# Almost Cohen-Macaulay bipartite graphs and connected in codimension two

Amir Mafi and Dler Naderi  Amir Mafi, Department of Mathematics, University Of
Kurdistan, P.O. Box: 416, Sanandaj, Iran<EMAIL_ADDRESS>Dler Naderi,
Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj,
Iran<EMAIL_ADDRESS>
###### Abstract.

In this paper we study almost Cohen-Macaulay bipartite graphs. Furthermore, we
prove that if $G$ is almost Cohen-Macaulay bipartite graph with at least one
vertex of positive degree, then there is a vertex of $\deg(v)\leq 2$. In
particular, if $G$ is an almost Cohen-Macaulay bipartite graph and $u$ is a
vertex of degree one of $G$ and $v$ its adjacent vertex, then
$G\setminus\\{v\\}$ is almost Cohen-Macaulay. Also, we show that an unmixed
Ferrers graph is almost Cohen-Macaulay if and only if it is connected in
codimension two. Moreover, we give some examples.

###### Key words and phrases:

Almost Cohen-Macaulay rings, Simplicial complex, Connectedness

###### 1991 Mathematics Subject Classification:

13C14, 13C05

## Introduction

Throughout this paper, we assume that $G$ is a finite simple graph (without
loops, multiplies edges and any isolated vertices) with vertex set $V(G)$ and
edge set $E(G)$. For $W\subseteq V(G)$ we denote by $G\setminus W$ the
subgraph of $G$ obtained by removing all vertices of $W$ from $G$. Moreover,
for any $v\in V(G)$ we denote by $N_{G}(v)$ the neighbor set of $v$ in $G$,
i.e. $N_{G}(v)=\\{u\in V(G)~{}|~{}\\{u,v\\}\in E(G)\\}$. The inclusive
neighborhood of $v\in V(G)$ is the set $N_{G}[v]$ consisting of $v$ and
vertices adjacent to $v$ in $G$, i.e. $N_{G}[v]=N_{G}(v)\cup\\{v\\}$.

Let $R=k[x_{1},\ldots,x_{n}]$ be the polynomial ring on $n$ variables over the
field $k$. We can associate to $G$ the ideal $I(G)$ of $R$ which is generated
by all the square-free monomials $x_{i}x_{j}$ such that $x_{i}$ is adjacent to
$x_{j}$. The ideal $I(G)$ is called the edge ideal of $G$. The complementary
simplicial complex of $G$ is defined by $\Delta_{G}=\\{F\subseteq
V(G)~{}|~{}F~{}\textit{is an independent set in}~{}G\\}$, where $F$ is an
independent set in $G$ if none of its elements are adjacent. Note that
$\Delta_{G}$ is precisely the Stanley-Reisner simplicial complex of $I(G)$,
i.e. $I_{\Delta_{G}}=I(G)$.

The graph $G$ is called Cohen-Macaulay (i.e. CM) if $R/I(G)$ is Cohen-
Macaulay. Cohen-Macaulay graphs were studied in several works (see [23] and
[5]). A complete classification of Cohen-Macaulay graphs does not exist.
However, all Cohen-Macaulay bipartite graphs have been characterized in a
combinatorial way by Herzog and Hibi in [12]. A graph $G$ is called bipartite,
if $V(G)=V_{1}\cup V_{2}$ with $V_{1}\cap V_{2}=\emptyset$ such that
$E(G)\subseteq V_{1}\times V_{2}$. It is easy to see that a graph $G$ is
bipartite if and only if it has no cycle of odd length. For a Cohen-Macaulay
bipartite graph $G$, Estrada and Villarreal [6] showed that
$G\setminus\\{u\\}$ is Cohen-Macaulay for some vertex $u\in V(G)$. As usual
$K_{m,n}$ will denote the complete bipartite graph containing every edge
joining $V_{1}$ and $V_{2}$, where $V_{1}$ and $V_{2}$ have $m$ and $n$
vertices respectively and it is easy to see that $K_{m,n}$ is Cohen-Macaulay
if and only if $m=n=1$.

A vertex cover of $G$ is a subset $C$ of $V(G)$ such that each edge has at
least one vertex in $C$. A minimal vertex cover $C$ of $G$ is a vertex cover
such that no proper subset of $C$ is a vertex cover of $G$. Observe that $C$
is a minimal vertex cover if and only if $V(G)\setminus C$ is a maximal
independent set. The graph $G$ is called unmixed if all its minimal vertex
covers are of the same cardinality. All unmixed bipartite graphs have been
characterized by Villarreal in [25].

Let $G$ be an unmixed bipartite graph with bipartition
$V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and $V_{2}=\\{y_{1},\ldots,y_{m}\\}$. Since
$G$ is unmixed, it follows that $\operatorname{ht}(I)=n=m$. The edges
$\\{x_{i},y_{i}\\}$ for $i=1,\ldots,n$ are called perfect matching edges of
$G$. By [10, Theorem 10.2], the height of $I(G)$ is equal to the maximum
number of independent lines in $G$, i.e. for any unmixed bipartite graph there
is a perfect matching. Therefore we may assume that $\\{x_{i},y_{i}\\}$ is an
edge of $G$ for all $i$. So each minimal vertex cover of $G$ is of the form
$\\{x_{i_{1}},\ldots,x_{i_{s}},y_{i_{s+1}},\ldots,y_{i_{n}}\\}$, where
$\\{i_{1},\ldots,i_{n}\\}=[n]$.

The graph $G$ is called almost Cohen-Macaulay (i.e. aCM) if $R/I(G)$ is aCM.
We say that $R/I(G)$ is aCM when $\operatorname{depth}R/I(G)\geq\dim
R/I(G)-1$. The aCM modules has been studied in [9, 15, 16, 14, 3, 19, 20, 21,
18].

In this paper we study almost Cohen-Macaulay bipartite graphs. Furthermore, we
prove that if $G$ is almost Cohen-Macaulay bipartite graph with at least one
vertex of positive degree, then there is a vertex of $\deg(v)\leq 2$. In
particular, if $G$ is an almost Cohen-Macaulay bipartite graph and $u$ is a
vertex of degree one of $G$ and $v$ its adjacent vertex, then
$G\setminus\\{v\\}$ is almost Cohen-Macaulay. Also, we show that an unmixed
Ferrers graph is almost Cohen-Macaulay if and only if it is connected in
codimension two. For any unexplained notion or terminology, we refer the
reader to [13] and [24] . Several explicit examples were performed with help
of the computer algebra systems Macaulay2 [7].

