Combinatoral Topology and Descompoibilities to Shellability

#### **Chapter 6**

## Vertex Decomposability of Path Complexes and Stanley' s Conjectures

*Seyed Mohammad Ajdani and Francisco Bulnes*

#### **Abstract**

Monomials are the link between commutative algebra and combinatorics. With a simplicial complex Δ, one can associate two square-free monomial ideals: the Stanley-Reisner ideal *I*<sup>Δ</sup> whose generators correspond to the non-face of Δ, or the facet ideal *I*(Δ) that is a generalization of edge ideals of graphs and whose generators correspond to the facets of Δ. The facet ideal of a simplicial complex was first introduced by Faridi in 2002. Let *G* be a simple graph. The edge ideal *I*(*G*) of a graph *G* was first considered by R. Villarreal in 1990. He studied algebraic properties of *I*(*G*) using a combinatorial language of *G*. In combinatorial commutative algebra, one can attach a monomial ideal to a combinatorial object. Then, algebraic properties of this ideal are studied using combinatorial properties of combinatorial object. One of interesting problems in combinatorial commutative algebra is the Stanley's conjectures. The Stanley's conjectures are studied by many researchers. Let *R* be a *<sup>n</sup>*-graded ring and *M* a *<sup>n</sup>*-graded *R*-module. Then, Stanley conjectured that depthð Þ *M* ≤sdepthð Þ *M :* He also conjectured that each Cohen-Macaulay simplicial complex is partition-able. In this chapter, we study the relation between vertex decomposability of some simplicial complexes and Stanley's conjectures.

**Keywords:** vertex decomposable, simplicial complex, Matroid, path

#### **1. Introduction**

Let *R* ¼ *K x*½ � 1, … , *xn* , where *K* is a field. Fix an integer *n*≥ *t*≥2 and let *G* be a directed graph. A sequence *xi*<sup>1</sup> , … , *xit* of distinct vertices is called a path of length *t* if there are *t* � 1 distinct directed edges *e*1, … ,*et*�<sup>1</sup> where *ej* is a directed edge from *xi <sup>j</sup>* to *xi <sup>j</sup>*þ<sup>1</sup> . Then, the path ideal of *G* of length *t* is the monomial ideal *It*ð Þ¼ *G xi*<sup>1</sup> … *xit* : *xi*<sup>1</sup> , … , *xit* ð is a path of length t in G) in the polynomial ring *R* ¼ *K x*½ � 1, … , *xn* . The distance *d x*ð Þ , *y* of two vertices *x* and *y* of a graph *G* is the length of the shortest path from *x* to *y*. The path complex Δ*t*ð Þ *G* is defined by

Δ*t*ð Þ¼ *G xi*<sup>1</sup> , … , *xit* f g : *xi*<sup>1</sup> , … , *xit* h i is a path of length *t* in *G :*

#### *Advanced Topics of Topology*

Path ideals of graphs were first introduced by Conca and De Negri [1, 2] in the context of monomial ideals of linear type. Recently, the path ideal of cycles has been extensively studied by several mathematicians. In [3], it is shown that *I*2ð Þ *Cn* is sequentially Cohen-Macaulay, if and only if, *n* ¼ 3 or *n* ¼ 5. Generalizing this result, in [4], it is proved that *It*ð Þ *Cn* , (*t*> 2), is sequentially Cohen-Macaulay, if and only if *n* ¼ *t* or *n* ¼ *t* þ 1 or *n* ¼ 2*t* þ 1. Also, the Betti numbers of the ideal *It*ð Þ *Cn* and *It*ð Þ *Ln* is computed explicitly in [5]. In particular, it has been shown that [6, 7]:

