# The facet ideals of matching complexes of line graphs

Guangjun Zhu${}^{{}^{*}}$, Hong Wang, Yijun Cui  Authors address: School of
Mathematical Sciences, Soochow University, Suzhou 215006, P.R. China
zhuguangjun@suda.edu.cn(Corresponding author:Guangjun Zhu),
651634806@qq.com(Hong Wang), 237546805@qq.com(Yijun Cui).

###### Abstract.

Let $L_{n}$ be a line graph with $n$ edges and ${\mathcal{F}}(L_{n})$ the
facet ideal of its matching complex. In this paper, we provide the irreducible
decomposition of ${\mathcal{F}}(L_{n})$ and some exact formulas for the
projective dimension and the regularity of ${\mathcal{F}}(L_{n})$.

###### Key words and phrases:

matching complex, irreducible decomposition, facet ideal, projective
dimension, regularity

###### 2010 Mathematics Subject Classification:

Primary 13F55; Secondary 13D05, 13C15,13D02

* Corresponding author

## Introduction

Graph complexes have provided an important link between combinatorics and
algebra, topology, and geometry (see e.g.,[4, 17, 23]). Some well-studied
examples are matching complexes and independence complexes of graphs. Let $G$
be a finite simple undirected graph (i.e., loopless and without multiple
edges) with the vertex set $V(G)$ and the edge set $E(G)$. A subset $M\subset
E(G)$ is a matching of $G$ if $M$ contains no adjacent edges. A collection of
all matchings of $G$ forms a simplicial complex $\mathcal{M}(G)$, whose vertex
set is the set of edges of $G$ and whose $k$-dimensional faces are matchings
of size $(k+1)$, we call $\mathcal{M}(G)$ the matching complex of $G$. A
subset $W\subset V(G)$ is called an independent set if no edge of $G$ has both
endpoints in $W$. A collection of all independent subsets of $G$ also forms a
simplicial complex $\text{Ind}(G)$, whose vertex set is $V(G)$ and whose face
set are all independent sets of $G$, we call $\text{Ind}(G)$ to be the
independent complex of $G$.

Topological and geometric properties of the matching complex and independence
complexes of some graphs has been studied by many authors (See [3, 6, 7, 8,
10, 18, 19, 22, 24]). The matching complex of a complete bipartite graph
$K_{m,n}$ was first introduced in the thesis of Garst [11] dealing with Tits
coset complexes. He showed that $\mathcal{M}(K_{m,n})$ is Cohen-Macaulay if
and only if $n\geq 2m-1$. Ziegler [24] strengthened this result by showing
that $\mathcal{M}(K_{m,n})$ is shellable if $n\geq 2m-1$. Consequently,
$\mathcal{M}(K_{m,n})$ has the homotopy type of a wedge of $(m-1)$-spheres
when $n\geq 2m-1$. Ehrenborg-Hetyei [8] and Engström [7] showed that the
independence complex $\text{Ind}(F)$ of a forest $F$ is either contractible or
is homotopy equivalent to a $(\gamma(F)-1)$-sphere, and is also
$\lfloor\frac{n-1}{2d}-1\rfloor$-connected, where $n$ is the number of
vertices, $d$ is the maximal degree of vertices, $\gamma(F)$ denotes the
domination number of $F$, and $\lfloor\frac{n-1}{2d}-1\rfloor$ is the largest
integer $\leq\frac{n-1}{2d}-1$.

Let $\Delta$ be a simplicial complex on the vertex set $V=\\{x_{i}\mid 1\leq
i\leq n\\}$, $\text{Facets}\,(\Delta)$ denote a collection of all facets
(maximal faces under inclusion) of $\Delta$. By identifying the vertex $x_{i}$
with the variable $x_{i}$ in the polynomial ring $S=k[x_{1},\dots,x_{n}]$ over
a field $k$, one can associate $\Delta$ with two squarefree monomial ideals of
$S$

${\mathcal{I}}_{\Delta}=(x_{i_{1}}\cdots
x_{i_{s}}\mid\\{x_{i_{1}},\ldots,x_{i_{s}}\\}\text{\ is not a face of \
}\Delta),$

and

${\mathcal{F}}(\Delta)=(x_{\ell_{1}}\cdots
x_{\ell_{s}}\mid\\{x_{\ell_{1}},\ldots,x_{\ell_{s}}\\}\in\text{Facets}\,(\Delta)).$

They are called the Stanley-Reisner ideal and the facet ideal of $\Delta$
respectively. The Stanley-Reisner ring of $\Delta$ is
$k[\Delta]=S/{\mathcal{I}}_{\Delta}$.

Some algebraic properties of the matching complexes $\mathcal{M}(K_{m,n})$
have been studied in [3, 10, 16]. Björner et al. in [3] showed that
$\operatorname{depth}\,(k[\mathcal{M}(K_{m,n})])=\min\\{m,n,\lfloor\frac{m+n+1}{3}\rfloor\\}$
for any $m,n$. Freidman-Hanlon in [10] showed that
$b_{r-1}(\mathcal{M}(K_{m,n}))\\\ =0$ if and only if $(m-r)(n-r)>r$, and
$b_{v-1}(\mathcal{M}(K_{m,n}))>0$ if and only if $n\geq 2m-4$ or
$(m,n)=(6,6),(7,7),(8,9)$, where $b_{i}(\mathcal{M}(K_{m,n}))$ is the $i$-th
Betti number of $\mathcal{M}(K_{m,n})$, it equals the rank of the homology
group $H_{i}(\mathcal{M}(K_{m,n}))$. Jiang et al. in [16] described the
irreducible decomposition of the facet ideal
${\mathcal{F}}(\mathcal{M}(K_{m,n}))$ of the matching complexe
$\mathcal{M}(K_{m,n})$ and provided some lower bounds for depth and regularity
of the facet ideal ${\mathcal{F}}(\Delta_{m,n})$, where $n\geq m$. They also
showed that these lower bounds can be obtained if $m\leq 3$.

In this article, we are interested in algebraic properties of the facet ideal
${\mathcal{F}}(L_{n})$ of the matching complex $\mathcal{M}(L_{n})$ of a line
graph $L_{n}$ with $n$ edges. By the definition of the matching complex and
the independence complex of a line graph, we can easily obtain that
$\mathcal{M}(L_{n})$ is isomorphic to $\text{Ind}\,(L_{n-1})$, and the
Stanley-Reisner ideal ${\mathcal{I}}_{\mathcal{M}(L_{n})}$ of
$\mathcal{M}(L_{n})$ is actually the edge ideal of a line graph with $n$
vertices, which was studied in [2, 13, 26]. This complex has an extremely
simple structure, but it is still worth discussing, as they appear naturally
in many situations. For some examples, see [17, 19]. The first author in [26,
Theorem 3.3] provided some exact formulas for the projective dimension and the
regularity of ${\mathcal{I}}_{\mathcal{M}(L_{n})}$ for any $n\geq 2$, she
showed that

$\operatorname{pd}\,({\mathcal{I}}_{\mathcal{M}(L_{n})})=\left\\{\begin{array}[]{ll}2p-1&\text{if}\
n=3p\text{\ or\ }n=3p+1,\\\ 2p&\text{if}\ n=3p+2,\end{array}\right.$

and

$\operatorname{reg}\,({\mathcal{I}}_{\mathcal{M}(L_{n})})=\left\\{\begin{array}[]{ll}p+1&\text{if}\
\ n=3p,\text{\ or\ }n=3p+1,\\\ p+2&\text{if}\ \ n=3p+2.\end{array}\right.$

See also [2, Corollary 4.15] and [13, Theorem 4.1]. In this paper, we will
provide the irreducible decomposition and some exact formulas for the
projective dimension and the regularity of the facet ideal
${\mathcal{F}}(L_{n})$ (see Theorem 2.6, Theorem 3.2 and Corollary 3.5).

Our paper is organized as follows. In the preliminary section, we collect
necessary terminology and results from the literature. In Section $2$, we give
the irreducible decomposition of the facet ideal ${\mathcal{F}}(L_{n})$ of a
line graph $L_{n}$ with $n$ edges. In Section $3$, we give some exact formulas
for projective dimension and the regularity of ${\mathcal{F}}(L_{n})$ by some
suitable short exact sequences.

##  1\. Preliminaries

In this section, we gather together the needed notations and basic facts,
which will be used throughout this paper. However, for more details, we refer
the reader to [5, 9, 14, 17].

