}$ $\displaystyle=\
\sum_{x_{s}\in\sigma}(-1)^{\alpha(\sigma;x_{s})}\big{[}y_{s}{\bf
f}((C\setminus y_{s})\cup x_{s};\sigma\setminus x_{s})-x_{s}{\bf
f}(C;\sigma\setminus x_{s})\big{]},$

as required.
Case 3.2. Let $y_{n}\in C$ and $x_{n}\notin\sigma$. Setting
$C_{1}=C\setminus\\{x_{i_{1}},\dots,x_{i_{t}},x_{n}\\}$, by the mapping cone
we see that $d_{i}^{J}({\bf
f}(C;\sigma))=d_{i}^{J_{1}}(x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot
y_{n}\cdot{\bf f}_{J_{1}}(C_{1};\sigma))$ has the required expression.
Case 3.3. Let $y_{n}\in C$ and $x_{n}\in\sigma$. Setting
$C_{3}=C\setminus\\{x_{i_{1}},\dots,x_{i_{t}},x_{j_{1}},\dots,x_{j_{p}},y_{n}\\}$,
$\sigma_{3}=\sigma\setminus x_{n}$ and ${\bf b}=x_{i_{1}}\cdots x_{i_{t}}\cdot
x_{j_{1}}\cdots x_{j_{p}}\cdot x_{n}y_{n}\cdot{\bf
f}_{J_{1,2}}(C_{3};\sigma_{3})$, by the mapping cone (Subsection 1.1, (ii)),
we have that

$d_{i}^{J}({\bf f}(C;\sigma))\ =\ -d_{i-1}^{J_{1,2}}({\bf b})+\psi_{i-1}({\bf
b}).$

Let us compute $-d_{i-1}^{J_{1,2}}({\bf b})$. Since ${\bf z}_{N[x_{n}]\cup
N[y_{n}]}=x_{i_{1}}\cdots x_{i_{t}}\cdot x_{j_{1}}\cdots x_{j_{p}}\cdot
x_{n}y_{n}$, we have

$\displaystyle-d_{i-1}^{J_{1,2}}({\bf b})\\!$ $\displaystyle=\\!-{\bf
z}_{N[x_{n}]\cup
N[y_{n}]}\\!\\!\sum_{x_{s}\in\sigma_{3}}\\!(-1)^{\alpha(\sigma_{3};x_{s})}\big{[}y_{s}{\bf
f}_{J_{1,2}}((C_{3}\setminus y_{s})\cup x_{s};\sigma_{3}\setminus
x_{s})-x_{s}{\bf f}_{J_{1,2}}(C_{3};\sigma_{3}\setminus x_{s})\big{]}$
$\displaystyle=-\sum_{x_{s}\in\sigma_{3}}(-1)^{\alpha(\sigma\setminus
x_{n};x_{s})}\big{[}y_{s}{\bf f}((C\setminus y_{s})\cup x_{s};\sigma\setminus
x_{s})-x_{s}{\bf f}(C;\sigma\setminus x_{s})\big{]}$
$\displaystyle=\phantom{-}\\!\\!\sum_{x_{s}\in\sigma\setminus
x_{n}}\\!\\!\\!(-1)^{\alpha(\sigma;x_{s})}\big{[}y_{s}{\bf f}((C\setminus
y_{s})\cup x_{s};\sigma\setminus x_{s})-x_{s}{\bf f}(C;\sigma\setminus
x_{s})\big{]},$

where the last equation follows from the fact that
$\alpha(\sigma;x_{s})=\alpha(\sigma\setminus x_{n};x_{s})+|\\{x_{n}\\}|$ for
all $x_{s}\in\sigma\setminus x_{n}$.

Whereas, for the term ${\psi}_{i-1}({\bf b})$, by the definition of the
comparison maps, we have

$\displaystyle{\psi}_{i-1}({\bf b})\ $ $\displaystyle=\ {\bf
z}_{N[x_{n}]}\cdot{\bf
f}_{J_{1}}(C_{3}\cup\\{x_{j_{1}},\dots,x_{j_{p}}\\};\sigma_{3})-{\bf
z}_{N[y_{n}]}\cdot{\bf
f}_{J_{2}}(C_{3}\cup\\{x_{i_{1}},\dots,x_{i_{t}}\\};\sigma_{3})$
$\displaystyle=\ y_{n}{\bf f}((C\setminus y_{n})\cup x_{n};\sigma\setminus
x_{n})-x_{n}{\bf f}(C;\sigma\setminus x_{n})$ $\displaystyle=\
(-1)^{\alpha(\sigma;x_{n})}\big{[}y_{n}{\bf f}((C\setminus y_{n})\cup
x_{n};\sigma\setminus x_{n})-x_{n}{\bf f}(C;\sigma\setminus x_{n})\big{]},$

where the last equation follows because
$\alpha(\sigma;x_{n})=|\varnothing|=0$. Hence, we see that

$d_{i}^{J}({\bf f}(C;\sigma))\ =\
\sum_{x_{s}\in\sigma}\\!(-1)^{\alpha(\sigma;x_{s})}\big{[}y_{s}{\bf
f}((C\setminus y_{s})\cup x_{s};\sigma\setminus x_{s})-x_{s}{\bf
f}(C;\sigma\setminus x_{s})\big{]},$

as desired. The induction is complete and the result follows.

######  Corollary 3.6.

Let $G$ be a Cohen–Macaulay very well–covered graph. Then

$\displaystyle\beta_{i}(I(G)^{\vee})\ $ $\displaystyle=\
\sum_{C\in\mathcal{C}(G)}\binom{|\mathcal{C}(G;C)|}{i},\ \ \ \ i\geq 0,$
$\displaystyle\operatorname{pd}(I(G)^{\vee})\ $ $\displaystyle=\
\max\big{\\{}\big{|}\mathcal{C}(G;C)\big{|}\ :\ C\in\mathcal{C}(G)\big{\\}}.$

In particular, the graded Betti numbers of $I(G)^{\vee}$ do not depend upon
the characteristic of the underlying field $K$.

Let $G$ be a graph and let $z_{i}z_{j},z_{k}z_{\ell}\in E(G)$ be a pair of
edges. We say that $z_{i}z_{j}$ and $z_{k}z_{\ell}$ are 3–disjoint if the
induced subgraph of $G$ on the vertex set $\\{z_{i},z_{j},z_{k},z_{\ell}\\}$
consists of two disjoint edges. The maximum size of a set of pairwise
$3$–disjoint edges in $G$ is denoted by $a(G)$ [29].

The next result holds.

######  Corollary 3.7.

Let $G$ be a Cohen–Macaulay very well–covered graph. Then

$a(G)=\operatorname{reg}(S/I(G))=\max\big{\\{}\big{|}\mathcal{C}(G;C)\big{|}\
:\ C\in\mathcal{C}(G)\big{\\}}=\operatorname{pd}(I(G)^{\vee}).$

###### Proof.

From [16, Proposition 8.1.10],
$\operatorname{pd}(I(G)^{\vee})=\operatorname{reg}(S/I(G))$. Thus the
assertion follows from [29, Lemma 3.4] and Corollary 3.6.