##  1\.  Preliminary

In this section, we recall some definitions and known results which is used in
this paper. Let $\Delta$ be a simplicial complex on the vertex set
$V=\\{x_{1},...,x_{n}\\}$. Every element of $\Delta$ is called a face of
$\Delta$ and a facet of $\Delta$ is a maximal face of $\Delta$ with respect to
inclusion. If all facets of $\Delta$ have the same cardinality, then $\Delta$
is called pure. For the simplicial complex $\Delta$, we may consider a square-
free monomial ideal $I=I_{\Delta}$ of $R$ which is generated by all minimal
nonface of $\Delta$ is called the Stanley-Reisner ideal of $\Delta$ and
$K[\Delta]=R/I_{\Delta}$ is called the Stanley-Reisner ring. For the
simplicial complex $\Delta$ and $F\in\Delta$, link of $F$ in $\Delta$ is
defined as $\operatorname{lk}_{\Delta}(F)=\\{G\in\Delta~{}|~{}G\cap
F=\emptyset,G\cup F\in\Delta\\}$. If $\Delta$ is a simplicial complex with
facets $F_{1},\ldots,F_{t}$, we denote $\Delta$ by $\langle
F_{1},\ldots,F_{t}\rangle$, and $\\{F_{1},\ldots,F_{t}\\}$ is called the facet
set of $\Delta$.

######  Proposition 1.1.

Let $\Delta_{1}$ and $\Delta_{2}$ be two simplicial complexes on $[n]$, and
let $\Delta=\Delta_{1}\cup\Delta_{2}$ and $\Gamma=\Delta_{1}\cap\Delta_{2}$.
Then there exists an exact sequence of the following form

$\displaystyle\ldots$
$\displaystyle\longrightarrow\tilde{H}_{l}(\Gamma;k)\longrightarrow\tilde{H}_{l}(\Delta_{1};k)\oplus\tilde{H}_{l}(\Delta_{2};k)\longrightarrow\tilde{H}_{l}(\Delta;k)$
$\displaystyle\longrightarrow\tilde{H}_{l-1}(\Gamma;k)\longrightarrow\tilde{H}_{l-1}(\Delta_{1};k)\oplus\tilde{H}_{l-1}(\Delta_{2};k)\longrightarrow\tilde{H}_{l-1}(\Delta;k)$
$\displaystyle\longrightarrow\ldots$

This sequence is called the reduced Mayer-Vietoris exact sequence ([13,
Proposition 5.1.8]).

######  Lemma 1.2.

([22, Lemma 2.5]) Let $x$ be a vertex of $G$ and $G^{{}^{\prime}}=G\setminus
N_{G}[x]$. Then $\Delta_{G^{{}^{\prime}}}=lk_{\Delta_{G}}\\{x\\}$. In
particular, $F$ is a facet of $\Delta_{G^{{}^{\prime}}}$ if and only if
$x\notin F$ and $F\cup\\{x\\}$ is a facet of $\Delta_{G}$.

The family of all unmixed bipartite graphs has been characterized in a
combinatorial way by Villarreal in the following result.

######  Theorem 1.3.

([25, Theorem 1.1]) Let $G$ be a bipartite graph without an isolated vertex.
Then $G$ is unmixed if and only if there is a partition
$V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and $V_{2}=\\{y_{1},\ldots,y_{n}\\}$ of
vertices of $G$ such that

  1. (1)

$\\{x_{i},y_{i}\\}\in E(G)$ for $1\leq i\leq n$ and

  2. (2)

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

In this case, such a partition and ordering is called a pure order of $G$. As
stated before, if $G$ is an unmixed bipartite graph on the vertex set
$V(G)=\\{x_{1},\ldots,x_{n}\\}\cup\\{y_{1},\ldots,y_{n}\\}$, then each of its
minimal vertex covers has the form
$\\{x_{i_{1}},\ldots,x_{i_{s}},y_{i_{s+1}},\ldots,y_{i_{n}}\\}$, where
$\\{i_{1},\ldots,i_{n}\\}=[n]$.

##  2\. Almost Cohen-Macaulay Bipartite graphs

We start this section by the following lemma.

######  Lemma 2.1.

Let Let G be a bipartite graph with bipartition
$V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and $V_{2}=\\{y_{1},\ldots,y_{m}\\}$ and let
$H=G\setminus N_{G}[x_{i_{1}},\ldots,x_{i_{s}}]$ be subgraph of $G$. Then

$\Delta_{H}=\operatorname{lk}_{\Delta_{G}}\\{x_{i_{1}},\ldots,x_{i_{s}}\\}$

###### Proof.

We use induction on $s$. When $s=1$, the result follows from Lemma 1.2. Now
suppose, inductively, that $s>1$ and the result has been proved for smaller
values of $s$. Let $K=G\setminus N_{G}[x_{i_{1}},\ldots,x_{i_{s-1}}]$, by
induction hypothesis
$\Delta_{K}=\operatorname{lk}_{\Delta_{G}}\\{x_{i_{1}},\ldots,x_{i_{s-1}}\\}$.
Set $\Gamma=\operatorname{lk}_{\Delta_{G}}\\{x_{i_{1}},\ldots,x_{i_{s-1}}\\}$.
Since $H=K\setminus N_{K}[x_{i_{s}}]$ and
$\operatorname{lk}_{\Gamma}\\{x_{i_{s}}\\}=\operatorname{lk}_{\Delta_{G}}\\{x_{i_{1}},\ldots,x_{i_{s}}\\}$,
we have
$\Delta_{H}=\operatorname{lk}_{\Delta_{G}}\\{x_{i_{1}},\ldots,x_{i_{s}}\\}$.

Let $I$ be a monomial ideal of $R$. We denote, as usual, by $G(I)$ the unique
minimal set of monomial generators of $I$. If $I$ is generated in a single
degree, then $I$ is said to be polymatroidal if for any two elements $u,v\in
G(I)$ such that $\deg_{x_{i}}(v)<\deg_{x_{i}}(u)$ there exists an index $j$
with $\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$ such that $x_{j}(u/x_{i})\in G(I)$. The
polymatroidal ideal $I$ is called matroidal if $I$ is generated by square-free
monomials (see [13]).

######  Lemma 2.2.

Let $K_{m,n}$ be complete bipartite graph. Then $K_{m,n}$ is aCM if and only
if $n\leq m\leq 2$.

###### Proof.

$(\Longleftarrow).$ This is obvious.

$(\Longrightarrow).$ Let $K_{m,n}$ be aCM complete bipartite graph and $n\leq
m$. So the edge ideal of $K_{m,n}$ is of the form
$I(G)=(x_{1},\ldots,x_{m})(y_{1},\ldots,y_{n})$. $I(G)$ is transversal
matroidal ideal of degree $2$. Therefore by [2] we have
$\operatorname{depth}(R/I)=1$. If $\dim(R/I)=1$, then $I(G)$ is CM and by [6,
Remark 2.2] we have $n=m=1$ . If $\dim(R/I)=2$, then
$\operatorname{ht}(I(G))=n+m-2$. Since $\operatorname{ht}(I(G))\leq n$, we
have $n+m-2\leq n$. Hence we must have $n\leq m\leq 2$.