Let $n$ be a positive integer and $[n]=\\{i\mid 1\leq i\leq n\\}$. In this
paper, we will assume that $L_{n}$ is a line graph with edge set $\\{x_{i}\mid
i\in[n]\\}$. For simplicity, we denote by ${\mathcal{F}}(L_{n})$ the facet
ideal of the matching complex $\mathcal{M}(L_{n})$ of $L_{n}$.

######  Example 1.1.

Let $L_{6}$ be a line graph with six edges $x_{1},\ldots,x_{6}$. Then the
facet ideal of its matching complex is

${\mathcal{F}}(L_{6})=(x_{1}x_{3}x_{5},x_{1}x_{3}x_{6},x_{1}x_{4}x_{6},x_{2}x_{4}x_{6},x_{2}x_{5}).$

A monomial ideal is called irreducible if it cannot be written as proper
intersection of two other monomial ideals. It is called reducible if it is not
irreducible. It is well known that a monomial ideal is irreducible if and only
if it is generated by pure powers of the variables, that is, it has the form
$(x_{i_{1}}^{a_{1}},\ldots,x_{i_{k}}^{a_{k}})$. The following lemma is a
fundamental fact.

######  Lemma 1.2.

([14, Theorem 1.3.1]) Let $I\subset S$ be a monomial ideal. Then there exists
a unique decomposition

$I=Q_{1}\cap\cdots\cap Q_{r}$

such that none of the $Q_{i}$ can be omitted in this intersection and each
$Q_{i}$ is an irreducible monomial ideal. In particular, if $I$ is squarefree,
then each $Q_{i}$ is a minimal (under inclusion) prime ideal over $I$.

This decomposition is called irredundant presentation of $I$ and each $Q_{i}$
is called an irreducible component of $I$.

######  Definition 1.3.

Let $\Delta$ be a simplicial complex on the vertex set $V$. A vertex cover of
$\Delta$ is a subset $C\subset V$ such that each facet of $\Delta$ has at
least one vertex in $C$. Such a vertex cover $C$ is called minimal if no
subset $C^{\prime}\subsetneq C$ is a vertex cover of $\Delta$.

We need the following lemma.

######  Lemma 1.4.

([9, Proposition 1.8]) Let $\Delta$ be a simplicial complex,
${\mathcal{F}}(\Delta)$ its facet ideal. Then $P=(x_{i_{1}},\ldots,x_{i_{s}})$
is a minimal prime ideal over ${\mathcal{F}}(\Delta)$ if and only if
$\\{x_{i_{j}}\mid 1\leq j\leq s\\}$ is a minimal vertex cover for $\Delta$.

Let $I\subset S$ be a non-zero homogeneous ideal and

$0\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p,j}(I)}\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p-1,j}(I)}\rightarrow\cdots\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{0,j}(I)}\rightarrow
I\rightarrow 0$

is a minimal graded free resolution of $I$, where $p\leq n$ and $S(-j)$ is an
$S$-module obtained by shifting the degrees of $S$ by $j$. The number
$\beta_{i,j}(I)$, the $(i,j)$-th graded Betti number of $I$, is an invariant
of $I$ that equals the minimal number of generators of degree $j$ in the $i$th
syzygy module of $I$. Of particular interest is the following invariants which
measure the ¡°size¡± of the minimal graded free resolution of $I$. The
projective dimension of $I$, denoted by $\operatorname{pd}\,(I)$, is defined
to be

$\mbox{pd}\,(I):=\mbox{max}\,\\{i\ |\ \beta_{i,j}(I)\neq 0\\}.$

The regularity of $I$, denoted by $\mbox{reg}\,(I)$, is defined by

$\mbox{reg}\,(I):=\mbox{max}\,\\{j-i\ |\ \beta_{i,j}(I)\neq 0\\}.$

In order to compute the projective dimension and regularity of a non-zero
homogeneous ideal, we shall use the following lemmas in this paper.

######  Lemma 1.5.

([15, Lemmas 2.2 and 3.2]) Let $S_{1}=k[x_{1},\dots,x_{m}]$,
$S_{2}=k[x_{m+1},\dots,x_{n}]$ be two polynomial rings and
$S=S_{1}\otimes_{k}S_{2}$, let $I\subset S_{1}$, $J\subset S_{2}$ be two non-
zero homogeneous ideals. Then

  * (1)

$\operatorname{pd}\,(I+J)=\operatorname{pd}\,(I)+\operatorname{pd}\,(J)+1$,

  * (2)

$\operatorname{reg}\,(I+J)=\operatorname{reg}\,(I)+\operatorname{reg}\,(J)-1$.

######  Lemma 1.6.

[12, Lemma 1.1 and Lemma 1.2] Let $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ be a short exact sequence of finitely generated graded
$S$-modules. Then

  * (1)

$\operatorname{reg}\,(B)\leq\max\,\\{\operatorname{reg}\,(A),\operatorname{reg}\,(C)\\}$,
the equality holds if $\operatorname{reg}\,(A)-1\neq\operatorname{reg}\,(C)$,

  * (2)

$\operatorname{reg}\,(C)\leq\max\,\\{\operatorname{reg}\,(A)-1,\operatorname{reg}\,(B)\\}$,
the equality holds if $\operatorname{reg}\,(A)\neq\operatorname{reg}\,(B)$,

  * (3)

$\operatorname{pd}\,(B)\leq\max\,\\{\operatorname{pd}\,(A),\operatorname{pd}\,(C)\\}$,
the equality holds if $\operatorname{pd}\,(A)+1\neq\operatorname{pd}\,(C)$,

  * (4)

$\operatorname{pd}\,(C)\leq\max\,\\{\operatorname{pd}\,(A)+1,\operatorname{pd}\,(B)\\}$,
the equality holds if $\operatorname{pd}\,(A)\neq\operatorname{pd}\,(B)$.

Let $\mathcal{G}(I)$ denote the minimal set of generators of a monomial ideal
$I\subset S$ and let $u\in S$ be a monomial, we set
$\mbox{supp}(u)=\\{x_{i}:x_{i}|u\\}$. If
$\mathcal{G}(I)=\\{u_{1},\ldots,u_{m}\\}$, we set
$\mbox{supp}(I)=\bigcup\limits_{i=1}^{m}\mbox{supp}(u_{i})$. The following
lemma is well known.

######  Lemma 1.7.

Let $I,J=(u)$ be two monomial ideals such that
$\mbox{supp}(u)\cap\mbox{supp}(I)=\emptyset$, where $u$ is a monomial of
degree $m$. Then

  * (1)

$\mbox{reg}\,(J)=m$,

  * (2)

$\mbox{pd}\,(uI)=\mbox{pd}\,(I)$,

  * (3)

$\mbox{reg}\,(uI)=\mbox{reg}\,(I)+m$.

##  2\. The irreducible decompositions of the facet ideals of the matching
complexes of line graphs

In this section, we will give the irreducible decomposition of the facet ideal
${\mathcal{F}}(L_{n})$ of the matching complex of a line graph $L_{n}$ with
$n$ edges.

Let $k,n$ be two positive integers with $n>k$, $L^{\prime}_{n-k}$ denote the
line graph with edge set $\\{x_{i}\mid i\in[n]\setminus[k]\\}$.

######  Remark 2.1.

Let $n$ be a positive integer, $L_{n}$ a line graph with edge set
$\\{x_{i}\mid i\in[n]\\}$. Let $\mathcal{M}(L_{n})$ be the matching complex of
$L_{n}$. Then $F\in\text{Facets}\,(\mathcal{M}(L_{n}))$ if and only if $F$
satisfies the following conditions:

  * (1)

If $n=1$, then $F=\\{x_{1}\\}$;

  * (2)

If $n=2$, then $F=\\{x_{1}\\}$ or $F=\\{x_{2}\\}$;

  * (3)

If $n=3$, then $F=\\{x_{1},x_{3}\\}$ or $F=\\{x_{2}\\}$;

  * (4)

If $n\geq 4$, then $F=\\{x_{i_{j}}\mid j\in[k]\\}$, where $i_{1}\in\\{1,2\\}$,
$i_{k}\in\\{n-1,n\\}$ and $i_{j}-i_{j-1}\in\\{2,3\\}$ for any $2\leq j\leq k$.