######  Remark 3.8.

The equality $a(G)=\operatorname{reg}(S/I(G))$ has been firstly proved in [29,
Lemma 3.4]. Here we give a new proof by Betti splittings. By [24, Lemma 2.2],
one always have $\operatorname{reg}(S/I(G))\geq a(G)$. It remains to prove
that $\operatorname{pd}(I(G)^{\vee})\leq a(G)$. Assume Setup 3.1.

By [12, Corollary 2.2] applied to the Betti splitting provided in Proposition
2.3, and by Lemma 3.3, we have

$\displaystyle\operatorname{pd}(I(G)^{\vee})\ $ $\displaystyle=\
\max\big{\\{}\operatorname{pd}(I(G_{1})^{\vee}),\
\operatorname{pd}(I(G_{2})^{\vee}),\
\operatorname{pd}(I(G_{3})^{\vee})+1\big{\\}}.$

By induction on $n=|V(G)|/2$, we may assume that
$\operatorname{pd}(I(G_{i})^{\vee})\leq a(G_{i})$ for $i=1,2,3$. Note that
$a(G_{i})\leq a(G)$, for $i=1,2,3$, clearly. Thus it remains to prove the
inequality $a(G_{3})+1\leq a(G)$. Let $D$ be a set of pairwise $3$–disjoint
edges of $G_{3}$ with $|D|=a(G_{3})$. Then $D\cup x_{n}y_{n}$ is again a set
of pairwise $3$–disjoint edges of $G$. Indeed, if $e=z_{k}z_{\ell}\in D$, then
$\\{z_{k},z_{\ell}\\}\subseteq V(G_{3})=V(G)\setminus\\{x_{i},y_{i}:x_{i}\in
N[x_{n}]\cup N[y_{n}]\\}$. Thus there can not be any edge connecting $z_{k}$
or $z_{\ell}$ with either $x_{n}$ or $y_{n}$. This shows that the induced
subgraph with vertex set $\\{z_{k},z_{\ell},x_{n},y_{n}\\}$ has only two
edges. Hence $a(G)\geq|D|+1=a(G_{3})+1$.

Let $M$ be a finitely generated $S$-module of dimension $d$, and let $P_{M}$
be the Hilbert polynomial of $M$ [4, 16]. Then,
$P_{M}(t)=\sum_{i=0}^{d-1}(-1)^{d-1-i}e_{d-1-i}\binom{t+i}{i}$,
$e_{d-1-i}\in{\mathbb{Q}}$, for all $i$. We define the multiplicity of $M$ as

$e(M)\ =\ \begin{cases}\hfil e_{0}&\text{if}\ d>0,\\\
\textup{length}(M)&\text{if}\ d=0.\end{cases}$

Now, we verify that $I(G)^{\vee}$ has the alternating sum property and thanks
to Corollary 3.6 we get a formula for the _multiplicity_ of $S/I(G)^{\vee}$,
$e(S/I(G)^{\vee})$.

We quote next definition from [35, Definition 4.1].

######  Definition 3.9.

Let $I$ be a monomial ideal in $S$ with $G(I)=\\{u_{1},\ldots,u_{m}\\}$ and
let $d=\min\\{\deg(u_{i}):i=1,\ldots,m\\}$. We say that $I$ has the
alternating sum property, if

$\sum_{i\geq 1}(-1)^{i}\beta_{i,i+j}(S/I)=\begin{cases}-1,&\mbox{for
$j=d$,}\\\ 0,&\mbox{for $j>d$.}\end{cases}$

######  Proposition 3.10.

Let $G$ be a Cohen–Macaulay very well–covered graph. Then

  1. (a)

$\sum_{i\geq 1}(-1)^{i}\beta_{i}(S/I(G)^{\vee})=-1$.

  2. (b)

$e(S/I(G)^{\vee})=|E(G)|=\frac{1}{2}\sum_{C\in\mathcal{C}(G)}\sum_{i=0}^{|\mathcal{C}(G;C)|}(-1)^{i+1}\binom{|\mathcal{C}(G;C)|}{i}(n+i)^{2}$.

###### Proof.

Let $|V(G)|=2n$.

(a). We proceed by induction on $n\geq 1$. For $n=1,2$ the only Cohen–Macaulay
very well–covered graphs are the following ones.

For each of these graphs, (a) holds. Let $n>2$. Using the same notation as in
Setup 3.1, from Proposition 2.3, Lemma 3.3 and by the inductive hypothesis on
the graphs $G_{1},G_{2},G_{3}$, we have that

$\displaystyle\sum_{i\geq 1}(-1)^{i}\beta_{i}(S/I(G)^{\vee})=\sum_{i\geq
0}(-1)^{i}\big{[}\beta_{i}(S/I(G_{1})^{\vee})+\beta_{i}(S/I(G_{2})^{\vee})+\beta_{i-1}(S/I(G_{3})^{\vee})\big{]}$
$\displaystyle=$ $\displaystyle\ \sum_{i\geq
0}(-1)^{i}\beta_{i}(S/I(G_{1})^{\vee})+\sum_{i\geq
0}(-1)^{i}\beta_{i}(S/I(G_{2})^{\vee})-\sum_{i\geq
0}(-1)^{i}\beta_{i}(S/I(G_{3})^{\vee})\ =$ $\displaystyle=$ $\displaystyle\ \
-1-1+1\ =\ -1.$

(b). Since $I(G)^{\vee}=\bigcap_{z_{i}z_{j}\in E(G)}(z_{i},z_{j})$ we see that
$I(G)^{\vee}$ has height two and, furthermore, $e(S/I(G)^{\vee})=|E(G)|$ by
[16, Corollary 6.2.3]. The first equality holds true.

On the other hand, by a formula of Peskine and Szpiro [31] (see also [16,
Corollary 6.1.7]), since $\beta_{i}(S/I(G)^{\vee})=\beta_{i-1}(I(G)^{\vee})$,
one has that

$\displaystyle e(S/I(G)^{\vee})$ $\displaystyle=$
$\displaystyle\frac{(-1)^{h}}{h!}\sum_{i=1}^{\operatorname{pd}(S/I(G)^{\vee})}(-1)^{i+1}\sum_{j=1}^{\beta_{i}(S/I(G)^{\vee})}a_{ij}^{h}=$
$\displaystyle=$
$\displaystyle\frac{(-1)^{h}}{h!}\sum_{i=0}^{\operatorname{pd}(I(G)^{\vee})}(-1)^{i+1}\sum_{j=1}^{\beta_{i}(I(G)^{\vee})}a_{ij}^{h},$

where $h=\textup{height}(I(G)^{\vee})$ and $a_{ij}$ are the shifts of the
$i$th free module of the minimal free resolution ${\mathbb{F}}$ of
$S/I(G)^{\vee}$, _i.e._ ,
$\bigoplus_{j=1}^{\beta_{i}(S/I(G)^{\vee})}S(-a_{ij})$. Since $I(G)^{\vee}$
has a $n$–linear resolution and $\textup{height}(I(G)^{\vee})=h=2$, from
Corollary 3.6, we have that