Estrada and Villarreal in [6] proved that if $G$ is a CM bipartite graph, then
there is a vertex $v\in V(G)$ such that $deg(v)=1$ and also they proved that
$G\setminus\\{u\\}$ is CM for some vertex $u\in V(G)$. Now we generalize that
results.

######  Theorem 2.3.

Let $G$ be an aCM bipartite graph with bipartition
$V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and $V_{2}=\\{y_{1},\ldots,y_{m}\\}$. If $G$
has at least one vertex of positive degree, then there is a vertex $v\in
V_{1}\cup V_{2}$ so that $\deg(v)\leq 2$.

###### Proof.

Since $G$ is aCM bipartite graph, by [18, Corollary 2.4] we may assume that
$m-1\leq n\leq m$. First assume $n=m-1$. We proceed by contradiction. Assume
$\deg(v)\geq 3$ for all $v\in V_{1}\cup V_{2}$. Let $v$ be a vertex of minimal
degree. We may assume $v=x_{n}$. If $\deg(x_{n})=n+1$, then $G=K_{n,n+1}$ is a
complete bipartite graph, but this is impossible by Lemma 2.2. Therefore we
may assume that $3\leq\deg(x_{n})\leq n$. Set $s=\deg(x_{n})$. Let
$N_{G}(x_{n})=\\{y_{n+1},\ldots,y_{n-s+2}\\}$. Let $\Delta$ be the simplicial
complex of independent sets of $G$, by Lemma 2.1 we have
$\Gamma=\operatorname{lk}_{\Delta}\\{x_{n}\\}$ is the simplicial complex of
independent sets of $G\setminus N_{G}[x_{n}]$. Since $\Delta$ is aCM, by [18,
Theorem 3.4] $\Gamma$ is C-M for $\\{x_{n}\\}\in\Delta\setminus\Delta(n)$ and
$\Gamma$ is aCM for $\\{x_{n}\\}\in\Delta(n)$.

If $\\{x_{n}\\}\in\Delta(n)$, then $\dim\Gamma=n-1$. Therefore we have
$n-1\leq|F|\leq n$ for all facet $F$ in $\Gamma$. We claim
$x_{n-s+2},\ldots,x_{n-1}$ are isolated vertex in $G\setminus N_{G}[x_{n}]$.
By contrary we assume one of them is not isolated vertex in $G\setminus
N_{G}[x_{n}]$, say $x_{n-1}$. So $\\{x_{n-1},y_{j}\\}\in E(G)$ for some $1\leq
j\leq n-s+1$. Therefore
$\\{y_{1},\ldots,y_{n-s+1},x_{n-s+2},\ldots,x_{n-2}\\}$ is maximal facet of
bipartite graph $G\setminus N_{G}[x_{n}]$, and this is contradiction, since
$|\\{y_{1},\ldots,y_{n-s+1},x_{n-s+2},\ldots,x_{n-2}\\}|=n-2$. So
$\deg(x_{n-s+2})=\ldots=\deg(x_{n-1})=s$ and
$N_{G}(x_{n-s+2})=\ldots=N_{G}(x_{n-1})=\\{y_{n+1},\ldots,y_{n-s+2}\\}$.
Consider the graph $H=G\setminus N_{G}[y_{1},\ldots,y_{n-s+1}]$. $H$ is aCM
bipartite graph, since by Lemma 2.1
$\operatorname{lk}_{\Delta}\\{y_{1},\ldots,y_{n-s+1}\\}$ is the simplicial
complexs of independent sets of $H$ but this is impossible because
$H=K_{s-1,s}$.

If $\\{x_{n}\\}\notin\Delta(n)$, then $\dim\Gamma=n-2$. Since $\Gamma$ is CM,
we have $|F|=n-1$ for all facet $F$ in $\Gamma$. We claim
$x_{n-s+2},\ldots,x_{n-1}$ are isolated vertex in $G\setminus N_{G}[x_{n}]$.
By contrary assume one of them is not isolated vertex in $G\setminus
N_{G}[x_{n}]$, say $x_{n-1}$. So $\\{x_{n-1},y_{j}\\}\in E(G)$ for some $1\leq
j\leq n-s+1$. Therefore
$\\{y_{1},\ldots,y_{n-s+1},x_{n-s+2},\ldots,x_{n-2}\\}$ and
$\\{x_{1},\ldots,x_{n-1}\\}$ are maximal facet of bipartite graph $G\setminus
N_{G}[x_{n}]$, and this is contradiction. So
$\deg(x_{n-s+2})=\ldots=\deg(x_{n-1})=s$ and
$N_{G}(x_{n-s+2})=\ldots=N_{G}(x_{n-1})=\\{y_{n+1},\ldots,y_{n-s+2}\\}$.
Consider the graph $H=G\setminus N_{G}[y_{1},\ldots,y_{n-s+1}]$. $H$ is aCM
bipartite graph, since by Lemma 2.1
$\operatorname{lk}_{\Delta}\\{y_{1},\ldots,y_{n-s+1}\\}$ is the simplicial
complexs of independent sets of $H$ but this is impossible because
$H=K_{s-1,s}$. If $|V_{1}|=|V_{2}|=m=n$, then the proof is exactely the same
of the above arguments.

######  Corollary 2.4.

Let $G$ be an aCM bipartite graph. Let $u$ be a vertex of degree one of $G$
and $v$ its adjacent vertex. Then $G\setminus\\{v\\}$ is aCM.

###### Proof.

Let $H=G\setminus\\{u,v\\}$ be subgraph of $G$, by Lemma 2.1 we have
$\operatorname{lk}_{\Delta}\\{u\\}$ is the simplicial complexs of independent
sets of $H$. By [18, Corollary 3.6], $\operatorname{lk}_{\Delta}\\{u\\}$ is
aCM. If $\Gamma=\langle F_{1},\ldots,F_{r}\rangle$ be simplicial complex of
independent set of $G\setminus\\{v\\}$, then $F_{i}=G_{i}\cup\\{u\\}$ such
that $G_{i}$ is a facet of $\operatorname{lk}_{\Delta}\\{u\\}$. Hence
$G\setminus\\{v\\}$ is aCM.

######  Corollary 2.5.

Let $G$ be an aCM bipartite graph. Let $u$ be a vertex of degree two of $G$
and $v,w$ its adjacent vertex. then $G\setminus\\{v,w\\}$ is aCM.

###### Proof.

Let $H=G\setminus\\{u,v,w\\}$ be subgraph of $G$, by Lemma 2.1 we have
$\operatorname{lk}_{\Delta}\\{u\\}$ is the simplicial complexs of independent
sets of $H$. By [18, Corollary 3.6], $\operatorname{lk}_{\Delta}\\{u\\}$ is
aCM. If $\Gamma=\langle F_{1},\ldots,F_{r}\rangle$ be simplicial complex of
independent set of $G\setminus\\{v,w\\}$, then $F_{i}=G_{i}\cup\\{u\\}$ such
that $G_{i}$ is a facet of $\operatorname{lk}_{\Delta}\\{u\\}$. Hence
$G\setminus\\{v,w\\}$ is aCM.