In particular, if $n\geq 4$, then $F\in\text{Facets}\,(\mathcal{M}(L_{n}))$ if
and only if
$F\setminus\\{x_{1}\\}\in\text{Facets}\,(\mathcal{M}(L^{\prime}_{n-2}))$ when
$x_{1}\in F$, or
$F\setminus\\{x_{2}\\}\in\text{Facets}\,(\mathcal{M}(L^{\prime}_{n-3}))$ when
$x_{2}\in F$. Similarly, if $n\geq 4$, then
$F\in\text{Facets}\,(\mathcal{M}(L_{n}))$ if and only if
$F\setminus\\{x_{n-1}\\}\in\text{Facets}\,(\mathcal{M}(L_{n-3}))$ when
$x_{n-1}\in F$, or
$F\setminus\\{x_{n}\\}\in\text{Facets}\,(\mathcal{M}(L_{n-2}))$ when $x_{n}\in
F$.

From the remark above, we have

######  Remark 2.2.

Let $n\geq 2$ be an integer, $L_{n}$ a line graph as in Remark 2.1. Let
${\mathcal{F}}(L_{n})$ be the facet ideal of the simplicial complex
$\mathcal{M}(L_{n})$. Then

$\mathcal{G}({\mathcal{F}}(L_{n}))=\mathcal{G}(x_{1}{\mathcal{F}}(L^{\prime}_{n-2}))\cup\mathcal{G}(x_{2}{\mathcal{F}}(L^{\prime}_{n-3}))=\mathcal{G}(x_{n}{\mathcal{F}}(L_{n-2}))\cup\mathcal{G}(x_{n-1}{\mathcal{F}}(L_{n-3})),$

where
$\\{x_{1}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-3}))=\\{x_{2}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-2}))=\\{x_{n}\\}\cap\text{supp}\,({\mathcal{F}}(L_{n-3}))=\\{x_{n-1}\\}\cap\text{supp}\,({\mathcal{F}}(L_{n-2}))=\emptyset$.
For the convenience of marking, we stipulate
${\mathcal{F}}(L^{\prime}_{-1})={\mathcal{F}}(L^{\prime}_{0})={\mathcal{F}}(L_{-1})={\mathcal{F}}(L_{0})=S$.

######  Lemma 2.3.

Let $k,n$ be two positive integers with $n>k\geq 3$ and
$C\subseteq\\{x_{i}\mid i\in[k-1]\\}$. Then $C$ is a minimal vertex cover of
$\mathcal{M}(L_{k})$ if and only if $C$ is a minimal vertex cover of
$\mathcal{M}(L_{n})$.

###### Proof.

It is enough to show that $C$ is a vertex cover of $\mathcal{M}(L_{k})$ if and
only if $C$ is a vertex cover of $\mathcal{M}(L_{n})$.

$(\Rightarrow)$ Let
$F=\\{x_{i_{1}},\ldots,x_{i_{j}},x_{i_{j+1}},\ldots,x_{i_{\ell}}\\}\in\text{Facets}\,(\mathcal{M}(L_{n}))$
with $1\leq i_{1}<\cdots<i_{j}\leq k<i_{j+1}<\cdots<x_{i_{\ell}}\leq n$. then
$i_{j+1}-i_{j}\in\\{2,3\\}$ by Remark 2.1. It follows that
$i_{j}\in\\{k-2,k-1,k\\}$. Take $F^{\prime}=\\{x_{i_{t}}\mid 1\leq t\leq
j\\}$.

Claim: $F^{\prime}\cap C\neq\emptyset$. Thus $F\cap C\neq\emptyset$, as
desired.

Indeed, if $i_{j}\in\\{k-1,k\\}$, then
$F^{\prime}\in\text{Facets}\,(\mathcal{M}(L_{k}))$. This implies that
$F^{\prime}\cap C\neq\emptyset$ because $C$ is a vertex cover of
$\mathcal{M}(L_{k})$. If $i_{j}=k-2$, then
$F^{\prime}\cup\\{x_{k}\\}\in\text{Facets}\,(\mathcal{M}(L_{k}))$. Thus
$F^{\prime}\cap C\neq\emptyset$ since $x_{k}\notin C$ and $C$ is a vertex
cover of $\mathcal{M}(L_{k})$.

$(\Leftarrow)$ Let $F\in\text{Facets}\,(\mathcal{M}(L_{k}))$, the $F$ can
always be extended to a facet $F^{\prime}$ in $\mathcal{M}(L_{n})$. Thus
$F^{\prime}\cap C\neq\emptyset$ since $C$ is a vertex cover of
$\mathcal{M}(L_{n})$. By hypothesis that
$C\subseteq\\{x_{1},\ldots,x_{k-1}\\}$, we obtain that if $x_{j}\in C\cap
F^{\prime}$ then $x_{j}\in F\cap C$, as desired.

By direct calculation, we obtain

######  Lemma 2.4.

Let $L_{n}$ be a line graph with edge set $\\{x_{i}\mid i\in[n]\\}$ where
$1\leq n\leq 5$. Then $P$ is a minimal prime over ${\mathcal{F}}(L_{n})$ if
and only if $P$ satisfies the following conditions:

  * (1)

If $n=1$, then $P=(x_{1})$;

  * (2)

If $n=2$, then $P=(x_{1},x_{2})$;

  * (3)

If $n=3$, then $P=(x_{1},x_{2})$ or $P=(x_{2},x_{3})$;

  * (4)

If $n=4$, then $P=(x_{1},x_{2})$ or $P=(x_{3},x_{4})$ or $P=(x_{1},x_{4})$;

  * (5)

If $n=5$, then $P=(x_{1},x_{2})$ or $P=(x_{4},x_{5})$ or
$P=(x_{2},x_{3},x_{4})$.

Let $\Omega_{n}$ be a collection of all minimal vertex covers of
$\mathcal{M}(L_{n})$ with $n\in[5]$. Now, we assume that $n\geq 6$ is an
integer. We write $n$ as $n=3p+d$, where $p\geq 2$ and $0\leq d\leq 2$. Set

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle\\{x_{1},x_{2}\\},\ \ \
A^{\prime}_{1}=\\{x_{n-1},x_{n}\\};$ $\displaystyle A_{i}$ $\displaystyle=$
$\displaystyle\\{x_{i},x_{i+1},x_{i+2}\\},\ \text{ for\ }2\leq i\leq n-3;$
$\displaystyle B_{j}$ $\displaystyle=$ $\displaystyle\\{x_{3\ell-2}\mid
1\leq\ell\leq j+1\\}\cup\\{x_{3j+2}\\},\ \text{ for\ }1\leq j\leq p-1;$
$\displaystyle B^{\prime}_{j}$ $\displaystyle=$
$\displaystyle\\{x_{n-3(\ell-1)}\mid 1\leq\ell\leq
j+1\\}\cup\\{x_{n-3j-1}\\},\ \text{ for\ }1\leq j\leq p-1;$ $\displaystyle D$
$\displaystyle=$ $\displaystyle\\{x_{3j+1}\mid 0\leq j\leq p\\},\text{\ if\
}n=3p+1.$ $\Omega_{n}=\left\\{\begin{array}[]{l@{\ \
}l}\\{A_{1},A^{\prime}_{1},A_{2},A_{3},B_{1},B^{\prime}_{1}\\}&\text{if\
}n=6,\\\
\\{A_{1},A^{\prime}_{1},A_{2},A_{3},A_{4},B_{1},B^{\prime}_{1},D\\}&\text{if\
}n=7,\end{array}\right.$

When $n\geq 8$. If $n=3p$ or $3p+1$, we set
$C_{k\ell}=\\{x_{k},x_{k+3\ell+2}\\}\cup\\{x_{k+3j-2}\mid 1\leq
j\leq\ell+1\\}$ for $k\in[n-3(\ell+1)]\setminus[1],\ell\in[p-2]$. If $n=3p+2$,
we set $C_{k\ell}=\\{x_{k},x_{k+3\ell+2}\\}\cup\\{x_{k+3j-2}\mid 1\leq
j\leq\ell+1\\}$, for $k\in[n-3(\ell+1)]\setminus[1],\ell\in[p-1]$. Put

$\Omega_{n}=\left\\{\begin{array}[]{l@{\ \
}l}\left\\{A_{i},A^{\prime}_{1},B_{j},B^{\prime}_{j},C_{k\ell}\mid\begin{subarray}{l}\text{for\
}i\in[n-3],\,j\in[p-1],\\\ k\in[n-3(\ell+1)]\setminus[1]\text{\ and\
}\ell\in[p-2]\end{subarray}\right\\}&\ \text{if $n=3p$},\\\
\left\\{A_{i},A^{\prime}_{1},B_{j},B^{\prime}_{j},C_{k\ell},D\mid\begin{subarray}{l}\text{for\
}i\in[n-3],\,j\in[p-1],\\\ k\in[n-3(\ell+1)]\setminus[1]\text{\ and\
}\ell\in[p-2]\end{subarray}\right\\}&\ \text{if $n=3p+1$},\\\
\left\\{A_{i},A^{\prime}_{1},B_{j},B^{\prime}_{j},C_{k\ell}\mid\begin{subarray}{l}\text{for\
}i\in[n-3],\,j\in[p-1],\\\ k\in[n-3(\ell+1)]\setminus[1]\text{\ and\
}\ell\in[p-1]\end{subarray}\right\\}&\ \text{if $n=3p+2$}.\end{array}\right.$

By Lemmas 1.4, 2.3 and the definition of $\Omega_{n}$, we have

######  Remark 2.5.

Let $s,n$ be two positive integers with $n>s\geq 3$. Then

$\\{C\in\Omega_{n}\mid
C\subseteq\\{x_{i}\,|\,i\in[s-1]\\}\\}\subseteq\Omega_{s}.$

######  Theorem 2.6.

Let $n$ be a positive integer, $L_{n}$ a line graph with edge set
$\\{x_{i}\mid i\in[n]\\}$. Then $P$ is a minimal prime over
${\mathcal{F}}(L_{n})$ if and only if $P$ is generated by $C$ where
$C\in\Omega_{n}$.