$\displaystyle e(S/I(G)^{\vee})\ $ $\displaystyle=\ \frac{1}{2}\sum_{i\geq
0}(-1)^{i+1}\sum_{C\in\mathcal{C}(G)}\binom{|\mathcal{C}(G;C)|}{i}(n+i)^{2}\
=$ $\displaystyle=\
\frac{1}{2}\sum_{C\in\mathcal{C}(G)}\sum_{i=0}^{|\mathcal{C}(G;C)|}(-1)^{i+1}\binom{|\mathcal{C}(G;C)|}{i}(n+i)^{2}.\quad\quad\square$

######  Example 3.11.

Let us consider the Cohen–Macaulay well–covered graph $G$ with $12$ vertices
in Example 2.4. Set $C_{1}=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\\}$,
$C_{2}=\\{y_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\\}$,
$C_{3}=\\{x_{1},y_{2},x_{3},x_{4},x_{5},x_{6}\\}$,
$C_{4}=\\{x_{1},x_{2},x_{3},x_{4},y_{5},x_{6}\\}$,
$C_{5}=\\{x_{1},x_{2},x_{3},x_{4},x_{5},y_{6}\\}$,
$C_{6}=\\{x_{1},x_{2},x_{3},x_{4},y_{5},y_{6}\\}$,
$C_{7}=\\{x_{1},x_{2},x_{3},y_{4},y_{5},y_{6}\\}$ and
$C_{8}=\\{x_{1},x_{2},y_{3},y_{4},y_{5},y_{6}\\}$. Then

$\displaystyle\begin{array}[]{ccc}\hline\cr\hline\cr\phantom{-}\mathcal{C}(G)\phantom{-}&\mathcal{C}(G;C)&|\mathcal{C}(G;C)|\\\
\hline\cr\hline\cr C_{1}&\varnothing&0\\\ C_{2}&\\{x_{1}\\}&1\\\
C_{3}&\\{x_{2}\\}&1\\\ C_{4}&\\{x_{5}\\}&1\\\ C_{5}&\\{x_{6}\\}&1\\\
C_{6}&\\{x_{5},x_{6}\\}&2\\\ C_{7}&\\{x_{4}\\}&1\\\ C_{8}&\\{x_{3}\\}&1\\\
\hline\cr\hline\cr\end{array}$

Using [15], one can verify that $\beta_{1}(S/I(G)^{\vee})=8$,
$\beta_{2}(S/I(G)^{\vee})=8$, $\beta_{3}(S/I(G)^{\vee})=1$. Hence,
$\sum_{i\geq 1}(-1)^{i}\beta_{i}(S/I(G)^{\vee})=-1$ and the alternating sum
property holds. Moreover,
$a(G)=\operatorname{reg}(S/I(G))=2=\max\big{\\{}\big{|}\mathcal{C}(G;C)\big{|}\
:\ C\in\mathcal{C}(G)\big{\\}}=\operatorname{pd}(I(G)^{\vee})$.

Finally, let us compute the multiplicity of $S/I(G)^{\vee}$. Firstly, we have
$e(S/I(G)^{\vee})=|E(G)|=20$. On the other hand, from the previous table, one
has

$\displaystyle\frac{1}{2}\sum_{C\in\mathcal{C}(G)}\sum_{i=0}^{|\mathcal{C}(G;C)|}(-1)^{i+1}\binom{|\mathcal{C}(G;C)|}{i}(6+i)^{2}=$
$\displaystyle=-\frac{1}{2}\binom{|\mathcal{C}(G;C_{1})|}{0}6^{2}+\frac{1}{2}\sum_{C\in\mathcal{C}(G)\setminus(C_{1}\cup
C_{6})}\sum_{i=0}^{|\mathcal{C}(G;C)|}(-1)^{i+1}\binom{|\mathcal{C}(G;C)|}{i}(6+i)^{2}+$
$\displaystyle+\frac{1}{2}\sum_{i=0}^{|\mathcal{C}(G;C_{6})|}(-1)^{i+1}\binom{|\mathcal{C}(G;C_{6})|}{i}(6+i)^{2}=$
$\displaystyle=\frac{1}{2}\Big{[}-\binom{0}{0}6^{2}+6\sum_{i=0}^{1}(-1)^{i+1}\binom{1}{i}(6+i)^{2}+\sum_{i=0}^{2}(-1)^{i+1}\binom{2}{i}(6+i)^{2}\Big{]}=$
$\displaystyle=\frac{1}{2}(-36+78-2)=20$

and Proposition 3.10 holds true.

##  4\. Homological shifts of vertex cover ideals of very well–covered graphs
and applications to Hibi ideals

In this section, we investigate the homological shift ideals of powers of the
vertex cover ideal of Cohen–Macaulay very well–covered graphs. Here is our
main result.

In what follows we refer to the notation in Setup 3.1.

######  Theorem 4.1.

Let $G$ be a Cohen–Macaulay very well–covered graph with $2n$ vertices. Then
$\textup{HS}_{k}(I(G)^{\vee})$ has linear quotients with respect to the
lexicographic order $>_{\operatorname{lex}}$ induced by
$x_{n}>y_{n}>x_{n-1}>y_{n-1}>\dots>x_{1}>y_{1}$, for all $k\geq 0$.

###### Proof.

We proceed by strong induction on $n=|V(G)|/2\geq 1$. For $n=1$,
$I(G)=(x_{1}y_{1})$ and $J=I(G)^{\vee}=(x_{1},y_{1})$. Hence,
$\textup{HS}_{0}(J)=J$, $\textup{HS}_{1}(J)=(x_{1}y_{1})$ have linear
quotients with respect to $>_{\operatorname{lex}}$.

Let $n>1$. By Proposition 2.3,

$I(G)^{\vee}\ =\ x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}\cdot
x_{n}I(G_{2})^{\vee}+x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\cdot
y_{n}I(G_{1})^{\vee}$ (6)

is a Betti splitting.

Set $f=x_{j_{1}}\cdots x_{j_{p}}\cdot x_{n}$ and $g=x_{i_{1}}\cdots
x_{i_{t}}\cdot y_{n}$. Since $\textup{HS}_{k}(wI)=w\cdot\textup{HS}_{k}(I)$
for all monomial ideals $I$ in $S$ and all non zero monomials $w\in S$, from
(6), Proposition 1.7 and Lemma 3.3, we have that

$\displaystyle\textup{HS}_{k}(I(G)^{\vee})\ =$ $\displaystyle\
fg\cdot\textup{HS}_{k-1}(I(G_{3})^{\vee})+f\cdot\textup{HS}_{k}(I(G_{2})^{\vee})+g\cdot\textup{HS}_{k}(I(G_{1})^{\vee}).$

Note that, since $|V(G_{j})|<|V(G)|$ ($j=1,2,3$), by the inductive hypothesis,
$\textup{HS}_{k}(I(G_{j})^{\vee})$ have linear quotients with respect to
$>_{\operatorname{lex}}$.