**Theorem 1.1** ([1, Corollary 5.15])**.** *Let n, t, p and d be integers such that n* ≥*t*≥2*, n* ¼ ð Þ *t* þ 1 *p* þ *d, where p*≥ 0 *and* 0≤*d*<ð Þ *t* þ 1 *. Then,*

i. *The projective dimension of the path ideal of a graph cycle Cn or line Ln is given by,*

$$\operatorname{pd}(I\_l(\mathbb{C}\_n)) = \begin{cases} 2p, & d \neq 0 \\ 2p - 1, & d = 0 \end{cases} \qquad \operatorname{pd}(I\_l(L\_n)) = \begin{cases} 2p - 1, & d \neq t \\ 2p, & d = t \end{cases} \tag{1}$$

ii. *The regularity of the path ideal of a graph cycle Cn or line Ln is given by,*

$$\begin{aligned} \text{reg}(I\_l(\mathbf{C}\_n)) &= (t - \mathbf{1})p + d + \mathbf{1} \\\\ \text{reg}(I\_l(L\_n)) &= \begin{cases} p(t - \mathbf{1}) + \mathbf{1}, & d < t \\\\ p(t - \mathbf{1}) + t, & d = t \end{cases} \end{aligned} \tag{2}$$

In [8] it has been shown that Δ*t*ð Þ *G* is a simplicial tree if *G* is a rooted tree and *t*≥ 2. One of interesting problems in combinatorial commutative algebra is the Stanley's conjectures. The Stanley's conjectures are studied by many researchers. Let *R* be a *<sup>n</sup>*-graded ring and *M* a *<sup>n</sup>*-graded *R*-module. Then, Stanley [9] conjectured that

$$\text{depth}(M) \le \text{sdepth}(M) \tag{3}$$

He also conjectured in [10] that each Cohen-Macaulay simplicial complex is partitionable. Herzog, Soleyman Jahan, and Yassemi in [11–14] showed that the conjecture about partitionability is a special case of the Stanley's first conjecture. In this chapter, we first study algebraic properties of Δ*t*ð Þ *Cn* . In Section 1, we recall some definitions and results, which will be needed later. In Section 2, for all *t*>2 we show that the following conditions are equivalent:


$$\mathbf{v}. \ n = t \text{ or } t + \mathbf{1}.$$

(see Theorem 2.6).

In Section 3, for all *t* ≥2 we show that Δ*t*ð Þ *G* is vertex decomposable if and only if *G* ¼ *H p*ð Þ , *n*, *q* or *G* ¼ *H p*ð Þ , *n* . In Section 4, vertex decomposability path complexes of Dynkin graphs are shown. In Section 5 as an application of our results, we show

that if *n* ¼ *t* or *t* þ 1 then Δ*t*ð Þ *Cn* is partitionable and Stanley's conjecture holds for *K*½ � Δ*t*ð Þ *Cn* and *K*½ � Δ*t*ð Þ *G* , where *G* ¼ *H p*ð Þ , *n*, *q* or *G* ¼ *H p*ð Þ , *n* .

#### **2. Preliminaries**

In this section, we recall some definitions and results which will be needed later. **Definition 2.1.** *A simplicial complex* Δ over a set of vertices *V* ¼ f g *x*1, … , *xn* , is a collection of subsets of *V*, with the property that:

a. f g *xi* ∈ Δ, for all *i*;

b. if *F* ∈ Δ, then all subsets of *F* are also in Δ (including the empty set).

An element of Δ is called a *face* of Δ and complement of a face *F* is *V*n*F* and it is denoted by *F<sup>c</sup>* . Also, the complement of the simplicial complex Δ ¼ h i *F*1, … , *Fr* is <sup>Δ</sup>*<sup>c</sup>* <sup>¼</sup> *<sup>F</sup><sup>c</sup>* 1, … , *F<sup>c</sup> r* � �. The *dimension* of a face *<sup>F</sup>* of <sup>Δ</sup>, dim *<sup>F</sup>*, is j j *<sup>F</sup>* � 1 where j j *<sup>F</sup>* is the number of elements of *F* and dim ∅ ¼ �1. The faces of dimensions 0 and 1 are called *vertices* and *edges*, respectively. A *non-face* of Δ is a subset *F* of *V* with *F* ∉ Δ. We denote by N ð Þ Δ , the set of all minimal non-faces of Δ. The maximal faces of Δ under inclusion are called *facets* of Δ. The *dimension* of the simplicial complex Δ, dim Δ, is the maximum of dimensions of its facets. If all facets of Δ have the same dimension, then Δ is called *pure*.