###### Proof.

Since ${\mathcal{F}}(L_{n})$ is a squarefree monomial ideal, we obtain by [14,
Corollary 1.3.6] that

${\mathcal{F}}(L_{n})=\bigcap\limits_{P\in\text{Min}\,({\mathcal{F}}(L_{n}))}P$

where $\text{Min}\,({\mathcal{F}}(L_{n}))$ is the set of minimal prime ideals
over ${\mathcal{F}}(L_{n})$. Hence it is enough to show that $\Omega_{n}$ is a
collection of all minimal vertex covers of $\mathcal{M}(L_{n})$ by Lemma 1.4.
We apply induction on $n$. Cases $1\leq n\leq 5$ follow from Lemma 2.4. Cases
$n=6,7$ are obtained by direct calculation. Suppose that $n\geq 8$ and that
the statement holds for any integer $6\leq t\leq n-1$. Now we prove that the
statement holds for $n$.

First, we will show that $C$ is a minimal vertex cover of $\mathcal{M}(L_{n})$
for any $C\in\Omega_{n}$. Let $T_{n}$ be the set of minimal vertex covers of
$\mathcal{M}(L_{n})$.

We distinguish into the following four cases:

(1) If $C=A_{1}$ or $C=A^{\prime}_{1}$, then it is obvious $C\in T_{n}$ from a
fact

$\mathcal{G}({\mathcal{F}}(L_{n}))=\mathcal{G}(x_{1}{\mathcal{F}}(L^{\prime}_{n-2}))\cup\mathcal{G}(x_{2}{\mathcal{F}}(L^{\prime}_{n-3}))=\mathcal{G}(x_{n}{\mathcal{F}}(L_{n-2}))\cup\mathcal{G}(x_{n-1}{\mathcal{F}}(L_{n-3})).$

(2) If $C\cap A^{\prime}_{1}=\emptyset$. Then one can easily see that

$C\notin\left\\{\begin{array}[]{l@{\ \
}l}\left\\{A^{\prime}_{1},A_{n-3},B_{p-1},B^{\prime}_{j},C_{n-3(\ell+1),\ell}\mid\begin{subarray}{c}\text{for\
}\ j\in[p-1],\\\ \ell\in[p-2]\end{subarray}\right\\}&\text{if\ }n=3p,\\\
\left\\{A^{\prime}_{1},A_{n-3},D,B^{\prime}_{j},C_{n-3(\ell+1),\ell}\mid\begin{subarray}{c}\text{for\
}\ j\in[p-1],\\\ \ell\in[p-2]\end{subarray}\right\\}&\text{if\ }n=3p+1,\\\
\left\\{A^{\prime}_{1},A_{n-3},B^{\prime}_{j},C_{n-3(\ell+1),\ell}\mid\text{for\
}j,\ell\in[p-1]\right\\}&\text{if\ }n=3p+2.\end{array}\right.$

By the definition of $\Omega_{n-1}$, it is easy to see that
$C\in\Omega_{n-1}$. Thus $C\in T_{n-1}$ by inductive hypothesis. Since
$C\subseteq\\{x_{i}\mid i\in[n-2]\\}$, we get $C\in T_{n}$ by Lemma 2.3.

(3) If $C\cap A_{1}=\emptyset$. Let $y_{i}=x_{n+1-i}$ for any $i\in[n]$, then
$L_{n}$ is not only a line graph with edges $y_{1},\ldots,y_{n}$, but also
satisfies $C\cap\\{y_{n-1},y_{n}\\}=\emptyset$. It follows that $C\in T_{n}$
from the proof of (2) above.

(4) If $C\cap A_{1}\neq\emptyset$ and $C\cap A^{\prime}_{1}\neq\emptyset$. One
can obtain that

$C\in\left\\{\begin{array}[]{l@{\ \
}l}\left\\{B_{p-1},B^{\prime}_{p-1}\right\\}&\text{if\ }n=3p,\\\
\left\\{D\right\\}&\text{if\ }n=3p+1,\\\ \left\\{C_{2,p-1}\right\\}&\text{if\
}n=3p+2.\end{array}\right.$

We divide into the following three cases:

(i) When $n=3p$. If $C=B_{p-1}$, then

$C=\\{x_{3\ell-2}\mid\ell\in[p]\\}\cup\\{x_{3p-1}\\}=\\{x_{3\ell-2}\mid\ell\in[p]\\}\cup\\{x_{n-1}\\}.$

Let $C^{\prime}=\\{x_{3\ell-2}\mid\ell\in[p]\\}$, then
$C=C^{\prime}\sqcup\\{x_{n-1}\\}$ and $C^{\prime}\in\Omega_{n-2}$ by the
definition of $\Omega_{n-2}$. Hence $C^{\prime}\in T_{n-2}$ by inductive
hypothesis. For any $F\in\text{Facets}\,(\mathcal{M}(L_{n}))$, it can be
written as $F_{1}\cup\\{x_{n}\\}$ or $F_{2}\cup\\{x_{n-1}\\}$, where
$F_{1}\in\text{Facets}\,(\mathcal{M}(L_{n-2}))$ and
$F_{2}\in\text{Facets}\,(\mathcal{M}(L_{n-3}))$. It follows that $C\in T_{n}$.

If $C=B^{\prime}_{p-1}$. Let $y_{i}=x_{n+1-i}$ for each $i$, then $L_{n}$ is
also a line graph with edges $y_{1},\ldots,y_{n}$. In this case, one has

$C=\\{y_{3\ell-2}\mid\ell\in[p]\\}\cup\\{y_{3p-1}\\}=\\{y_{3\ell-2}\mid\ell\in[p]\\}\cup\\{y_{n-1}\\}.$

It follows that $C\in T_{n}$ from the above proof.

(ii) If $n=3p+1$, then $C=D=\\{x_{3j-2}\mid j\in[p+1]\\}$. Let
$C^{\prime\prime}=\\{x_{3j-2}\mid j\in[p]\\}$, then
$C=C^{\prime\prime}\sqcup\\{x_{n}\\}$ and $C^{\prime\prime}\in\Omega_{n-3}$ by
the definition of $\Omega_{n-3}$. By inductive hypothesis, we obtain
$C^{\prime\prime}\in T_{n-3}$. It follows that $C\in T_{n}$ from the proof of
(i).