######  Lemma 2.6.

Let $G$ be an unmixed bipartite graph with bipartition
$V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and $V_{2}=\\{y_{1},\ldots,y_{n}\\}$ and let
$N_{G}(x_{n})=\\{y_{n},\ldots,y_{n-i+1}\\}$. Then
$\operatorname{lk}_{\Delta_{G}}\\{y_{n},\ldots,y_{n-i+1}\\}$ is subsimplicial
complex of $\operatorname{lk}_{\Delta_{G}}\\{x_{n}\\}$.

###### Proof.

Let $H=G\setminus N_{G}[x_{n}]$ and $K=G\setminus
N_{G}[y_{n},\ldots,y_{n-i+1}]$ be subgraphs of $G$, by Lemma 2.1, we have
$\operatorname{lk}_{\Delta_{G}}\\{x_{n}\\}$ and
$\operatorname{lk}_{\Delta_{G}}\\{y_{n},\ldots,y_{n-i+1}\\}$ are the
simplicial complexs of independent sets of $H$ and $K$ respectively. If
$F\in\operatorname{lk}_{\Delta_{G}}\\{y_{n},\ldots,y_{n-i+1}\\}$, then
$y_{j}\notin F$ for all $n-i+1\leq j\leq n$ and
$F\cup\\{y_{n},\ldots,y_{n-i+1}\\}\in{\Delta_{G}}$. This implies that
$F\cup\\{y_{n},\ldots,y_{n-i+1}\\}$ is an independent set of $G$. So
$(F\cup\\{y_{n},\ldots,y_{n-i+1}\\})\cap
N_{G}(\\{y_{n},\ldots,y_{n-i+1}\\})=\emptyset$. But this means that
$F\subseteq V(K)\subseteq V(H)$ because $V(K)=V(G)\setminus
N_{G}[y_{n},\ldots,y_{n-i+1}]$. Since
$\\{y_{n},\ldots,y_{n-i+1}\\}=N_{G}(x_{n})$ and $x_{n}\in
N_{G}(y_{n},\ldots,y_{n-i+1})$, we have $(F\cup
N_{G}(x_{n}))\cap\\{x_{n}\\}=\emptyset$. Thus $(F\cup\\{x_{n}\\})\cap
N_{G}(x_{n})=\emptyset$. Hence
$F\in\operatorname{lk}_{\Delta_{G}}\\{x_{n}\\}$.

######  Lemma 2.7.

Let $G$ be an unmixed bipartite graph with pure order of vertices
$\\{x_{1},\ldots,x_{n}\\}\cup\\{y_{1},\ldots,y_{n}\\}$ and let
$N_{G}(x_{i})=\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$. Then

  1. (i)

$G\setminus N_{G}[x_{i}]$ is unmixed bipartite subgraph of $G$.

  2. (ii)

$x_{i_{1}},\ldots,x_{i_{r_{i}}}$ are isolated vertices in $G\setminus
N_{G}[x_{i}]$.

In particular if $x_{i}$ is a vertex of minimal degree, then

  1. (iii)

$G\setminus
N_{G}[\\{y_{1},\ldots,y_{n}\\}\setminus\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}]$
is complete bipartite graph with bipartition
$\\{x_{i},x_{i_{1}},\ldots,x_{i_{r_{i}}}\\}\cup\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$

  2. (iv)

$N_{G}(y_{i})=N_{G}(y_{i_{1}})=\ldots=N_{G}(y_{i_{r_{i}}})$

###### Proof.

$(i)$ Since $\operatorname{lk}_{\Delta_{G}}\\{x_{i}\\}$ is the simplicial
complexs of independent sets of $G\setminus N_{G}[x_{i}]$ and any link of a
pure complex is pure, we have $G\setminus N_{G}[x_{i}]$ is unmixed bipartite
subgraph of $G$.
$(ii)$ If $x_{j}$ for some $i_{1}\leq j\leq i_{r_{i}}$ is not isolated in
$G\setminus N_{G}[x_{i}]$, then there exists an integer
$k\in\\{1,\ldots,n\\}\setminus\\{i_{1},\ldots,i_{r_{i}}\\}$ such that
$x_{j}y_{k}\in E(G\setminus N_{G}[x_{i}])$. Therefore
$\\{y_{1},\ldots,y_{n}\\}\setminus\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$
is a minimal vertex cover for $G\setminus N_{G}[x_{i}]$ and there is also a
minimal vertex cover for $G\setminus N_{G}[x_{i}]$ containing
$(\\{x_{1},\ldots,x_{n}\\}\setminus\\{x_{i},x_{i_{1}},\ldots,x_{i_{r_{i}}}\\})\cup\\{x_{j}\\}$,
which is a contradiction since $G\setminus N_{G}[x_{i}]$ is unmixed.
$(iii)$ Since $x_{i_{1}},\ldots,x_{i_{r_{i}}}$ are isolated vertices in
$G\setminus N_{G}[x_{i}]$, we have $\deg x_{i_{j}}\leq r_{i}+1$ and
$N_{G}(x_{i_{j}})\subseteq N_{G}(x_{i})$ for all $i_{1}\leq j\leq i_{r_{i}}$.
By hypothesis $x_{i}$ is a vertex of minimal degree, therefore $\deg
x_{i_{j}}\geq r_{i}+1$ for all $i_{1}\leq j\leq i_{r_{i}}$. Hence $G\setminus
N_{G}[\\{y_{1},\ldots,y_{n}\\}\setminus\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}]$
is complete bipartite graph with bipartition
$\\{x_{i},x_{i_{1}},\ldots,x_{i_{r_{i}}}\\}\cup\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$.
$(iv)$ Let $x_{k}\in N_{G}(y_{i_{j}})$ for some
$j\in\\{i,i_{1},\ldots,i_{r_{i}}\\}$, say $x_{k}\in N_{G}(y_{i_{1}})$.
Therefore $\\{x_{k},y_{i_{1}}\\}\in E(G)$. Since
$N_{G}(x_{i_{1}})=\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$, by pure order
of vertices we have $\\{x_{k},y_{i_{j}}\\}\in E(G)$ for all
$j\in\\{i,i_{1},\ldots,i_{r_{i}}\\}$. Hence
$N_{G}(y_{i})=N_{G}(y_{i_{1}})=\ldots=N_{G}(y_{i_{r_{i}}})$ .