Let Fð Þ¼ Δ *F*1, … , *Fq* � � be the facet set of <sup>Δ</sup>. It is clear that <sup>F</sup>ð Þ <sup>Δ</sup> determines Δ completely and we write Δ ¼ *F*1, … , *Fq* � �. A simplicial complex with only one facet is called a *simplex*. A simplicial complex Γ is called a *subcomplex* of Δ, if Fð Þ Γ ⊂ Fð Þ Δ .

For *v*∈*V*, the subcomplex of Δ obtained by removing all faces *F* ∈ Δ with *v*∈*F* is denoted by Δn*v*. That is,

$$
\Delta \backslash \upsilon = \langle F \in \Delta : \quad \upsilon \notin F \rangle. \tag{4}
$$

The *link* of a face *F* ∈ Δ, denoted by linkΔð Þ *F* , is a simplicial complex on *V* with the faces, *G* ∈ Δ such that, *G* ∩ *F* ¼ ∅ and *G* ∪ *F* ∈ Δ. The link of a vertex *v*∈*V* is simply denoted by linkΔð Þ*v* .

$$\text{link}\_{\Delta}(\boldsymbol{\nu}) = \{ F \in \Delta \, : \, \, \boldsymbol{\nu} \notin F, \quad F \cup \{ \boldsymbol{\nu} \} \in \Delta \}. \tag{5}$$

Let Δ be a simplicial complex over *n* vertices f g *x*1, … , *xn* . For *F* ⊂ f g *x*1, … , *xn* , we set:

$$\mathfrak{x}\_F = \prod\_{\mathfrak{x}\_i \in F} \mathfrak{x}\_i. \tag{6}$$

We define the *facet ideal* of Δ, denoted by *I*ð Þ Δ , to be the ideal of *S* generated by f g *xF* : *F* ∈ Fð Þ Δ . The *non-face ideal* or the *Stanley-Reisner ideal* of Δ, denoted by *I*Δ, is the ideal of *S* generated by square-free monomials f g *xF* : *F* ∈ N ð Þ Δ . Also, we call *K*½ � Δ ≔ *S=I*<sup>Δ</sup> the *Stanley-Reisner ring* of Δ.

**Definition 2.2.** A simplicial complex Δ on f g *x*1, … , *xn* is said to be a matroid if, for any two facets *F* and *G* of Δ and any *xi* ∈ *F*, there exists a *x <sup>j</sup>* ∈ *G* such that ð Þ *F*nf g *xi* ∪ *x <sup>j</sup>* � � is a facet of Δ.

**Definition 2.3.** A simplicial complex Δ is recursively defined to be *vertex decomposable*, if it is either a simplex, or else has some vertex *v* so that,

a. Both Δn*v* and linkΔð Þ*v* are vertex decomposable, and

b. No face of linkΔð Þ*v* is a facet of Δn*v*.

A vertex *v* which satisfies in condition (b) is called a *shedding vertex*.

**Definition 2.4.** A simplicial complex Δ is *shellable*, if the facets of Δ can be ordered *F*1, … , *Fs* such that, for all 1≤*i* <*j*≤*s*, there exists some *v*∈*F <sup>j</sup>*n*Fi* and some *l* ∈f g 1, … , *j* � 1 with *F <sup>j</sup>*n*Fl* ¼ f g*v* .

A simplicial complex Δ is called disconnected, if the vertex set *V* of Δ is a disjoint union *V* ¼ *V*<sup>1</sup> ∪*V*<sup>2</sup> such that no face of Δ has vertices in both *V*<sup>1</sup> and *V*2. Otherwise, Δ is connected. It is well known that