(iii) If $n=3p+2$, then $C=C_{2,p-1}=\\{x_{2},x_{3p+1}\\}\cup\\{x_{3j}\mid
j\in[p]\\}=\\{x_{2},x_{n-1}\\}\cup\\{x_{3j}\mid j\in[p]\\}$. Let
$C^{\prime\prime\prime}=\\{x_{3j}\mid j\in[p]\\}$. According to (ii), we
obtain that $C^{\prime\prime\prime}$ is a minimal vertex cover of
$\mathcal{M}(L^{\prime\prime}_{n-4})$, where $L^{\prime\prime}_{n-4}$ is a
line graph with edge set $\\{x_{i}\mid 3\leq i\leq n-2\\}$. Note that

$\displaystyle\mathcal{G}({\mathcal{F}}(L_{n}))$ $\displaystyle=$
$\displaystyle\mathcal{G}(x_{1}{\mathcal{F}}(L^{\prime}_{n-2}))\cup\mathcal{G}(x_{2}{\mathcal{F}}(L^{\prime}_{n-3}))$
$\displaystyle=$
$\displaystyle\mathcal{G}(x_{1}x_{n}{\mathcal{F}}(L^{\prime\prime}_{n-4}))\cup\mathcal{G}(x_{1}x_{n-1}{\mathcal{F}}(L^{\prime\prime}_{n-5}))\cup\mathcal{G}(x_{2}{\mathcal{F}}(L^{\prime}_{n-3})),$

thus for any $F\in\text{Facets}\,(\mathcal{M}(L_{n}))$, it can be written in
one of the following three different forms:

$(a)\ \ F_{1}\cup\\{x_{1},x_{n}\\},\ (b)\ \ F_{2}\cup\\{x_{1},x_{n-1}\\},\
(c)\ \ F_{3}\cup\\{x_{2}\\},$

where $F_{1}\in\text{Facets}\,(\mathcal{M}(L^{\prime\prime}_{n-4}))$,
$F_{2}\in\text{Facets}\,(\mathcal{M}(L^{\prime\prime}_{n-5}))$,
$F_{3}\in\text{Facets}\,(\mathcal{M}(L^{\prime}_{n-3}))$,
$L^{\prime\prime}_{n-4}$, $L^{\prime\prime}_{n-5}$ and $L^{\prime}_{n-3}$ are
line graphs with $\\{x_{i}\mid 3\leq i\leq n-2\\}$, $\\{x_{i}\mid 3\leq i\leq
n-3\\}$, and $\\{x_{i}\mid 4\leq i\leq n\\}$ as edge sets respectively. Hence
$C\in T_{n}$.

Next, we will show that any vertex cover $C$ of $\mathcal{M}(L_{n})$ must
contain some element in $\Omega_{n}$. Thus $\Omega_{n}$ is exactly a
collection of all minimal vertex covers of $\mathcal{M}(L_{n})$, as desired.

We divide into the following two cases:

(1) If $A^{\prime}_{1}\cap C=\emptyset$, then $C\subseteq\\{x_{i}\mid
i\in[n-2]\\}$. It follows that $C$ is a vertex cover of $\mathcal{M}(L_{n-1})$
by Lemma 2.3. By induction hypothesis, we obtain $C^{\prime}\subseteq C$ for
some $C^{\prime}\in\Omega_{n-1}$. According to Remark 2.5, one has
$C^{\prime}\in\Omega_{n}$ because of $C^{\prime}\subseteq\\{x_{i}\mid
i\in[n-2]\\}$, as desired.

(2) If $A^{\prime}_{1}\cap C\neq\emptyset$, then $C$ satisfies one of the
following three conditions:

$(i)\ A^{\prime}_{1}\cap C=A^{\prime}_{1},\ \ (ii)\ A^{\prime}_{1}\cap
C=\\{x_{n}\\},\ \ (iii)\ A^{\prime}_{1}\cap C=\\{x_{n-1}\\}.$

$(i)$ If $A^{\prime}_{1}\cap C=A^{\prime}_{1}$, obviously $C\supseteq
A^{\prime}_{1}$.

$(ii)$ If $A^{\prime}_{1}\cap C=\\{x_{n}\\}$. Set

$q=\max\\{\ell\mid\,x_{n-3s}\in C\ \text{for any\ }0\leq s\leq\ell\\}.$

It follows that $\\{x_{n-3s}\mid 0\leq s\leq q\\}\subseteq C$.

We consider the following three cases:

(a) If $n-3q=1$, then $q=p$ from the notation of $n$. In this case,
$C\supseteq D$.

(b) If $n-3q\geq 2$ and there exists some $x_{n-3s-1}\in C$ with $s\in[q]$,
then

$C\supseteq\\{x_{n-3j}\mid 0\leq j\leq s\\}\cup\\{x_{n-3s-1}\\}.$

Hence

$C\supseteq\left\\{\begin{array}[]{l@{\ \ }l}A_{1}&\text{if\ }n=3q+2,\\\
B^{\prime}_{s}&\text{otherwise}.\end{array}\right.$

(c) If $n-3q\geq 2$ and $x_{n-3s-1}\notin C$ for any $s\in[q]$, then $n-3q\geq
6$. Indeed, we choose $F=\\{x_{n+2-3j}\mid\,j\in[q+1]\\}$, then $F\cap
C=\emptyset$. If $n-3q=2$ or $3$, then
$F\in\text{Facets}\,(\mathcal{M}(L_{n}))$, contradicting with the assumption
that $C$ is a vertex cover of $\mathcal{M}(L_{n})$. If $n-3q=4$ or $5$, then
$F\cup\\{x_{n-3q-3}\\}\in\text{Facets}\,(\mathcal{M}(L_{n}))$, and
$x_{n-3q-3}\notin C$ by the choice of $q$, a contradiction again.

Let $C^{\prime}=C\setminus\\{x_{i}\mid n-3q-2\leq i\leq n\\}$. Claim:
$C^{\prime}$ is a vertex cover of $\mathcal{M}(L_{n-3q-3})$. It follows that
$C^{\prime}\supseteq\Gamma$ for some $\Gamma\in\Omega_{n-3q-3}$ by induction
hypothesis. Note a fact that $x_{n-3q-3}\notin C$, which implies
$\Gamma\subseteq\\{x_{i}|i\in[n-3q-4]\\}$. Hence $\Gamma\in\Omega_{n}$ by
Lemma 2.3.

The proof of Claim: If the assertion does not hold, then there exists some
$F^{\prime}\in\text{Facets}\,(\mathcal{M}(L_{n-3q-3}))$ such that
$F^{\prime}\cap C^{\prime}=\emptyset$. According to the definition of matching
complex, we get $\max\\{i\mid x_{i}\in F^{\prime}\\}=n-3q-3,\text{or}\
n-3q-4$. Choose $\Lambda=F^{\prime}\cup
F=F^{\prime}\cup\\{x_{n+2-3j}\mid\,j\in[q+1]\\}$. Then
$\Lambda\in\text{Facets}\,(\mathcal{M}(L_{n}))$, and $\Lambda\cap C=\emptyset$
by the supposition that $x_{n-3s-1}\notin C$ for any $s\in[q]$ and
$F^{\prime}\cap C^{\prime}=\emptyset$. It contradicts with the assumption that
$C$ is a vertex cover of $\mathcal{M}(L_{n})$.

$(iii)$ If $A^{\prime}_{1}\cap C=\\{x_{n-1}\\}$, then $C^{\prime}$ is a vertex
cover of $\mathcal{M}(L_{n-2})$ by Remark 2.1, where
$C^{\prime}=C\setminus\\{x_{n-1}\\}$. Hence $C^{\prime}\supseteq\Gamma$ for
some $\Gamma\in\Omega_{n-2}$ by induction hypothesis. We divide into the
following two cases:

(a)If $x_{n-2}\notin\Gamma$, then, by Lemma 2.3, we have $\Gamma\in\Omega_{n}$
because of $\Gamma\subseteq\\{x_{i}\mid\,i\in[n-3]\\}$, as desired.

(b) If $x_{n-2}\in\Gamma$, then, from the definition of $\Omega_{n-2}$ and a
fact $\Gamma\in\Omega_{n-2}$, it can be obtained that

$\Gamma\in\left\\{\begin{array}[]{l@{\ \
}l}\left\\{A^{\prime\prime}_{1},B^{\prime\prime}_{j},D^{\prime}\mid\,j\in[p^{\prime}-1]\right\\}&\text{if\
}n=3p\\\
\left\\{A^{\prime\prime}_{1},B^{\prime\prime}_{j}\mid\,j\in[p^{\prime}-1]\right\\}&\text{if\
}n=3p+1\ \text{or}\ 3p+2,\end{array}\right.$

where $A^{\prime\prime}_{1}=\\{x_{n-3},x_{n-2}\\}$,
$B^{\prime\prime}_{j}=\\{x_{n+1-3\ell}\mid\ell\in[j+1]\\}\cup\\{x_{n-3(j+1)}\\}$
for any $j\in[p^{\prime}-1]$ with $p^{\prime}=\left\\{\begin{array}[]{l@{\ \
}l}p-1&\text{if\ }n=3p\ \text{or}\ 3p+1\\\ p&\text{if\
}n=3p+2\end{array}\right.$, and $D^{\prime}=\\{x_{3\ell-2}\mid\ell\in[p-1]\\}$
when $n=3p$.