######  Lemma 2.8.

Let $G$ be an unmixed bipartite graph with pure order of vertices
$\\{x_{1},\ldots,x_{n}\\}\cup\\{y_{1},\ldots,y_{n}\\}$ and let $H=G\setminus
N_{G}[x_{i}]$ and $K=H\setminus N_{G}[\\{x_{i_{1}},\ldots,x_{i_{r_{i}}}\\}]$
be subgraphs of $G$ such that $x_{i}$ is a vertex of minimal degree and
$N_{G}(x_{i})=\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$. Then $F$ is a facet
of $\Delta_{G}$ if and only if $F$ can be written as

  1. (1)

$F=F^{\prime}\cup\\{x_{i}\\}$, where $F^{\prime}$ is a facet of $\Delta_{H}$.

  2. (2)

$F=F^{\prime\prime}\cup\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$, where
$F^{\prime\prime}$ is a facet of $\Delta_{K}$.

###### Proof.

Let $F$ be a facet of $\Delta_{G}$. Since $G$ is unmixed bipartite graph, it
follows that if $x_{i}\notin F$, then $y_{i}\in F$. By Lemma 2.7 part $(iv)$,
we have $N_{G}(y_{i})=N_{G}(y_{i_{1}})=\ldots=N_{G}(y_{i_{r_{i}}})$ and so
$y_{i_{1}},\ldots,y_{i_{r_{i}}}\in F$. Set
$F^{\prime\prime}=F\setminus\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$. We
must show that $F^{\prime\prime}$ is a facet of $\Delta_{K}$. First notice
that $F^{\prime\prime}$ is an independent set of $K$, because $E(K)\subseteq
E(G)$. Let $L$ be a facet of $\Delta_{K}$ containing $F^{\prime\prime}$. Since
$L\cup\\{y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\\}$ is an independent set of
$G$, and $G$ is unmixed, we obtain that
$|L|+r_{i}+1\leq|F|=|F^{\prime\prime}|+r_{i}+1$. Hence $F^{\prime\prime}=L$.

On the other hand if $x_{i}\in F$, then we have
$y_{i},y_{i_{1}},\ldots,y_{i_{r_{i}}}\notin F$. Set
$F^{\prime}=F\setminus\\{x_{i}\\}$. $F^{\prime}$ is a facet of $\Delta_{H}$,
since $\Delta_{H}=\operatorname{lk}_{\Delta_{G}}{\\{x_{i}\\}}$. The converse
also follows readily by using the similar arguments.

######  Theorem 2.9.

Let $G$ be an unmixed bipartite graph with bipartition
$V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and $V_{2}=\\{y_{1},\ldots,y_{n}\\}$ and let
$H=G\setminus N_{G}[x_{n}]$ and $K=G\setminus N_{G}[y_{n}]$ be subgraphs of
$G$ such that $N_{G}(x_{n})=\\{y_{n}\\}$. Then $H$ and $K$ are aCM graphs if
and only if $G$ is aCM graph.

###### Proof.

$(\Longleftarrow).$ Let $\Delta=\langle F_{1},F_{2},\ldots,F_{m}\rangle$ be
the simplicial complex of independent sets of $G$. By Lemma 2.1 we have
$\operatorname{lk}_{\Delta}\\{x_{n}\\}$ and
$\operatorname{lk}_{\Delta}\\{y_{n}\\}$ are the simplicial complexs of
independent sets of $H$ and $K$ respectively. Therefore by[18, Corollary 3.6],
$H$ and $K$ are aCM.

$(\Longrightarrow).$ By [18, Theorem 3.4], it is enough to show that
$\tilde{H}_{i}(\operatorname{lk}_{\Delta}F;k)=0$ for all $F\in\Delta(n-1)$ and
for all $i<\dim\operatorname{lk}_{\Delta}F-1$. We proceed by induction on
$\dim\operatorname{lk}_{\Delta}F$. The case
$\dim\operatorname{lk}_{\Delta}F=1$ is clear. Assume that
$\dim\operatorname{lk}_{\Delta}F>1$ and the assertion holds for
$\dim\operatorname{lk}_{\Delta}F=n-1$. Since for each $l$,
$\\{x_{l},y_{l}\\}\in E(G)$ and $G$ is unmixed, every facet $F_{k}$ has
exactely one of $x_{l}$ and $y_{l}$. Thus either $x_{l}\in F_{k}$ or $y_{l}\in
F_{k}$. Since $N_{G}(x_{n})=\\{y_{n}\\}$, we can assume $F_{i}$ contains
$x_{n}$ such that $y_{n}\notin F_{i}$ for $i=1,\ldots,s$ and $F_{i}$ contains
$y_{n}$ such that $x_{n}\notin F_{i}$ for $i=s+1,\ldots,m$. Let
$\Delta_{1}=\langle F_{1},\ldots,F_{s}\rangle$ and $\Delta_{2}=\langle
F_{s+1},\ldots,F_{m}\rangle$. Then $\Delta=\Delta_{1}\cup\Delta_{2}$. We claim
$\Delta_{1}\cap\Delta_{2}=\operatorname{lk}_{\Delta}\\{y_{n}\\}$. It is
obvious that $\operatorname{lk}_{\Delta}\\{y_{n}\\}$ is a subsimplicial
complex of $\Delta_{1}\cap\Delta_{2}$, since by Lemma 2.6,
$\operatorname{lk}_{\Delta}\\{y_{n}\\}$ is subsimplicial complex of
$\operatorname{lk}_{\Delta}\\{x_{n}\\}$. Now assume
$F\in\Delta_{1}\cap\Delta_{2}$. If $F\in\Delta_{1}$, then $F\subseteq F_{i}$
for some $i$ in $\\{1,\ldots,s\\}$. By Lemma 2.8, there exist a facet
$F^{\prime}_{i}$ in $\operatorname{lk}_{\Delta}\\{x_{n}\\}$ such that
$F\subseteq F^{\prime}_{i}\cup\\{x_{n}\\}$. By similar argument there exist a
facet $F^{\prime}_{j}$ in $\operatorname{lk}_{\Delta}\\{y_{n}\\}$ such that
$F\subseteq F^{\prime}_{j}\cup\\{y_{n}\\}$. Hence $F\subseteq
F^{\prime}_{i}\cap F^{\prime}_{j}\in\operatorname{lk}_{\Delta}\\{y_{n}\\}$.
Now by using the Mayer-Vietoris exact sequence we have
$\tilde{H}_{i}({\Delta};k)=0$ for $i<n-2$. Hence $\Delta$ is aCM.

######  Theorem 2.10.

Let $G$ be an unmixed bipartite graph with pure order of vertices
$\\{x_{1},\ldots,x_{n}\\}\cup\\{y_{1},\ldots,y_{n}\\}$. Let $H=G\setminus
N_{G}[x_{n}]$ and $K=G\setminus N_{G}[y_{n},y_{n-1}]$ be subgraphs of $G$ such
that $N_{G}(x_{n})=\\{y_{n},y_{n-1}\\}$. If $H$ is aCM and $K$ is CM, then $G$
is aCM.