Matroid ) vertex decomposable ) shellable ) Cohen‐Macaulay

**Definition 2.5.** Given a simplicial complex Δ on *V*, we define Δ<sup>∨</sup>, the *Alexander dual* of Δ, by

$$
\Delta^\vee = \{ V \backslash F : \quad F \notin \Delta \}.\tag{7}
$$

It is known that for the complex <sup>Δ</sup> one has *<sup>I</sup>*Δ<sup>∨</sup> <sup>¼</sup> *<sup>I</sup>* <sup>Δ</sup>*<sup>c</sup>* ð Þ. Let *<sup>I</sup>* 6¼ 0 be a homogeneous ideal of *S* and be the set of non-negative integers. For every *i*∈ ∪ f g0 , one defines:

$$t\_i^S(I) = \max\left\{ j : \quad \rho\_{i,j}^S(I) \neq \mathbf{0} \right\} \tag{8}$$

where *β<sup>S</sup> i*,*j* ð Þ*I* is the *i*, *j*-th graded Betti number of *I* as an *S*-module. The *Castelnuovo-Mumford regularity* of *I* is given by

$$\text{reg}(I) = \sup \{ t\_i^S(I) - i \, : \, \, i \in Z \}. \tag{9}$$

We say that the ideal *I* has a *d-linear resolution*, if *I* is generated by homogeneous polynomials of degree *d* and *β<sup>S</sup> i*,*j* ðÞ¼ *I* 0, for all *j* 6¼ *i* þ *d* and *i* ≥0. For an ideal that has a *d*-linear resolution, the Castelnuovo-Mumford regularity would be *d*. If *I* is a graded ideal of *S*, then we write ð Þ *Id* for the ideal generated by all homogeneous polynomials of degree *d* belonging to *I*.

**Definition 2.6.** A graded ideal *I* is componentwise linear if ð Þ *Id* has a linear resolution for all *d*.

Also, we write *I*½ � *<sup>d</sup>* for the ideal generated by the squarefree monomials of degree *d* belonging to *I*.

**Definition 2.7.** A graded *S*-module *M* is called *sequentially Cohen-Macaulay* (over *K*), if there exists a finite filtration of graded *S*-modules,

$$0 = M\_0 \subset M\_1 \subset \cdots \subset M\_r = M \tag{10}$$

such that each *Mi=Mi*�<sup>1</sup> is Cohen-Macaulay, and the Krull dimensions of the quotients are increasing:

$$\dim(M\_1/M\_0) < \dim(M\_2/M\_1) < \dots < \dim(M\_r/M\_{r-1}).\tag{11}$$

The Alexander dual allows us to make a bridge between (sequentially) Cohen-Macaulay ideals and (componetwise) linear ideals.

*Vertex Decomposability of Path Complexes and Stanley's Conjectures DOI: http://dx.doi.org/10.5772/intechopen.101083*

**Definition 2.8** (Alexander duality)**.** For a square-free monomial ideal *I* ¼ *M*1, … , *Mq* <sup>⊂</sup>*<sup>S</sup>* <sup>¼</sup> *K x*½ � 1, … , *xn* , the *Alexander dual* of *<sup>I</sup>*, denoted by *<sup>I</sup>* <sup>∨</sup>, is defined to be:

$$I^\vee = P\_{M\_1} \cap \dots \cap P\_{M\_q} \tag{12}$$

where, *PMi* is prime ideal generated by *x <sup>j</sup>* : *x <sup>j</sup>*j*Mi* .

**Theorem 2.1** ([7, Proposition 8.2.20], [5, Theorem 3])**.** *Let I be a square-free monomial ideal in S* ¼ *K x*½ � 1, … , *xn .*


**Remark 2.1.** Two special cases, we will be considering in this paper, are when *G* is a cycle *Cn*, or a line graph *Ln* on vertices f g *x*1, … , *xn* with edges

$$\begin{aligned} E(\mathbf{C}\_n) &= \{ \{ \mathbf{x}\_1, \mathbf{x}\_2 \}, \{ \mathbf{x}\_2, \mathbf{x}\_3 \}, \dots, \{ \mathbf{x}\_{n-1}, \mathbf{x}\_n \}, \{ \mathbf{x}\_n, \mathbf{x}\_1 \} \}; \\ E(L\_n) &= \{ \{ \mathbf{x}\_1, \mathbf{x}\_2 \}, \{ \mathbf{x}\_2, \mathbf{x}\_3 \}, \dots, \{ \mathbf{x}\_{n-1}, \mathbf{x}\_n \} \} \end{aligned}$$

**Remark 2.2.** All Cohen-Macaulay simplicial complexes of positive dimension are connected.

#### **3. Vertex decomposability path complexes of cycles**

As the main result of this section, it is shown that Δ*t*ð Þ *Cn* is matroid, vertex decomposable, shellable, and Cohen-Macaualay if and only if *n* ¼ *t* or *n* ¼ *t* þ 1. For the proof, we shall need the following lemmas and propositions.