Let

$\displaystyle G(\textup{HS}_{k-1}(I(G_{3})^{\vee}))\ $ $\displaystyle=\
\big{\\{}w_{1}>_{\operatorname{lex}}w_{2}>_{\operatorname{lex}}\dots>_{\operatorname{lex}}w_{q}\big{\\}},$
(7) $\displaystyle G(\textup{HS}_{k}(I(G_{2})^{\vee}))\ $ $\displaystyle=\
\big{\\{}v_{1}\,>_{\operatorname{lex}}\,v_{2}>_{\operatorname{lex}}\dots>_{\operatorname{lex}}v_{r}\big{\\}},$
(8) $\displaystyle G(\textup{HS}_{k}(I(G_{1})^{\vee}))\ $ $\displaystyle=\
\big{\\{}u_{1}>_{\operatorname{lex}}u_{2}\\!>_{\operatorname{lex}}\dots>_{\operatorname{lex}}u_{s}\big{\\}},$
(9)

thus $G(\textup{HS}_{k}(I(G)^{\vee}))$ is ordered as follows:

$fgw_{1}>_{\operatorname{lex}}\dots>_{\operatorname{lex}}fgw_{q}\ \
>_{\operatorname{lex}}\ \
fv_{1}>_{\operatorname{lex}}\dots>_{\operatorname{lex}}fv_{r}\ \
>_{\operatorname{lex}}\ \
gu_{1}>_{\operatorname{lex}}\dots>_{\operatorname{lex}}gu_{s}.$

We prove that such an order is an admissible order of
$\textup{HS}_{k}(I(G)^{\vee})$. For our purpose, we need to show that all the
following three colon ideals

$\displaystyle(fgw_{1},\dots,fgw_{\ell-1}):fgw_{\ell},\,\,\,\,\ell\in\\{2,\dots,q\\},$
(10)
$\displaystyle\big{(}fg\textup{HS}_{k-1}(I(G_{3})^{\vee}),fv_{1},\dots,fv_{\ell-1}\big{)}:fv_{\ell},\,\,\,\,\ell\in\\{1,\dots,r\\},$
(11)
$\displaystyle\big{(}fg\textup{HS}_{k-1}(I(G_{3})^{\vee}),f\textup{HS}_{k}(I(G_{2})^{\vee}),gu_{1},\dots,gu_{\ell-1}\big{)}:gu_{\ell},\,\,\,\,\ell\in\\{1,\dots,s\\},$
(12)

are generated by variables.

First Colon Ideal. Let us consider the colon ideal in (10). Since,

$(fgw_{1},\dots,fgw_{\ell-1}):fgw_{\ell}=(w_{1},\dots,w_{\ell-1}):w_{\ell},$

the assertion follows from the fact that $\textup{HS}_{k-1}(I(G_{3})^{\vee})$
has linear quotients with respect to the order in (7).
Second Colon Ideal. Let us consider the colon ideal in (11).
Set
$P=\big{(}fg\textup{HS}_{k-1}(I(G_{3})^{\vee}),fv_{1},\dots,fv_{\ell-1}\big{)}:fv_{\ell}$.
One can observe that

$P=(fg\textup{HS}_{k-1}(I(G_{3})^{\vee})):fv_{\ell}+(v_{1},\dots,v_{\ell-1}):v_{\ell},$

with $(v_{1},\dots,v_{\ell-1}):v_{\ell}$ generated by variables. Indeed,
$\textup{HS}_{k}(I(G_{2})^{\vee})$ has linear quotients with respect to the
order in (8).
Thus, if we show that each generator of
$(fg\textup{HS}_{k-1}(I(G_{3})^{\vee})):fv_{\ell}$ is divided by a variable of
$P$, we conclude that $P$ is generated by variables, as wanted. The colon
ideal $(fg\textup{HS}_{k-1}(I(G_{3})^{\vee})):fv_{\ell}$ is generated by the
monomials

$\frac{\textup{lcm}(fgw_{j},fv_{\ell})}{fv_{\ell}},\ \ \ j=1,\dots,q.$

Fix $j\in\\{1,\dots,q\\}$. By Construction 3.4, we have that

$\displaystyle fgw_{j}\ $ $\displaystyle=\ w{\bf x}_{\sigma},$ $\displaystyle
w\in G(I(G)^{\vee}),\ y_{n}\in\operatorname{supp}(w),\
\sigma\subseteq\mathcal{C}(G;\operatorname{supp}(w)),\ x_{n}\in\sigma,\
|\sigma|=k,$ $\displaystyle fv_{\ell}\ $ $\displaystyle=\ v{\bf x}_{\tau},$
$\displaystyle v\in G(I(G)^{\vee}),\ x_{n}\in\operatorname{supp}(v),\
\tau\subseteq\mathcal{C}(G;\operatorname{supp}(v)),\ x_{n}\notin\tau,\
|\tau|=k.$

Let $h=\textup{lcm}(w{\bf x}_{\sigma},v{\bf x}_{\tau})/(v{\bf x}_{\tau})$. If
$\deg(h)=1$, $h$ is a variable and there is nothing to prove. Suppose
$\deg(h)>1$. Let us consider the following integer

$i\ =\ \min\Big{\\{}i\ :\ z_{i}\ \text{divides }h=\frac{\textup{lcm}(w{\bf
x}_{\sigma},v{\bf x}_{\tau})}{v{\bf x}_{\tau}},\
z_{i}\in\\{x_{i},y_{i}\\}\Big{\\}}.$ (13)