###### Proof.

Let $\Delta=\langle F_{1},F_{2},\ldots,F_{m}\rangle$ be the simplicial complex
of independent sets of $G$. By Lemma 2.1, we have
$\operatorname{lk}_{\Delta}\\{x_{n}\\}$ and
$\operatorname{lk}_{\Delta}\\{y_{n},y_{n-1}\\}$ are simplicial complexs of
independent sets of $H$ and $K$, respectively.

By [18, Theorem 3.4], it is enough to show that
$\tilde{H}_{i}(\operatorname{lk}_{\Delta}F;k)=0$ for all $F\in\Delta(n-1)$ and
for all $i<\dim\operatorname{lk}_{\Delta}F-1$. We proceed by induction on
$\dim\operatorname{lk}_{\Delta}F$. The case
$\dim\operatorname{lk}_{\Delta}F=1$ is clear. Assume that
$\dim\operatorname{lk}_{\Delta}F>1$ and the assertion holds for
$\dim\operatorname{lk}_{\Delta}F=n-1$. Since
$N_{G}(x_{n})=\\{y_{n},y_{n-1}\\}$ and $G$ is unmixed, every facet $F_{k}$ has
exactely one of $x_{n}$ and $y_{i}$ for $n-1\leq i\leq n$. Now by Lemma 2.7,
we can assume $F_{i}$ contains $x_{n}$ and none of
$y_{j}\in\\{y_{n},y_{n-1}\\}$ for $i=1,\ldots,s$ and also
$\\{y_{n},y_{n-1}\\}\subseteq F_{j}$ for $j=s+1,\ldots,m$ such that
$x_{n}\notin F_{j}$. Let $\Delta_{1}=\langle F_{1},\ldots,F_{s}\rangle$ and
$\Delta_{2}=\langle F_{s+1},\ldots,F_{m}\rangle$. Then
$\Delta=\Delta_{1}\cup\Delta_{2}$. By Lemma 2.1,
$\operatorname{lk}_{\Delta}\\{y_{n},y_{n-1}\\}$ is subsimplicial complex of
$\operatorname{lk}_{\Delta}\\{x_{n}\\}$. We claim
$\Delta_{1}\cap\Delta_{2}=\operatorname{lk}_{\Delta}\\{y_{n},y_{n-1}\\}$. It
is obvious that $\operatorname{lk}_{\Delta}\\{y_{n},y_{n-1}\\}$ is a
subsimplicial complex of $\Delta_{1}\cap\Delta_{2}$. Now assume
$F\in\Delta_{1}\cap\Delta_{2}$. If $F\in\Delta_{1}$, then $F\subseteq F_{i}$
for some $i$ in $\\{1,\ldots,s\\}$. By Lemma 2.8, there exist a facet
$F^{\prime}_{i}$ in $\operatorname{lk}_{\Delta}\\{x_{n}\\}$ such that
$F\subseteq F^{\prime}_{i}\cup\\{x_{n}\\}$. By similar argument there exist a
facet $F^{\prime}_{j}$ in $\operatorname{lk}_{\Delta}\\{y_{n},y_{n-1}\\}$ such
that $F\subseteq F^{\prime}_{j}\cup\\{y_{n},y_{n-1}\\}$. Hence $F\subseteq
F^{\prime}_{i}\cap
F^{\prime}_{j}\in\operatorname{lk}_{\Delta}\\{y_{n},y_{n-1}\\}$. By using the
Mayer-Vietoris exact sequence, it therefore follows that $\Delta$ is aCM.

The following example shows that the condition of Cohen-Macualyness of $K$ is
essential.

######  Example 2.11.

Let $G$ be the following unmixed bipartite graph.

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

$H=G\setminus N_{G}[x_{6}]$ and $K=G\setminus N_{G}[\\{y_{5},y_{6}\\}]$ are
aCM, but $G$ is not aCM.

##  3\. Almost Cohen-Macaulay and connected in codimension two

We recall the concept of connected in codimension $k$, for a topological space
is defined by Hartshoren in [11], for any non-negative integer $k$. For a
monomial ideal $I$, considering the Zariski topology on
$\operatorname{Spec}(R/I)$, we get that the closed subsets in this topology
are the sets
$V(J)=\\{\mathfrak{q}\in\operatorname{Spec}(R)~{}|~{}J\subseteq\mathfrak{q}\\}$,
where $J\subseteq I$ is an ideal of $R$ and the irreducible components of
$\operatorname{Spec}(R/I)$ are the closed sets $V(\mathfrak{p})$, where
$\mathfrak{p}$ is a minimal prime ideal of $I$. $\operatorname{Spec}(R/I)$
with this topology is a connected space. By [11, Proposition 1.1], the ideal
$I$ is called connected in codimension $k$, if $V(\mathfrak{p})$ and
$V(\mathfrak{q})$ are irreducible comnponents of $\operatorname{Spec}(R/I)$,
then there is a finite sequence
$V(\mathfrak{p})=V(\mathfrak{p}_{1}),V(\mathfrak{p}_{2}),\ldots,V(\mathfrak{p}_{r})=V(\mathfrak{q})$
of irreducible components of $\operatorname{Spec}(R/I)$, such that for each
$i=1,2,\ldots,r-1$, $V(\mathfrak{p}_{i})\cap V(\mathfrak{p}_{i+1})$ is of
codimension $\leq k$ in $\operatorname{Spec}(R/I)$. Since the codimension of
$V(\mathfrak{p})$ is equal to
$\operatorname{ht}(\mathfrak{p})-\operatorname{ht}(I)$ for all prime ideals
$\mathfrak{p}\supseteq I$, a monomial ideal $I\subset R$ is connected in
codimension $k$, if for any pair of distinct minimal prime ideals
$\mathfrak{p}$ and $\mathfrak{q}$, there exists a sequence of minimal prime
ideals $\mathfrak{p}=\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}=\mathfrak{q}$
such that
$\operatorname{ht}(\mathfrak{p}_{i}+\mathfrak{p}_{i+1})\leq\operatorname{ht}(I)+k$.
As $\mathfrak{p}_{i}+\mathfrak{p}_{i+1}$ is again a prime ideal, one can
replace the height with the number of generators.

The definition for a simplicial complex $\Delta$ should be organized in such a
way that the Stanley-Reisner ideal of $\Delta$, becomes connected in
codimension $k$, i.e. A simplicial complex $\Delta$ is said to be connected in
codimension $k$, if for any two facets $F$ and $G$ of $\Delta$, there is a
sequence of facets $F=F_{1},F_{2},...,F_{r}=G$ such that $\dim(F_{i}\cap
F_{i+1})\geq\dim\Delta-k$, for each $1\leq i\leq r-1$. By the above
definition, if a monomial ideal $I$ is connected in codimension one, then it
is equidimensional i.e. all minimal prime ideals of $I$ have the same height.
In particular, if $I$ is square-free monomial ideal connected in codimension
one then it is unmixed i.e. all prime ideals of $\operatorname{Ass}(I)$ have
the same height, (see also [1, Definition 3.1]).