If $\Gamma=A^{\prime\prime}_{1}$, then
$\Gamma\cup\\{x_{n-1}\\}=A_{n-3}\in\Omega_{n}$; If
$\Gamma=B^{\prime\prime}_{j}$ for some $j\in[p^{\prime}-1]$, then
$\Gamma\cup\\{x_{n-1}\\}=C_{n-3j-3,\,j}\in\Omega_{n}$; If $n=3p$ and
$\Gamma=D^{\prime}$, thus $\Gamma\cup\\{x_{n-1}\\}=B_{p-1}\in\Omega_{n}$.

In all cases it follows that $\Gamma\cup\\{x_{n-1}\\}\in\Omega_{n}$, as
desired. This finishes the proof.

Given an ideal $I\subset S$, we set

$\text{bight}\,(I)=\sup\\{\operatorname{height}\,(P)\mid P\text{ is a minimal
prime ideal of }S\text{ over }I\\}.$

As a consequence of the above theorem, we have

######  Corollary 2.7.

Let $n$ be a positive integer, $L_{n}$ a line graph with edge set
$\\{x_{i}\mid i\in[n]\\}$. Then

  1. (a)

$\operatorname{height}\,({\mathcal{F}}(L_{2}))=1$ and
$\operatorname{height}\,({\mathcal{F}}(L_{n}))=2$ for any $n\geq 3$;

  2. (b)

$\text{{\em
bight}}\,({\mathcal{F}}(L_{n}))=\left\\{\begin{array}[]{ll}p+1&\text{if}\ \
n=3p\text{\ or\ }n=3p+1,\\\ p+2&\text{if}\ \ n=3p+2.\end{array}\right.$

##  3\. Projective dimension and regularity of facet ideals of the matching
complexes of line graphs

In this section, we will provide some formulas for the projective dimension
and regularity of the facet ideal ${\mathcal{F}}(L_{n})$ of the matching
complex of a line graph $L_{n}$ with n edges.

######  Lemma 3.1.

Let $n\geq 5$ be an integer, $L_{n}$ a line graph with edge set $\\{x_{i}\mid
i\in[n]\\}$. Let ${\mathcal{F}}(L_{n})$ be the facet ideal of the simplicial
complex $\mathcal{M}(L_{n})$. Then

${\mathcal{F}}(L_{n})=J_{n}+K_{n}\ \text{and \ }J_{n}\cap
K_{n}=x_{1}x_{2}P_{n},$

where $J_{n}=x_{1}{\mathcal{F}}(L^{\prime}_{n-2})$,
$K_{n}=x_{2}{\mathcal{F}}(L^{\prime}_{n-3})$,
$P_{n}=x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+x_{3}x_{5}{\mathcal{F}}(L^{\prime}_{n-6})$
and we stipulate
${\mathcal{F}}(L^{\prime}_{-1})={\mathcal{F}}(L^{\prime}_{0})=S$.

###### Proof.

It’s obvious that ${\mathcal{F}}(L_{n})=J_{n}+K_{n}$ by Remark 2.2. By direct
calculation, we obtain $J_{5}\cap
K_{5}=x_{1}x_{2}(x_{3}x_{5},x_{4})=x_{1}x_{2}P_{5}$ and $J_{6}\cap
K_{6}=x_{1}x_{2}(x_{3}x_{5},x_{4}x_{6})=x_{1}x_{2}P_{6}$, as desired.

Assume that $n\geq 7$. By repeated using Remark 2.2, for all $1\leq i\leq 4$,
we have

${\mathcal{F}}(L^{\prime}_{n-i})=x_{i+1}{\mathcal{F}}(L^{\prime}_{n-i-2})+x_{i+2}{\mathcal{F}}(L^{\prime}_{n-i-3}),$

where
$\\{x_{i+1}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-i-3}))=\\{x_{i+2}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-i-2}))=\emptyset$.
One has

$J_{n}\cap
K_{n}=x_{1}x_{2}({\mathcal{F}}(L^{\prime}_{n-2})\cap{\mathcal{F}}(L^{\prime}_{n-3})).$

By comparing the expressions of $J_{n}\cap K_{n}$, we get
$P_{n}={\mathcal{F}}(L^{\prime}_{n-2})\cap{\mathcal{F}}(L^{\prime}_{n-3})$. It
follows that

$\displaystyle P_{n}$ $\displaystyle=$
$\displaystyle(x_{3}{\mathcal{F}}(L^{\prime}_{n-4})+x_{4}{\mathcal{F}}(L^{\prime}_{n-5}))\cap(x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+x_{5}{\mathcal{F}}(L^{\prime}_{n-6}))$
$\displaystyle=$ $\displaystyle
x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+(x_{3}{\mathcal{F}}(L^{\prime}_{n-4}))\cap(x_{5}{\mathcal{F}}(L^{\prime}_{n-6}))+(x_{4}{\mathcal{F}}(L^{\prime}_{n-5}))\cap(x_{5}{\mathcal{F}}(L^{\prime}_{n-6}))$
$\displaystyle=$ $\displaystyle
x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+(x_{3}{\mathcal{F}}(L^{\prime}_{n-4}))\cap(x_{5}{\mathcal{F}}(L^{\prime}_{n-6}))$
$\displaystyle=$ $\displaystyle
x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+[x_{3}(x_{5}{\mathcal{F}}(L^{\prime}_{n-6})+x_{6}{\mathcal{F}}(L^{\prime}_{n-7})]\cap(x_{5}{\mathcal{F}}(L^{\prime}_{n-6}))$
$\displaystyle=$ $\displaystyle
x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+x_{3}x_{5}{\mathcal{F}}(L^{\prime}_{n-6})+(x_{3}x_{6}{\mathcal{F}}(L^{\prime}_{n-7}))\cap(x_{5}{\mathcal{F}}(L^{\prime}_{n-6}))$
$\displaystyle=$ $\displaystyle
x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+x_{3}x_{5}{\mathcal{F}}(L^{\prime}_{n-6})+x_{3}x_{5}x_{6}({\mathcal{F}}(L^{\prime}_{n-7})\cap{\mathcal{F}}(L^{\prime}_{n-6}))$
$\displaystyle=$ $\displaystyle
x_{4}{\mathcal{F}}(L^{\prime}_{n-5})+x_{3}x_{5}{\mathcal{F}}(L^{\prime}_{n-6}).$

where the penultimate equality holds because of
$\\{x_{3},x_{6}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-6}))=\\{x_{5}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-7}))=\emptyset$.

######  Theorem 3.2.

Let $n$ be a positive integer, $L_{n}$ a line graph with edge set
$\\{x_{i}\mid i\in[n]\\}$. Let ${\mathcal{F}}(L_{n})$ be the facet ideal of
the simplicial complex $\mathcal{M}(L_{n})$. Then

$\operatorname{pd}\,({\mathcal{F}}(L_{n}))\leq\left\\{\begin{array}[]{ll}p&\text{if}\
n=3p\text{\ or\ }n=3p+1,\\\ p+1&\text{if}\ n=3p+2,\end{array}\right.$

and

$\operatorname{reg}\,({\mathcal{F}}(L_{n}))=\left\\{\begin{array}[]{ll}1&\text{if}\
\ n=1,\\\ 2p&\text{if}\ \ n=3p,\text{\ or\ }n=3p+1\text{\ with \ }p>1,\\\
2p+1&\text{if}\ \ n=3p+2.\end{array}\right.$

Moreover, if $n\geq 5$, then

$\operatorname{pd}(P_{n})\leq\left\\{\begin{array}[]{ll}p-1&\text{if}\
n=3p\text{\ or\ }n=3p+1,\\\ p&\text{if}\ \ n=3p+2,\end{array}\right.$

and

$\operatorname{reg}(P_{n})=\left\\{\begin{array}[]{ll}2p-1&\text{if}\
n=3p\text{\ or\ }n=3p+1,\\\ 2p&\text{if}\ n=3p+2,\end{array}\right.$

where $P_{n}$ as defined in Lemma 3.1.