Since $\deg(h)>1$ and $y_{n}$ divides $h$, we have that $i<n$. From Lemma 2.2,
we have that $y_{i}$ divides at least one of the monomials $w{\bf
x}_{\sigma}$, $v{\bf x}_{\tau}$. Indeed, if $y_{i}$ does not divide any of
these monomials, then $x_{i}$ will divide both these monomials, and
consequentially, $z_{i}\in\\{x_{i},y_{i}\\}$ does not divide $h$. Against our
assumption.
Now, let $N[y_{i}]=\\{x_{k_{1}},\dots,x_{k_{b}},x_{i},y_{i}\\}$. We claim that
$x_{k_{1}}\cdots x_{k_{b}}$ divides both monomials $w$, $v$. Indeed, $x_{i}$
divides at least one of the monomials $w$, $v$. For instance, say $x_{i}$
divides $w{\bf x}_{\sigma}$, then either $x_{i}$ divides $w$ or
$x_{i}(w/y_{i})\in G(I(G)^{\vee})$. In both cases, $x_{k_{1}}\cdots x_{k_{b}}$
must divide $w$, since $w$ or $x_{i}(w/y_{i})$ are minimal vertex covers. The
same reasoning works if $x_{i}$ divides $v{\bf x}_{\tau}$. Thus
$x_{k_{1}}\cdots x_{k_{b}}$ must divide $w$ or $v$. Without loss of
generality, suppose that $x_{k_{1}}\cdots x_{k_{b}}$ divides $w$. Then, by
Construction 3.4, none of the variables $y_{k_{1}},\dots,y_{k_{b}}$ divides
$w$. By Characterization 1.12(iii), $k_{1}<\dots<k_{b}<i$. Thus, by the
meaning of $i$, $z_{d}\in\\{x_{d},y_{d}\\}$ does not divide $h$ for all
$d=k_{1},\dots,k_{b}$. Hence, we see that $x_{k_{1}}\cdots x_{k_{b}}$ divides
both the monomials $w$, $v$. Now, we distinguish two cases.
Case 1. Suppose $x_{i}y_{i}$ divides one between the monomials $w{\bf
x}_{\sigma}$ and $v{\bf x}_{\tau}$, and that $y_{i}$ divides the other one.
Suppose, for instance, that $x_{i}y_{i}$ divides $w{\bf x}_{\sigma}$ and that
$y_{i}$ divides $v{\bf x}_{\tau}$. Then, $x_{i}(v/y_{i}){\bf
x}_{\tau}>_{\operatorname{lex}}v{\bf x}_{\tau}$. Moreover, $x_{i}(v/y_{i})\in
G(I(G)^{\vee})$, because $x_{k_{1}}\cdots x_{k_{b}}$ divides $v$ and so
$\operatorname{supp}(x_{i}(v/y_{i}))$ is again a minimal vertex cover of $G$.
We claim that $x_{i}(v/y_{i}){\bf x}_{\tau}\in\textup{HS}_{k}(I(G)^{\vee})$.
Indeed for all $c\in\tau$,
$N[y_{c}]\setminus\\{x_{c},y_{c}\\}\subseteq\operatorname{supp}(v)$. Since
$N[y_{c}]\setminus\\{x_{c},y_{c}\\}$ is a subset of
$X=\\{x_{1},\dots,x_{n}\\}$ and
$\operatorname{supp}(x_{i}(v/y_{i}))=(\operatorname{supp}(v)\setminus
y_{i})\cup x_{i}$, we see that $N[y_{c}]\setminus\\{x_{c},y_{c}\\}$ is again a
subset of $\operatorname{supp}(x_{i}(v/y_{i}))$. Thus, for all $c\in\tau$,
$x_{c}x_{i}(v/y_{i})/y_{c}\in G(I(G)^{\vee})$, _i.e._ ,
$\tau\subseteq\mathcal{C}(G;\operatorname{supp}(x_{i}(v/y_{i})))$. Finally,
$x_{i}(v/y_{i}){\bf x}_{\tau}\in\textup{HS}_{k}(I(G)^{\vee})$. Moreover,

$\frac{\textup{lcm}(x_{i}(v/y_{i}){\bf x}_{\tau},v{\bf x}_{\tau})}{v{\bf
x}_{\tau}}=x_{i}\ \ \text{and}\ \ x_{i}\ \ \text{divides}\ \ h.$

It follows that $h$ is divided by the variable $x_{i}$ belonging to $P$, as
desired.
Otherwise, suppose $x_{i}y_{i}$ divides $v{\bf x}_{\tau}$ and $y_{i}$ divides
$w{\bf x}_{\sigma}$. Then, $x_{i}(w/y_{i}){\bf
x}_{\sigma}>_{\operatorname{lex}}w{\bf x}_{\sigma}$ and, as before,
$x_{i}(w/y_{i}){\bf x}_{\sigma}\in G(\textup{HS}_{k}(I(G)^{\vee}))$. Moreover,

$h^{\prime}=\frac{\textup{lcm}(x_{i}(w/y_{i}){\bf x}_{\sigma},v{\bf
x}_{\tau})}{v{\bf x}_{\tau}}=h/x_{i}\ \ \text{divides}\ \ h$

and so $h^{\prime}\in P$ and $\deg(h^{\prime})<\deg(h)$. Hence, we can iterate
the previous reasoning by considering the integer $i^{\prime}$ arising from
$h^{\prime}$, as in formula (13). In such a situation it is $i^{\prime}>i$.
Case 2. Suppose $x_{i}$ divides one of the monomials $w{\bf x}_{\sigma}$,
$v{\bf x}_{\tau}$ and $y_{i}$ divides the other one. Suppose $x_{i}$ divides
$w{\bf x}_{\sigma}$ and $y_{i}$ divides $v{\bf x}_{\tau}$. Then
$x_{i}(v/y_{i}){\bf x}_{\tau}>_{\operatorname{lex}}v{\bf x}_{\tau}$, and
arguing as before, one gets $x_{i}(v/y_{i}){\bf
x}_{\tau}\in\textup{HS}_{k}(I(G)^{\vee})$. Moreover,

$\frac{\textup{lcm}(x_{i}(v/y_{i}){\bf x}_{\tau},v{\bf x}_{\tau})}{v{\bf
x}_{\tau}}=x_{i}\ \ \text{and}\ \ x_{i}\ \ \text{divides}\ \ h.$

Therefore, $h$ is divided by the variable $x_{i}$ belonging to $P$, as
desired.
Otherwise, suppose $x_{i}$ divides $v{\bf x}_{\tau}$ and $y_{i}$ divides
$w{\bf x}_{\sigma}$. It follows that $x_{i}(w/y_{i}){\bf x}_{\sigma}\in
G(\textup{HS}_{k}(I(G)^{\vee}))$ and $x_{i}(w/y_{i}){\bf
x}_{\sigma}>_{\operatorname{lex}}w{\bf x}_{\sigma}$. Moreover,

$h^{\prime}=\frac{\textup{lcm}(x_{i}(w/y_{i}){\bf x}_{\sigma},v{\bf
x}_{\tau})}{v{\bf x}_{\tau}}=h/x_{i}\ \ \text{divides}\ \ h.$

Thus $h^{\prime}\in P$ and $\deg(h^{\prime})<\deg(h)$. In such a case, we
iterate the reasoning made above by considering the integer $i^{\prime}$
arising from $h^{\prime}$ as in (13), and this time $i^{\prime}>i$. It is
clear that iterating the reasoning above we get that $P$ is generated by
variables.
Third Colon Ideal. Set
$P\\!=\\!\big{(}fg\textup{HS}_{k-1}(I(G_{3})^{\vee}),f\textup{HS}_{k}(I(G_{2})^{\vee}),gu_{1},\dots,gu_{\ell-1}\big{)}\\!:\\!gu_{\ell}$.
One can observe that $P$ is generated by the monomials

$\frac{\textup{lcm}(fgw_{i},gu_{\ell})}{gu_{\ell}},\ \
\frac{\textup{lcm}(fv_{j},gu_{\ell})}{gu_{\ell}},\ \
\frac{\textup{lcm}(gu_{h},gu_{\ell})}{gu_{\ell}},\ \ $

with $i\in[q]$, $j\in[r]$ and $h\in[\ell-1]$. To prove that $P$ is generated
by variables it is enough to show that any of the monomials above is divided
by a variable $z_{b}\in P$. Note that $\textup{HS}_{k}(I(G_{1})^{\vee})$ has
linear quotients with respect to the order in (9), thus the colon ideal
$(gu_{1},\dots,gu_{\ell-1}):gu_{\ell}=(u_{1},\dots,u_{\ell-1}):u_{\ell}$ is
generated by variables. Thus,
$\textup{lcm}(gu_{h},gu_{\ell})/(gu_{\ell})\in(u_{1},\dots,u_{\ell-1}):u_{\ell}$
is divided by a variable of $P$. For the other two type of monomial generators
it suffices to repeat the same argument as in the Second Colon Ideal case.