**Lemma 3.1.** *Let* Δ*t*ð Þ *Pn be a simplicial complex on the path Pn* ¼ f g *x*1, … , *xn and* 2≤ *t*≤*n. Then* Δ*t*ð Þ *Pn is vertex decomposable.*

*Proof.* If *t* ¼ *n*, then Δ*n*ð Þ *Pn* is a simplex which is vertex decomposable. Let 2≤ *t*<*n* then one has

$$\Delta\_{\mathbf{t}}(\mathbf{P}\mathbf{n}) = \langle \{\mathbf{x}\_1, \dots, \mathbf{x}\_{\mathbf{t}}\}, \{\mathbf{x}\_2, \dots, \mathbf{x}\_{\mathbf{t}+1}\}, \dots, \{\mathbf{x}\_{\mathbf{n}-\mathbf{t}+1}, \dots, \mathbf{x}\_{\mathbf{n}}\} \rangle. \tag{13}$$

So Δ*t*ð Þn *Pn xn* ¼ f g *x*1, … , *xt* , f g *x*2, … , *xt*þ<sup>1</sup> , … , *xn*�*<sup>t</sup>* h i f g , … , *xn*�<sup>1</sup> . Now, we use induction on the number of vertices of *Pn* and by induction hypothesis Δ*t*ð Þn *Pn xn* is vertex decomposable. On the other hand, it is clear that *link*<sup>Δ</sup>*t*ð Þ *Pn* f g *xn* ¼ h i f g *xn*�*t*þ1, … , *xn*�<sup>1</sup> . Thus, *link*<sup>Δ</sup>*t*ð Þ *Pn* f g *xn* is a simplex which is not a facet of Δ*t*ð Þn *Pn xn*. Therefore, Δ*t*ð Þ *Pn* is vertex decomposable.

**Lemma 3.2.** *Let* Δ2ð Þ *Cn be a simplicial complex on x*f g 1, … , *xn . Then* Δ2ð Þ *Cn is vertex decomposable.*

*Proof.* Since Δ2ð Þ¼ *Cn* h i f g *x*1, *x*<sup>2</sup> , f g *x*2, *x*<sup>3</sup> , … , f g *xn*�1, *xn* , f g *xn*, *x*<sup>1</sup> then we have

$$
\Delta\_2(C\_n) \backslash \mathfrak{x}\_n = \langle \{ \mathfrak{x}\_1, \mathfrak{x}\_2 \}, \{ \mathfrak{x}\_2, \mathfrak{x}\_3 \}, \dots, \{ \mathfrak{x}\_{n-2}, \mathfrak{x}\_{n-1} \} \rangle. \tag{14}
$$

By Lemma 2.1 Δ2ð Þn *Cn xn* is vertex decomposable. Also, it is trivial that link<sup>Δ</sup>2ð Þ *Cn* f g *xn* ¼ h i f g *xn*�<sup>1</sup> , f g *x*<sup>1</sup> is vertex decomposable and no face of link<sup>Δ</sup>2ð Þ *Cn* f g *xn* is a facet of Δ2ð Þn *Cn xn*. Therefore, Δ2ð Þ *Cn* is vertex

decomposable. □ **Lemma 3.3.** *Let* Δ*t*ð Þ *Cn be a simplicial complex on x*f g 1, … , *xn and* 3 ≤*t*≤*n* � 2*. Then* Δ*t*ð Þ *Cn is not Cohen-Macaulay.*