Thus we can rewrite the following result:

######  Theorem 3.1.

([8, Theorem 1.3]) Let $G$ be a bipartite graph with at least four vertices.
Then $G$ is a connected in codimension one if and only if $G$ is a CM graph.

######  Theorem 3.2.

Let $\Delta$ be $(d-1)$-dimensional ($d\geq 3$) aCM simplicial complex. Then
it is connected in codimension two.

###### Proof.

We argue by induction on $d$. If $d=3$, then by [18, Corollary 3.5] $\Delta$
is connected. Therefore for arbitrary facets $F$ and $E$, there exists a
sequence of facets $F=F_{0},F_{1},...,F_{n-1},F_{n}=E$ such that $F_{i}\cap
F_{i+1}\neq\emptyset$. So $\dim(F_{i}\cap F_{i+1})\geq 0=\dim\Delta-2$. Hence
$\Delta$ is connected in codimension two. Now we assume that $d>3$. Let $F$
and $E$ be two facets of $\Delta$. Since $\Delta$ is aCM and $d\geq 3$, we
have $\Delta$ is connected by [18, Lemma 3.2 and Theorem 3.4]. Therefore there
exists a sequence of facets $F=F_{0},F_{1},...,F_{n-1},F_{n}=E$ such that
$F_{i}\cap F_{i+1}\neq\emptyset$. Let $x_{i}$ be a vertex belonging to
$F_{i}\cap F_{i+1}$. Since $\Delta$ is aCM,
$\operatorname{lk}_{\Delta}\\{{x_{i}}\\}$ is CM for
$\\{x_{i}\\}\in\Delta\setminus\Delta(d-1)$ and
$\operatorname{lk}_{\Delta}\\{{x_{i}}\\}$ is aCM for
$\\{x_{i}\\}\in\Delta(d-1)$ by [18, Corollary 3.6]. By [13, Lemma 9.1.12] and
working with induction on the dimension of $\Delta$, we may assume that
$\operatorname{lk}_{\Delta}\\{{x_{i}}\\}$ is connected in codimension one for
$\\{x_{i}\\}\in\Delta\setminus\Delta(d-1)$ and
$\operatorname{lk}_{\Delta}\\{{x_{i}}\\}$ is connected in codimension two for
$\\{x_{i}\\}\in\Delta(d-1)$ respectively. Thus
$F_{i}^{\prime}:=F_{i}\setminus\\{x_{i}\\}$ and
$F_{i+1}^{\prime}:=F_{i+1}\setminus\\{x_{i}\\}$ are facets of
$\operatorname{lk}_{\Delta}\\{{x_{i}}\\}$ and therefore there exists a
sequence of facets
$F_{i}^{\prime}=H_{0}^{\prime},H_{1}^{\prime},\ldots,H_{r-1}^{\prime},H_{r}^{\prime}=F_{i+1}^{\prime}$
of $\operatorname{lk}_{\Delta}\\{{x_{i}}\\}$ such that $|H_{j}^{\prime}\cap
H_{j+1}^{\prime}|\geq d-3$. Set $H_{j}=H_{j}^{\prime}\cup\\{x_{i}\\}$. So
there exists a sequence of facets
$F_{i}=H_{0},H_{1},\ldots,H_{r-1},H_{r}=F_{i+1}$ of $\Delta$, where all
$H_{j}$ contain $x_{i}$ with $|H_{j}|\geq d-1$, such that $|H_{j}\cap
H_{j+1}|\geq d-2$. Composing all these sequences of facets which we have
between each $F_{i}$ and $F_{i+1}$ yields the desired sequence between $F$ and
$E$. This completes the proof.

As before if the monomial ideal $I$ is connected in codimension one, then $I$
is equidimensional. But the following example says that for connected in
codimensional two this is false.

######  Example 3.3.

Let $G$ be the following graph. Then

$I(G)=(x_{1}y_{1},x_{1}y_{2},x_{2}y_{1},x_{2}y_{2},x_{2}y_{3},x_{3}y_{1},x_{3}y_{2},x_{3}y_{3},x_{3}y_{4},x_{4}y_{1},x_{4}y_{2},x_{4}y_{3},x_{4}y_{4}).$

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

By Macaulay $2$, we have

$\displaystyle\operatorname{Ass}(R/I)=$
$\displaystyle\\{\mathfrak{p}_{1}=(x_{1},x_{2},x_{3},x_{4}),\mathfrak{p}_{2}=(x_{2},x_{3},x_{4},y_{1},y_{2}),\mathfrak{p}_{3}=(x_{3},x_{4},y_{1},y_{2},y_{3}),$
$\displaystyle\mathfrak{p}_{4}=(y_{1},y_{2},y_{3},y_{4})\\}.$

By considering the sequence
$\mathfrak{p}_{1},\mathfrak{p}_{2},\mathfrak{p}_{3},\mathfrak{p}_{4}$ of
minimal prime ideals of $I(G)$, we have
$\operatorname{ht}(\mathfrak{p}_{i}+\mathfrak{p}_{i+1})\leq
6=\operatorname{ht}(I)+2$ for $1\leq i\leq 3$. Therefore for any pair of
distinct minimal prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ in $I(G)$,
there exists a sequence of minimal prime ideals
$\mathfrak{p}=\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}=\mathfrak{q}$ such that
$\operatorname{ht}(\mathfrak{p}_{i}+\mathfrak{p}_{i+1})\leq
6=\operatorname{ht}(I)+2$. Therefore $I(G)$ is connected in codimension $2$.
But $I(G)$ is not unmixed.

######  Theorem 3.4.

Let $G$ be an unmixed aCM bipartite graph with vertex partition $V_{1}\cup
V_{2}$. Then the vertices $V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and
$V_{2}=\\{y_{1},\ldots,y_{n}\\}$ can be labeled such that:

  1. (i)

$\\{x_{i},y_{i}\\}$ are edges for $i=1,\ldots,n$;

  2. (ii)

if $\\{x_{i},y_{j}\\}$ is an edge, then $i\leq j+1$;

  3. (iii)

if $\\{x_{i},y_{j}\\}$ and $\\{x_{j},y_{k}\\}$ are edges, then
$\\{x_{i},y_{k}\\}$ is an edge.