As an immediate consequence we have the following classification.

######  Theorem 4.2.

Let $G$ be a very well–covered graph with $2n$ vertices. Then, the following
conditions are equivalent:

  1. (i)

$G$ is Cohen–Macaulay.

  2. (ii)

$I(G)^{\vee}$ has homological linear quotients.

  3. (iii)

$I(G)^{\vee}$ has homological linear resolution.

Let $G$ be a finite simple graph with vertex set
$V(G)=X=\\{x_{1},\dots,x_{n}\\}$. The whisker graph $G^{*}$ of $G$ is the
graph with vertex set
$V(G^{*})=\\{x_{1},\dots,x_{n}\\}\cup\\{y_{1},\dots,y_{n}\\}$ and edge set
$E(G^{*})=E(G)\cup\\{x_{i}y_{i}:i=1,\dots,n\\}$. In other words, the whisker
graph of $G$ is obtained by adding to each vertex $x_{i}$ a whisker, that
means that for each vertex $x_{i}\in V(G)$ we add a new vertex $y_{i}$ and an
edge $x_{i}y_{i}$ connecting these two vertices.

######  Corollary 4.3.

Let $G$ be any simple graph with $n$ vertices. Then the vertex cover ideal of
the whisker graph $G^{*}$ of $G$ has homological linear quotients.

###### Proof.

It is easy to see that $G^{*}$ satisfies the conditions (i)–(v) of
Characterization 1.12. Thus $G^{*}$ is a Cohen–Macaulay very well–covered
graph and the result follows from the previous theorem.

Our experiments and the results above, suggest the following conjecture.

######  Conjecture 4.4.

Let $G$ be a Cohen–Macaulay very well–covered graph with 2n vertices. Then
$\textup{HS}_{k}((I(G)^{\vee})^{\ell})$ has linear quotients with respect to
the lexicographic order induced by
$x_{n}>y_{n}>x_{n-1}>y_{n-1}>\dots>x_{1}>y_{1}$, for all $k\geq 0$, and all
$\ell\geq 1$.

Note that our conjecture would also imply that $I(G)^{\vee}$ has linear
powers.

At present it seems too difficult to prove our conjecture in full generality.
Therefore we concentrate our attention on the subclass of Cohen–Macaulay
bipartite graphs.

For this purpose, we need to recall what an Hibi ideal is [22].

Let $(P,\succeq)$ be a finite partially ordered set (a poset, for short) and
set $P=\\{p_{1},\ldots,p_{n}\\}$. A poset ideal of $P$ is a subset
$\mathcal{I}$ of $P$ such that if $p_{i}\in P$, $p_{j}\in\mathcal{I}$ and
$p_{i}\preceq p_{j}$, then $p_{i}\in\mathcal{I}$ [17]. To any poset ideal
$\mathcal{I}$ of $P$, we associate the squarefree monomial
$u_{\mathcal{I}}=(\prod_{p_{i}\in\mathcal{I}}x_{i})(\prod_{p_{i}\in
P\setminus\mathcal{I}}y_{i})$ in the polynomial ring
$S=K[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$. Then the Hibi ideal (associated
to $P$) is the monomial ideal of $S$ defined as follows:

$H_{P}\ =\ \big{(}u_{\mathcal{I}}\ :\ \mathcal{I}\ \text{is a poset ideal of}\
P\big{)}.$

As an immediate consequence of Theorem 4.1 we have the next result.

######  Corollary 4.5.

Let $(P,\succeq)$ be a finite poset. Then $H_{P}$ has homological linear
quotients.

###### Proof.

By [16, Lemma 9.1.11], the Alexander dual $H_{P}^{\vee}$ may be seen as the
edge ideal of a Cohen–Macaulay bipartite graph $G_{P}$. In particular $G_{P}$
is a Cohen–Macaulay very well–covered graph. Hence, seeing $H_{P}$ as the
cover ideal of $G_{P}$, the result follows immediately from Theorem 4.1.

Now, we turn to the powers of an Hibi ideal. We denote by $\mathcal{J}(P)$ the
distributive lattice consisting of all poset ideals of $(P,\succeq)$ ordered
by inclusion.

From now on, with abuse of language but with the aim of simplifying the
notation, we identify each $p_{i}\in P$ with the variable $x_{i}$, $i\in[n]$.

######  Construction 4.6.

Let $(P,\succeq)$ be a finite poset. For any integer $\ell\geq 1$, we
construct a new poset $(P(\ell),\succeq_{\ell})$ defined as follows:

  1. -

$P(\ell)\ =\ \big{\\{}x_{i,1},x_{i,2},\dots,x_{i,\ell}\ :\
i=1,\dots,n\big{\\}}$,

  2. -

and $x_{i,r}\succeq_{\ell}x_{j,s}$ if and only if $x_{i}\succeq x_{j}$ and
$r\geq s$.

By [22, pag 99] we have the following useful property.

######  Lemma 4.7.

Let $(P,\succeq)$ be a finite poset. Each minimal monomial generator of
$H_{P}^{\ell}$ posses a unique expression of the form

$u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}},\ \ \
\text{with}\ \ \
\mathcal{I}_{1}\subseteq\mathcal{I}_{2}\subseteq\dots\subseteq\mathcal{I}_{\ell},\
\ \ \mathcal{I}_{i}\in\mathcal{J}(P),\ \ \ i=1,\dots,\ell.$

Now, we need the technique of polarization discussed in Subsection 1.2. Let
$(P,\succeq)$ be a finite poset and $\ell\geq 1$ be a positive integer. Note
that $X$ and $\varnothing$ are both poset ideals of $P$, thus both
$u_{X}=x_{1}x_{2}\cdots x_{n}$ and $u_{\varnothing}=y_{1}y_{2}\cdots y_{n}$
belong to $H_{P}$. Thus, we have that
$u_{X}^{\ell}=x_{1}^{\ell}x_{2}^{\ell}\cdots x_{n}^{\ell}$ and
$u_{\varnothing}^{\ell}=y_{1}^{\ell}y_{2}^{\ell}\cdots y_{n}^{\ell}$ belong to
$H_{P}^{\ell}$. Therefore the polynomial ring in which $H_{P}^{\ell}$ lives is

$R=K[x_{i,j},y_{i,j}\ :\ i=1,\dots,n,\ j=1,\dots,\ell].$

In order to preserve the structure of Hibi ideals, we innocuously _modify_
polarization. More precisely, let $1\leq k\leq\ell$ and $i\in[n]$, then we set

$\displaystyle(x_{i}^{k})^{\wp}\ $ $\displaystyle=\ x_{i,1}x_{i,2}\cdots
x_{i,\ell},$ $\displaystyle(y_{i}^{k})^{\wp}\ $ $\displaystyle=\
y_{i,\ell}y_{i,\ell-1}\cdots y_{i,\ell+1-k},$

and extend the polarization of an arbitrary monomial in the obvious way. In
other words, with respect to the usual polarization, we are just applying the
relabeling of the variables $y_{i,j}\mapsto y_{i,\ell+1-j}$ for $i=1,\dots,n$
and $j=1,\dots,\ell$.