###### Proof.

Cases $1\leq n\leq 3$ are clear by Lemma 1.5 and case $n=4$ follows from [26,
Theroem 3.7]. Assume that $n\geq 5$. Consider the following short exact
sequence

$None$ $0\longrightarrow J_{n}\cap K_{n}\longrightarrow J_{n}\oplus
K_{n}\longrightarrow{\mathcal{F}}(L_{n})\longrightarrow 0,$

where $J_{n}$ and $K_{n}$, as defined in Lemma 3.1, satisfy $J_{n}\cap
K_{n}=x_{1}x_{2}P_{n}$. Applying Lemma 1.6 (2), (4) and Lemma 1.7 (2), (3) to
the short exact sequence above, we obtain

$\displaystyle\operatorname{pd}\,({\mathcal{F}}(L_{n}))$ $\displaystyle\leq$
$\displaystyle\max\,\\{\operatorname{pd}\,(J_{n}\cap
K_{n})+1,\operatorname{pd}\,(J_{n}),\operatorname{pd}\,(K_{n})\\}$
$\displaystyle=$
$\displaystyle\max\,\\{\operatorname{pd}\,(P_{n})+1,\operatorname{pd}\,({\mathcal{F}}(L^{\prime}_{n-2})),\operatorname{pd}\,({\mathcal{F}}(L^{\prime}_{n-3}))\\},$

and

$\displaystyle\operatorname{reg}\,({\mathcal{F}}(L_{n}))$ $\displaystyle\leq$
$\displaystyle\max\,\\{\operatorname{reg}\,(J_{n}\cap
K_{n})-1,\operatorname{reg}\,(J_{n}),\operatorname{reg}\,(K_{n})\\}$
$\displaystyle=$
$\displaystyle\max\,\\{2+\operatorname{reg}\,(P_{n})-1,1+\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-2})),1+\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-3}))\\},$
$\displaystyle=$ $\displaystyle
1+\max\,\\{\operatorname{reg}\,(P_{n}),\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-2})),\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-3}))\\},$

where equality holds if
$\operatorname{reg}\,(P_{n})\neq\max\,\\{\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-2})),\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-3}))\\}-1$.

Since $L^{\prime}_{n-k}$ is also a line graph with edge set $\\{x_{i}\mid
i\in[n]\setminus[k]\\}$ for any positive integers $n>k$, we have
$\operatorname{pd}\,({\mathcal{F}}(L^{\prime}_{n-k}))=\operatorname{pd}\,({\mathcal{F}}(L_{n-k}))$
and
$\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-k}))=\operatorname{reg}\,({\mathcal{F}}(L_{n-k}))$.
Thus, from the formulas for the projection dimension and regularity above, we
obtain

$\operatorname{pd}\,({\mathcal{F}}(L_{n}))\leq\max\,\\{\operatorname{pd}\,(P_{n})+1,\operatorname{pd}\,({\mathcal{F}}(L_{n-2})),\operatorname{pd}\,({\mathcal{F}}(L_{n-3}))\\},\hskip
71.13188pt(1)$

and

$\operatorname{reg}\,({\mathcal{F}}(L_{n}))\leq
1+\max\,\\{\operatorname{reg}\,(P_{n}),\operatorname{reg}\,({\mathcal{F}}(L_{n-2})),\operatorname{reg}\,({\mathcal{F}}(L_{n-3}))\\},\hskip
62.59596pt(2)$

where equality holds if
$\operatorname{reg}\,(P_{n})\neq\max\,\\{\operatorname{reg}\,({\mathcal{F}}(L_{n-2})),\operatorname{reg}\,({\mathcal{F}}(L_{n-3}))\\}-1$.

We will prove these assertions by induction on $n$.

By direct calculation, one has $P_{5}=(x_{4},x_{3}x_{5})$,
$P_{6}=(x_{4}x_{6},x_{3}x_{5})$, ${\mathcal{F}}(L_{2})=(x_{1},x_{2})$,
${\mathcal{F}}(L_{3})=(x_{1}x_{3},x_{2})$ and
${\mathcal{F}}(L_{4})=(x_{1}x_{3},x_{1}x_{4},x_{2}x_{4})$. It follows that

$\operatorname{pd}\,(P_{5})=\operatorname{pd}\,(P_{6})=\operatorname{pd}\,({\mathcal{F}}(L_{2}))=\operatorname{pd}\,({\mathcal{F}}(L_{3}))=\operatorname{pd}\,({\mathcal{F}}(L_{4}))=1,$
$\operatorname{reg}\,(P_{5})=2,\ \operatorname{reg}\,(P_{6})=3,\
\operatorname{reg}\,({\mathcal{F}}(L_{2}))=1,\
\operatorname{reg}\,({\mathcal{F}}(L_{2}))=\operatorname{reg}\,({\mathcal{F}}(L_{4}))=2.$

Hence, according to the formulas $(1)$ and $(2)$, we have

$\operatorname{pd}\,({\mathcal{F}}(L_{5}))\leq 2,\ \
\operatorname{pd}\,({\mathcal{F}}(L_{6}))\leq 2,\ \
\operatorname{reg}\,({\mathcal{F}}(L_{5}))=3,\ \
\operatorname{reg}\,({\mathcal{F}}(L_{6}))=4.$

This proves the assertion for $n=5$ and $n=6$.

If $n=7$, then $P_{7}=(x_{3}x_{5}x_{7},x_{4}x_{6},x_{4}x_{7})$. It follows
that $(P_{7}:x_{4})=(x_{6},x_{7})$ and
$(P_{7},x_{4})=(x_{3}x_{5}x_{7},x_{4})$. One has
$\operatorname{pd}\,((P_{7}:x_{4}))=\operatorname{pd}\,((P_{7},x_{4}))=1$,
$\operatorname{reg}\,((P_{7}:x_{4}))=1$ and
$\operatorname{reg}\,((P_{7},x_{4}))=3$ by Lemma 1.5 (1) and (3). Applying
Lemma 1.6 to the following short exact sequence

$0\longrightarrow(P_{7}:x_{4})(-1)\stackrel{{\scriptstyle\cdot
x_{4}}}{{\longrightarrow}}P_{7}\longrightarrow(P_{7},x_{4})\longrightarrow 0,$

we obtain $\operatorname{pd}\,(P_{7})=1$ and $\operatorname{reg}\,(P_{7})=3$.
By inductive hypothesis, we have

$\operatorname{pd}\,({\mathcal{F}}(L_{4}))\leq 1,\
\operatorname{pd}\,({\mathcal{F}}(L_{5}))\leq 2,\
\operatorname{reg}\,({\mathcal{F}}(L_{4}))=2,\
\operatorname{reg}\,({\mathcal{F}}(L_{5}))=3.$

It follows that $\operatorname{pd}\,({\mathcal{F}}(L_{7}))\leq 2$ and
$\operatorname{reg}\,({\mathcal{F}}(L_{7}))=4$ by formulas $(1)$ and $(2)$, as
desired.

Assume that $n\geq 8$. Set $M_{n}=x_{4}{\mathcal{F}}(L^{\prime}_{n-5})$ and
$N_{n}=x_{3}x_{5}{\mathcal{F}}(L^{\prime}_{n-6})$. One has

$None$
$\operatorname{pd}\,(M_{n})=\operatorname{pd}\,({\mathcal{F}}(L_{n-5}))\leq
p-1,\ \
\operatorname{pd}\,(N_{n})=\operatorname{pd}\,({\mathcal{F}}(L_{n-6}))\leq
p-2,$ $\displaystyle\hskip 28.45274pt\operatorname{reg}\,(M_{n})$
$\displaystyle=$ $\displaystyle
1+\operatorname{reg}\,({\mathcal{F}}(L_{n-5}))=1+\left\\{\begin{array}[]{ll}2p-3&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ 2(p-1)&\text{if}\ n=3p+2\end{array}\right.$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}2p-3&\text{if}\
n=3p\\\ 2p-2&\text{if}\ n=3p+1\\\ 2p-1&\text{if}\
n=3p+2,\end{array}\right.\hskip 165.02606pt(4)$
$\displaystyle\operatorname{reg}\,(N_{n})$ $\displaystyle=$ $\displaystyle
2+\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-6}))=2+\left\\{\begin{array}[]{ll}2(p-2)&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ 2(p-2)+1&\text{if}\ n=3p+2\end{array}\right.$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}2p-2&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ 2p-1&\text{if}\ n=3p+2.\end{array}\right.\hskip
116.65646pt(5)$

Since
$\\{x_{4}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-6}))=\\{x_{3},x_{5}\\}\cap\text{supp}\,({\mathcal{F}}(L^{\prime}_{n-5}))=\emptyset$,
one has

$P_{n}=M_{n}+N_{n}\ \ \text{and\ \ }M_{n}\cap
N_{n}=x_{3}x_{4}x_{5}({\mathcal{F}}(L^{\prime}_{n-5})\cap{\mathcal{F}}(L^{\prime}_{n-6})).$

Combining Lemma 1.7 (2) and (3), the expressions of $M_{n}\cap N_{n}$ and the
inductive hypothesis, we obtain