###### Proof.

Since $G$ is unmixed bipartite graph, by Theorem 1.3 we have
$\\{x_{i},y_{i}\\}$ are edges for $i=1,\ldots,n$ and if $\\{x_{i},y_{j}\\}$
and $\\{x_{j},y_{k}\\}$ are edges, then $\\{x_{i},y_{k}\\}$ is an edge of $G$.
Let $\Delta$ be the simplicial complex with $I_{\Delta}=I(G)$. Then $V_{1}$
and $V_{2}$ are facets of $\Delta$, while $\\{x_{i},y_{i}\\}$ are not faces of
$\Delta$. By Theorem 3.2, $\Delta$ is connected in codimension two, it follows
that there is a sequence of facets $F_{1},\ldots,F_{s}$ of $\Delta$ with
$F_{1}=V_{1}$ and $F_{s}=V_{2}$ such that $|F_{k-1}\cap F_{k}|\geq n-2$ for
$k=1,\ldots,s$. Then $|F_{2}\setminus F_{1}|\leq 2$, say $F_{2}\setminus
F_{1}=\\{y_{1}\\}$ or $F_{2}\setminus F_{1}=\\{y_{1},y_{2}\\}$. Therefore
$F_{2}=\\{y_{1},x_{2},\ldots,x_{n}\\}$ or
$F_{2}=\\{y_{1},y_{2},x_{3},\ldots,x_{n}\\}$ because $\\{x_{1},y_{1}\\}$ and
$\\{x_{2},y_{2}\\}$ are not faces and $\Delta$ is pure. A similar argument as
above implies that $|F_{k}\setminus F_{k-1}|\leq 2$ for $k=1,\cdots,s$ and
hence we may assume that $F_{k}=\\{y_{1},\ldots,y_{i},x_{i+1},\ldots,x_{n}\\}$
for $k=1,\ldots,s$ and for some $i$ such that if $i>j+1$, then
$\\{x_{i},y_{j}\\}$ is a face of $\Delta$. Therefore, if $i>j+1$, then
$\\{x_{i},y_{j}\\}$ is not an edge of $G$. In other word, if
$\\{x_{i},y_{j}\\}$ is an edge of $G$, then $i\leq j+1$.

The following example shows that the converse of Theorem 3.4 is not true.

######  Example 3.5.

Let $G$ be the following graph which is unmixed and connected in codimension
two and also if $\\{x_{i},y_{j}\\}\in E(G)$, then $i\leq j+1$. But $G$ is not
aCM, since $\dim(R/I)=6$ and $\operatorname{depth}(R/I)=4$.

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

By Macaulay $2$, we have

$\displaystyle\operatorname{Ass}(R/I)=$
$\displaystyle\\{(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}),(x_{1},x_{2},x_{3},x_{4},y_{5},y_{6}),(x_{1},x_{2},x_{5},x_{6},y_{3},y_{4}),$
$\displaystyle(x_{1},x_{2},y_{3},y_{4},y_{5},y_{6}),(y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})\\}.$

It is easy to check that for any pair of distinct minimal prime ideals
$\mathfrak{p}$ and $\mathfrak{q}$ in $I(G)$, there exists a sequence of
minimal prime ideals
$\mathfrak{p}=\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}=\mathfrak{q}$ such that
$\operatorname{ht}(\mathfrak{p}_{i}+\mathfrak{p}_{i+1})\leq
8=\operatorname{ht}(I)+2$. Therefore $I(G)$ is connected in codimension two.

Given a positive integer $n$, and an $n$-tuple
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$ of positive integers such that
$m=\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}$, the Ferrers graph
$G=G_{\lambda}$ associated to $\lambda$ defined by Corso and Nagel [4] and it
is the bipartite graph with bipartition $V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and
$V_{2}=\\{y_{1},\ldots,y_{m}\\}$ such that if $(x_{i},y_{j})$ is an edge of
$G$, then so is $(x_{r},y_{s})$ for $1\leq r\leq i$ and $1\leq s\leq j$.
Suppose that $x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}$ are indeterminates over
the field $k$. The edge ideal of $G$ in the polynomial ring
$R=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}]$ denoted by $I_{\lambda}$.

In [4], it is shown that

(1)
$I_{\lambda}=\cap_{i=1}^{n+1}(x_{1},\ldots,x_{i-1},y_{1},\ldots,y_{\lambda_{i}})$

where, by convenience of notation, we have set $\lambda_{n+1}=0$. Set
$c_{1}=1$, and suppose that

$\lambda_{1}=\ldots=\lambda_{c_{2}-1}>\lambda_{c_{2}}=\ldots=\lambda_{c_{3}-1}>\lambda_{c_{3}}=\ldots=\lambda_{c_{k}-1}>\lambda_{c_{k}}=\ldots=\lambda_{n}.$

Finally set $c_{k+1}=n+1$. Then a minimal prime decomposition of $I_{\lambda}$
can be obtained as follows, by omitting redundant terms from (1):

(2)
$I_{\lambda}=\cap_{i=1}^{k+1}(x_{1},\ldots,x_{c_{i}-1},y_{1},\ldots,y_{\lambda_{c_{i}}})$

######  Theorem 3.6.

Let $G$ be a unmixed Ferrers graph with associated partition
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})$ and let $I_{\lambda}$
be the edge ideal in $R=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ associated
to $G$. Then $I_{\lambda}$ is aCM if and only if $I_{\lambda}$ is connected in
codimension two.

###### Proof.

$(\Longrightarrow).$ This is obvious by Theorem 3.2.
$(\Longleftarrow).$ Let $V_{1}=\\{x_{1},\ldots,x_{n}\\}$ and
$V_{2}=\\{y_{1},\ldots,y_{n}\\}$. Since $I_{\lambda}$ is the Ferrers ideal, by
[4, Corollary 2.2], we have $\operatorname{ht}(I)=\min\\{\min_{1\leq j\leq
n}\\{\lambda_{j}+j-1\\},n\\}$ and $\operatorname{pd}(R/I)=\max_{1\leq j\leq
n}\\{\lambda_{j}+j-1\\}$. By [18, Proposition 2.3] it is enough to show that
$n\leq\lambda_{j}+j-1\leq n+1$ for $j=1,\ldots,n$. By [4, Corollary 2.6], we
have

$\displaystyle\operatorname{Ass}(R/I)=\\{(y_{1},\ldots,y_{n}),(x_{1},\ldots,x_{c_{2}-1},y_{1},\ldots,y_{\lambda_{c_{2}}}),\ldots,(x_{1},\ldots,x_{n})\\}$

Since $I_{\lambda}$ is unmixed we have, $\lambda_{c_{i}}+c_{i}-1=n$. Also
since $I_{\lambda}$ is connected in codimension two, we have
$\lambda_{c_{i}}-\lambda_{c_{i-1}}\leq 2$ and $c_{i}-c_{i-1}\leq 2$. Therefore
for any $j$, there exists $\lambda_{c_{k}}$ such that
$\lambda_{c_{k}}=\lambda_{j}$ and $c_{k}\leq j\leq c_{k}+1$. Therefore
$\max_{1\leq j\leq n}\\{\lambda_{j}+j-1\\}\leq n+1$. Hence
$\operatorname{ht}(I)+1\geq\operatorname{pd}(R/I)$.

——————————————————————————–

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