######  Example 4.8.

Consider the poset $(P,\succeq)$ with $P=\\{x_{1},x_{2},x_{3}\\}$, $x_{3}\succ
x_{1}$ and $x_{3}\succ x_{2}$. The poset $(P,\succeq)$ and the distributive
lattice $\mathcal{J}(P)$ are depicted below:

$x_{1}$$x_{3}$$x_{2}$$\\{x_{1}\\}$$\\{x_{1},x_{2}\\}$$\\{x_{1},x_{2},x_{3}\\}$$\\{x_{2}\\}$$\varnothing$$(P,\succeq)$$\mathcal{J}(P)$

The poset $(P(2),\succeq_{2})$ and the distributive lattice
$\mathcal{J}(P(2))$ are the following ones:

$x_{1,1}$$x_{2,1}$$x_{1,2}$$x_{2,2}$$x_{3,1}$$x_{3,2}$$(P(2),\succeq_{2})$$\\{x_{1,1},x_{1,2},x_{2,1},x_{2,2},x_{3,1},x_{3,2}\\}$$\\{x_{1,1},x_{1,2},x_{2,1},x_{2,2},x_{3,1}\\}$$\\{x_{1,1},x_{1,2},x_{2,1},x_{3,1}\\}$$\\{x_{1,1},x_{2,1},x_{2,2},x_{3,1}\\}$$\\{x_{1,1},x_{1,2},x_{2,1},x_{2,2}\\}$$\\{x_{1,1},x_{1,2},x_{2,1}\\}$$\\{x_{1,1},x_{2,1},x_{2,2}\\}$$\\{x_{1,1},x_{2,1},x_{3,1}\\}$$\\{x_{1,1},x_{1,2}\\}$$\\{x_{1,1},x_{2,1}\\}$$\\{x_{2,1},x_{2,2}\\}$$\\{x_{1,1}\\}$$\\{x_{2,1}\\}$$\varnothing$$\mathcal{J}(P(2))$

We have that

$\displaystyle H_{P}\ $ $\displaystyle=\ (x_{1}x_{2}x_{3},\ x_{1}x_{2}y_{3},\
x_{1}y_{2}y_{3},\ y_{1}x_{2}y_{3},\ y_{1}y_{2}y_{3}),$ $\displaystyle
H_{P(2)}\ $ $\displaystyle=\
\big{(}x_{1,1}x_{1,2}x_{2,1}x_{2,2}x_{3,1}x_{3,2},\
x_{1,1}x_{1,2}x_{2,1}x_{2,2}x_{3,1}y_{3,2},\
x_{1,1}x_{1,2}x_{2,1}x_{2,2}y_{3,1}y_{3,2},$ $\displaystyle\phantom{=\
\big{(}.}x_{1,1}x_{1,2}x_{2,1}y_{2,2}x_{3,1}y_{3,2},\
x_{1,1}y_{1,2}x_{2,1}y_{2,2}x_{3,1}y_{3,2},\
x_{1,1}y_{1,2}x_{2,1}x_{2,2}x_{3,1}y_{3,2},$ $\displaystyle\phantom{=\
\big{(}.}x_{1,1}x_{1,2}x_{2,1}y_{2,2}y_{3,1}y_{3,2},\
x_{1,1}y_{1,2}x_{2,1}y_{2,2}y_{3,1}y_{3,2},\
x_{1,1}y_{1,2}x_{2,1}x_{2,2}y_{3,1}y_{3,2},$ $\displaystyle\phantom{=\
\big{(}.}x_{1,1}x_{1,2}y_{2,1}y_{2,2}y_{3,1}y_{3,2},\
y_{1,1}y_{1,2}x_{2,1}x_{2,2}y_{3,1}y_{3,2},\
x_{1,1}y_{1,2}y_{2,1}y_{2,2}y_{3,1}y_{3,2},$ $\displaystyle\phantom{=\
\big{(}.}y_{1,1}y_{1,2}x_{2,1}y_{2,2}y_{3,1}y_{3,2},\
y_{1,1}y_{1,2}y_{2,1}y_{2,2}y_{3,1}y_{3,2}\big{)}.$

One can easily verify that $(H_{P}^{2})^{\wp}=H_{P(2)}$ with respect to our
modified polarization. For instance, consider
$(x_{1}y_{2}y_{3})(y_{1}x_{2}y_{3})\in G(H_{P}^{2})$. Then

$\big{(}(x_{1}y_{2}y_{3})(y_{1}x_{2}y_{3})\big{)}^{\wp}\ =\
(x_{1}y_{1}x_{2}y_{2}y_{3}^{2})^{\wp}\ =\
x_{1,1}y_{1,2}x_{2,1}y_{2,2}y_{3,1}y_{3,2}\ =\ u_{\mathcal{I}}\in
G(H_{P(2)}),$

where $\mathcal{I}=\\{x_{1,1},x_{2,1}\\}\in\mathcal{J}(P(2))$.

######  Theorem 4.9.

Let $(P,\succeq)$ be a finite poset. Then, for any $\ell\geq 1$,

$(H_{P}^{\ell})^{\wp}\ =\ H_{P(\ell)}.$

###### Proof.

By Lemma 4.7, $H_{P}^{\ell}$ is minimally generated by the monomials

$u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}},\ \ \
\text{with}\ \ \
\mathcal{I}_{1}\subseteq\mathcal{I}_{2}\subseteq\dots\subseteq\mathcal{I}_{\ell},\
\ \ \mathcal{I}_{j}\in\mathcal{J}(P),\ \ \ j=1,\dots,\ell,$

while $H_{P(\ell)}\subset K\big{[}x_{i,j},y_{i,j}:i\in[n],j\in[\ell]\big{]}$
is generated by the squarefree monomials

$u_{\mathcal{I}}\ =\
\big{(}\prod_{x_{i,j}\in\mathcal{I}}x_{i,j}\big{)}\big{(}\prod_{x_{i,j}\in
P(\ell)\setminus\mathcal{I}}y_{i,j}\big{)},\ \ \ \ \
\mathcal{I}\in\mathcal{J}(P(\ell)).$

Hence, to get the assertion, we must show that for all
$u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}}\in
G(H_{P}^{\ell})$, the monomial $(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots
u_{\mathcal{I}_{\ell}})^{\wp}$ is equal to $u_{\mathcal{I}}$, for some poset
ideal $\mathcal{I}\in\mathcal{J}(P(\ell))$, and conversely, given any
$u_{\mathcal{I}}\in G(H_{P(\ell)})$, with
$\mathcal{I}\in\mathcal{J}(P(\ell))$, then there exist
$\mathcal{I}_{1}\subseteq\mathcal{I}_{2}\subseteq\dots\subseteq\mathcal{I}_{\ell}$,
$\mathcal{I}_{i}\in\mathcal{J}(P)$, $i=1,\dots,\ell$, such that
$u_{\mathcal{I}}=(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots
u_{\mathcal{I}_{\ell}})^{\wp}$.