$\displaystyle\operatorname{pd}\,(M_{n}\cap N_{n})$ $\displaystyle=$
$\displaystyle\operatorname{pd}\,({\mathcal{F}}(L^{\prime}_{n-5})\cap{\mathcal{F}}(L^{\prime}_{n-6}))=\operatorname{pd}\,({\mathcal{F}}(L^{\prime}_{(n-3)-2})\cap{\mathcal{F}}(L^{\prime}_{(n-3)-3}))$
$\displaystyle=$
$\displaystyle\operatorname{pd}\,(P_{n-3})\leq\left\\{\begin{array}[]{ll}p-2&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ p-1&\text{if}\ \ n=3p+2\hskip
113.81102pt(6)\end{array}\right.$ $\displaystyle\operatorname{reg}\,(M_{n}\cap
N_{n})$ $\displaystyle=$ $\displaystyle
3+\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{n-5})\cap{\mathcal{F}}(L^{\prime}_{n-6}))$
$\displaystyle=$ $\displaystyle
3+\operatorname{reg}\,({\mathcal{F}}(L^{\prime}_{(n-3)-2})\cap{\mathcal{F}}(L^{\prime}_{(n-3)-3}))$
$\displaystyle=$ $\displaystyle
3+\operatorname{reg}\,(P_{n-3})=3+\left\\{\begin{array}[]{ll}2(p-1)-1&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ 2(p-1)&\text{if}\ n=3p+2\end{array}\right.$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}2p&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ 2p+1&\text{if}\ n=3p+2\hskip
170.71652pt(7)\end{array}\right.$

Applying Lemma 1.6 (2), (4) and formulas $(3)\sim(7)$ to the following short
exact sequence

$None$ $0\longrightarrow M_{n}\cap N_{n}\longrightarrow M_{n}\oplus
N_{n}\longrightarrow P_{n}\longrightarrow 0,$

we obtain

$\displaystyle\operatorname{pd}\,(P_{n})$ $\displaystyle\leq$
$\displaystyle\max\,\\{\operatorname{pd}\,(M_{n}\cap
N_{n})+1,\operatorname{pd}\,(M_{n}),\operatorname{pd}\,(N_{n})\\}$
$\displaystyle\leq$ $\displaystyle\left\\{\begin{array}[]{ll}p-1&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ p&\text{if}\ \ n=3p+2\end{array}\right.\hskip
147.95424pt(8)$ $\displaystyle\operatorname{reg}\,(P_{n})$ $\displaystyle=$
$\displaystyle\max\,\\{\operatorname{reg}\,(M_{n}\cap
N_{n})-1,\operatorname{reg}\,(M_{n}),\operatorname{reg}\,(N_{n})\\}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}2p-1&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ 2p&\text{if}\ n=3p+2\end{array}\right.\hskip
142.26378pt(9)$

Combining formulas (1), (2), (7), (8) and inductive hypothesis, one has

$\operatorname{pd}\,({\mathcal{F}}(L_{n}))\leq\left\\{\begin{array}[]{ll}p&\text{if}\
n=3p\text{\ or\ }n=3p+1\\\ p+1&\text{if}\ n=3p+2,\end{array}\right.$

and

$\operatorname{reg}\,({\mathcal{F}}(L_{n}))=\left\\{\begin{array}[]{ll}2p&\text{if}\
\ n=3p,\text{\ or\ }n=3p+1\\\ 2p+1&\text{if}\ \ n=3p+2.\end{array}\right.$

This finishes the proof.

######  Lemma 3.3.

([21, Corollary 3.33]) Let $I\subset S$ be a squarefree monomial ideal, then

$\text{{\em bight}}\,(I)\leq\operatorname{pd}\,(I)+1.$

By Corollary 2.7, Theorem 3.2 and the lemma above, we have

######  Corollary 3.4.

Let $n$ be a positive integer, $L_{n}$ defined as in Lemma 3.1. Then

$\operatorname{pd}\,({\mathcal{F}}(L_{n}))=\left\\{\begin{array}[]{ll}p&\text{if}\
\ n=3p\text{\ \ or\ \ }n=3p+1\\\ p+1&\text{if}\ \ n=3p+2.\end{array}\right.$

An immediate consequence of the above corollary is the following corollary

######  Corollary 3.5.

Let $n$ be a positive integer, $L_{n}$ defined as in Lemma 3.1. Then

$\operatorname{depth}\,({\mathcal{F}}(L_{n}))=\left\\{\begin{array}[]{ll}2p&\text{if}\
\ n=3p\\\ 2p+1&\text{if}\ \ n=3p+1\text{\ \ or\ \ }n=3p+2.\end{array}\right.$

###### Proof.

It follows from Auslander-Buchsbaum formula (see [5, Theorem 1.3.3]).

Acknowledgments

This research is supported by the National Natural Science Foundation of China
(No.11271275) and by foundation of the Priority Academic Program Development
of Jiangsu Higher Education Institutions.

## References

  * [1] C. A. Athanasiadis, Decompositions and connectivity of matching and chessboard complexes, Discrete Comput. Geom., 31 (3) (2004), 395-403.
  * [2] A. Alilooee, S. Faridi, Graded Betti numbers of path ideals of cycles and lines, J. Algebra Appl., 17 (01) (2017), 1850011-1-17.
  * [3] A. Björner, L. Lovász, S. T. Vrećica, and R.T. Z̆ivaljević, Chessboard complexes and matching complexes, J. London Math. Soc., 49 (1994), 25-39.
  * [4] S. Bouc, Homologie de certains ensembles de 2-sous-groups des groupes symmétriques, J. Algebra, 150 (1992), 158-186.
  * [5] W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised Edition, Cambridge University Press, 1998.
  * [6] A. Engström, Independence complexes of claw-free graphs,Eur. J. Combin., 29 (2008), 234-241.
  * [7] A. Engström, Complexes of directed trees and independence complexes, Discrete Math., 309 (2009), 3299-3309.
  * [8] R. Ehrenborga and G. Hetyei, The topology of the independence complex, Eur. J. Combin., 27 (2006), 906-923.
  * [9] S. Faridi, The facet ideal of a simplicial complex, Manuscripta math., 109 (2002), 159-174.
  * [10] J. Freidman and P. Hanlon, On the Betti Numbers of Chessboard Complexes, J. Algebraic Combin., 8 (1998), 193-203.
  * [11] P.F. Garst, Cohen-Macaulay complexes and group actions, Ph.D. Thesis, University of Wisconsin, Madison, WI, 1979.
  * [12] H. T. Hà, N. V. Trung, and T. N. Trung, Depth and regularity of powers of sums of ideals, Math. Z., 282 (3-4) (2016), 819-838.
  * [13] J. He and A. Van Tuyl, Algebraic properties of the path ideal of a tree, Comm. Algbra, 38(5) (2010), 1725-1742.
  * [14] J. Herzog and T. Hibi, Monomial ideals, Springer-Verlag, New York (2011).
  * [15] L. T. Hoa and N. D. Tam, On some invariants of a mixed product of ideals, Arch. Math., 94 (4) (2010), 327-337.
  * [16] C. Jiang, Y. Zhao, H. Wang and G. Zhu, The facet ideals of chessboard complexes, submitted.
  * [17] J. Jonsson, Simplicial Complexes of Graphs, Lecture Notes in Math., vol. 1928, Springer, 2008.
  * [18] Kazuhiro Kawamura, Independence complexes of chordal graphs, Discrete Math., 310 (2010), 2204-2211.
  * [19] K. Kawamura, Homotopy types of independence complexes of forests, Contrib. Discrete Math., 5 (2) (2010), 67-75.
  * [20] M. Marietti, D. Testa, Cores of simplicial complexes, Discrete Comput. Geom., 40 (2008), 444-468.
  * [21] S. Morey, R.H. Villarreal, Edge ideals: algebraic and combinatorial properties, Progress in commutative algebra, 1, 85¨C126, de Gruyter, Berlin, 2012.
  * [22] V. Reiner and J. Roberts, Minimal resolutions and the homology of matching and chessboard complexes, J. Algebraic Combin., 11 (2) (2000), 135-154.
  * [23] M. L. Wachs, Topology of matching, chessboard, and general bounded degree graph complexes, Algebra Universalis, 49 (4) (2003), 345-385.
  * [24] G. M. Ziegler, Shellability of chessboard complexes, Israel J. Math., 87 (1994), 97-110.
  * [25] X. Zheng, Resolutions of facet ideals, Comm. Algebra, 32 (2004), 2301-2324.
  * [26] G. Zhu, Projective dimension and regularity of the path ideal of the line graph, J. Algebra Appl., 17 (4) (2018), 1850068-1-15.