Let $u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}}\in
G(H_{P}^{\ell})$, with
$\mathcal{I}_{1}\subseteq\mathcal{I}_{2}\subseteq\dots\subseteq\mathcal{I}_{\ell}$.
Set $k_{i}=\big{|}\\{r\ :\ x_{i}\in\mathcal{I}_{r}\\}\big{|}$ and
$m_{i}=\big{|}\\{s\ :\ x_{i}\notin\mathcal{I}_{s}\\}\big{|}$. Therefore,
$k_{i}+m_{i}=\ell$ and

$(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}})^{\wp}\
=\ \big{(}\prod_{x_{i}\in
P}\prod_{r=1}^{k_{i}}x_{i,r}\big{)}\big{(}\prod_{x_{i}\in
P}\prod_{s=k_{i}+1}^{\ell}y_{i,s}\big{)}.$

We claim that $\mathcal{I}=\big{\\{}x_{i,r}\ :\ r=1,\dots,k_{i}\big{\\}}$ is a
poset ideal of $P(\ell)$. From this, it will follow that
$(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots
u_{\mathcal{I}_{\ell}})^{\wp}=u_{\mathcal{I}}\in G(H_{P(\ell)})$, as wanted.
Indeed, let $x_{i,r}\in\mathcal{I}$ and $x_{j,s}\in P(\ell)$ such that
$x_{j,s}\preceq_{\ell}x_{i,r}$. We must prove that $x_{j,s}\in\mathcal{I}$. By
definition of the order $\succeq_{\ell}$ we have $x_{j}\preceq x_{i}$ and
$r\geq s$. If $j=i$, then also $x_{i,s}\in\mathcal{I}$, by the polarization
technique. Let $j\neq i$. We note that $k_{j}\geq k_{i}$. Indeed, for any $c$
such that $x_{i}\in\mathcal{I}_{c}$, one has that $x_{j}\in\mathcal{I}_{c}$,
since $x_{j}\preceq x_{i}$. Thus, if $x_{i,r}$ divides
$(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}})^{\wp}$,
then $x_{j,r}$ divides $(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots
u_{\mathcal{I}_{\ell}})^{\wp}$. For any $d\leq r$, $x_{j,d}$ divides
$(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}})^{\wp}$,
by the polarization technique. Since $s\leq r$, then $x_{j,s}$ divides
$(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots u_{\mathcal{I}_{\ell}})^{\wp}$,
as wanted.

Conversely, let $\mathcal{I}\in\mathcal{J}(P(\ell))$ be a poset ideal and
$u_{\mathcal{I}}\in G(H_{P(\ell)})$. Set

$\mathcal{I}_{k}=\big{\\{}x_{i}\ :\ x_{i,\ell+1-k}\in\mathcal{I}\big{\\}},\ \
\ \text{for}\ \ \ k=1,\dots,\ell.$

We claim that the sets $\mathcal{I}_{j}$ are poset ideals of $P$ and that
$\mathcal{I}_{1}\subseteq\mathcal{I}_{2}\subseteq\dots\subseteq\mathcal{I}_{\ell}$.
From this, it will follow that
$u_{\mathcal{I}}=(u_{\mathcal{I}_{1}}u_{\mathcal{I}_{2}}\cdots
u_{\mathcal{I}_{\ell}})^{\wp}\in G((H_{P}^{\ell})^{\wp})$, as wanted.

Fix $k\in[\ell]$ and let $x_{i}\in\mathcal{I}_{k}$ and $x_{j}\preceq x_{i}$.
We must prove that $x_{j}\in\mathcal{I}_{k}$, too. Since $x_{j}\preceq x_{i}$,
then $x_{j,\ell+1-k}\preceq_{\ell}x_{i,\ell+1-k}$ by definition of the order
$\succeq_{\ell}$. Thus $x_{j,\ell+1-k}\in\mathcal{I}$ and consequently
$x_{j}\in\mathcal{I}_{k}$, as wanted.

Now, let $k\in[\ell]$ with $k<\ell$. We prove that
$\mathcal{I}_{k}\subseteq\mathcal{I}_{k+1}$. As a consequence, it follows that
$\mathcal{I}_{1}\subseteq\mathcal{I}_{2}\subseteq\dots\subseteq\mathcal{I}_{\ell}$,
as desired. Let $x_{i}\in\mathcal{I}_{k}$, then
$x_{i,\ell+1-k}\in\mathcal{I}$. Since $\ell+1-k>\ell+1-(k+1)$, we get that
$x_{i,\ell+1-k}\succeq_{\ell}x_{i,\ell+1-(k+1)}$ and so
$x_{i,\ell+1-(k+1)}\in\mathcal{I}$, too. Thus $x_{i}\in\mathcal{I}_{k+1}$. The
assertion follows.

The following key lemma proved by Olteanu [30, Proposition 5.3] follows by a
more general result stated by Jahan [23, Lemma 3.3].

######  Lemma 4.10.

Let $I\subset S$ be a monomial ideal. Then $I$ has linear quotients with
admissible order $u_{1}>u_{2}>\dots>u_{m}$ of $G(I)$ if and only if $I^{\wp}$
has linear quotients with admissible order
$u_{1}^{\wp}>u_{2}^{\wp}>\dots>u_{m}^{\wp}$ of $G(I^{\wp})$.

The previous lemma obviously also holds with respect to our modified
polarization. Finally, we obtain the next result which gives a positive answer
to Conjecture 4.4 for the class of bipartite graphs.

######  Corollary 4.11.

Let $G$ be a Cohen–Macaulay bipartite graph with $2n$ vertices. Then
$\textup{HS}_{k}((I(G)^{\vee})^{\ell})$ has linear quotients with respect to
the lexicographic order induced by
$x_{n}>y_{n}>x_{n-1}>y_{n-1}>\dots>x_{1}>y_{1}$, for all $k\geq 0$ and all
$\ell\geq 1$.

###### Proof.

By [16, Theorem 9.1.3], there exists a poset $(P,\succeq)$ such that
$I(G)^{\vee}=H_{P}$. Let $\ell\geq 1$. Then by Theorem 4.9,
$(H_{P}^{\ell})^{\wp}=H_{P(\ell)}$. By Theorem 4.1, for all $k\geq 0$,
$\textup{HS}_{k}(H_{P(\ell)})$ has linear quotients with respect to the
lexicographic order induced by

$x_{n,\ell}>y_{n,\ell}>x_{n,\ell-1}>y_{n,\ell-1}>\dots>x_{n,1}>y_{n,1}>x_{n-1,\ell}>y_{n-1,\ell}>\dots>x_{1,1}>y_{1,1}.$

By Lemma 1.8,
$\textup{HS}_{k}(H_{P(\ell)})=\textup{HS}_{k}((H_{P}^{\ell})^{\wp})=\textup{HS}_{k}(H_{P}^{\ell})^{\wp}$.
Finally, applying Lemma 4.10, we obtain that
$\textup{HS}_{k}(H_{P}^{\ell})=\textup{HS}_{k}((I(G)^{\vee})^{\ell})$ has
linear quotients with respect to the lexicographic order induced by
$x_{n}>y_{n}>x_{n-1}>y_{n-1}>\dots>x_{1}>y_{1}$, as wanted.

Acknowledgment. We thank the referees for their helpful suggestions, that
allowed us to improve the quality of the paper.

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