*Proof.* It suffices to show that *I*<sup>Δ</sup>*t*ð Þ *Cn* <sup>∨</sup> has not a linear resolution. Since *I*<sup>Δ</sup>*t*ð Þ *Cn* <sup>∨</sup> ¼ *<sup>I</sup>* <sup>Δ</sup>*t*ð Þ *Cn <sup>c</sup>* ð Þ, then one can easily check that *<sup>I</sup>*<sup>Δ</sup>*t*ð Þ *Cn* <sup>∨</sup> <sup>¼</sup> *In*�*t*ð Þ *Cn* . By Theorem 1.1 we have

$$\text{reg}\left(I\_{\Delta\_{\mathbb{C}}(\mathbb{C}\_{\mathbb{R}})^{\vee}}\right) = (n - t - \mathbf{1})p + d + \mathbf{1}.\tag{15}$$

Since 3 ≤*t*≤*n* � 2 then one has *reg I*<sup>Δ</sup>*t*ð Þ *Cn* <sup>∨</sup> 6¼ *<sup>n</sup>* � *<sup>t</sup>* and by Theorem 2.1 <sup>Δ</sup>*t*ð Þ *Cn* is not Cohen-Macaulay.

**Proposition 3.1.** *Let* Δ*t*ð Þ *Cn be a simplicial complex on x*f g 1, … , *xn and t*≥ 3*. Then* Δ*t*ð Þ *Cn is vertex decomposable if and only if n* ¼ *t or t* þ 1*.*

*Proof.* By Lemma 3.3, it suffices to show that if *n* ¼ *t* or *t* þ 1, then Δ*t*ð Þ *Cn* is vertex decomposable. If *n* ¼ *t*, then Δ*n*ð Þ *Cn* is a simplex which is vertex decomposable.

If *t* ¼ *n* � 1, then we have

$$\Delta\_{n-1}(\mathbb{C}\_n) = \langle \{\mathbf{x}\_1, \dots, \mathbf{x}\_{n-1}\}, \{\mathbf{x}\_2, \dots, \mathbf{x}\_n\}, \{\mathbf{x}\_3, \dots, \mathbf{x}\_n, \mathbf{x}\_1\}, \dots, \{\mathbf{x}\_n, \mathbf{x}\_1, \dots, \mathbf{x}\_{n-2}\} \rangle.$$

Now, we use induction on the number of vertices of *Cn* and show that Δ*<sup>n</sup>*�<sup>1</sup>ð Þ *Cn* is vertex decomposable. It is clear that Δ*<sup>n</sup>*�<sup>1</sup>ð Þn *Cn xn* ¼ h i f g *x*1, … , *xn*�<sup>1</sup> is a simplex, which is vertex decomposable.

On the other hand,

$$\text{link}\_{\Delta\_{n-1}(\mathbb{C}\_{\pi})} \{ \mathbf{x}\_{\pi} \} = \langle \{ \mathbf{x}\_1, \dots, \mathbf{x}\_{n-2} \}, \dots, \{ \mathbf{x}\_{n-1}, \mathbf{x}\_1, \dots, \mathbf{x}\_{n-3} \} \rangle = \Delta\_{n-2} (\mathbf{C}\_{n-1}).\tag{16}$$

By induction hypothesis link<sup>Δ</sup>*n*�1ð Þ *Cn* f g *xn* is vertex decomposable. It is easy to see that no face of link<sup>Δ</sup>*n*�1ð Þ *Cn* f g *xn* is a facet of Δ*<sup>n</sup>*�<sup>1</sup>ð Þn *Cn xn*. Therefore, Δ*<sup>n</sup>*�<sup>1</sup>ð Þ *Cn* is vertex decomposable.

**Proposition 3.2.** Δ2ð Þ *Cn is a matroid if and only if n* ¼ 3 *or* 4*.*

*Proof.* If *n* ¼ 3 or 4, then it is easy to see that Δ2ð Þ *Cn* is a matroid. Now, we prove the converse. It suffices to show that Δ2ð Þ *Cn* is not a matroid for all *n* ≥5. We consider two facets f g *x*1, *x*<sup>2</sup> and f g *xn*�1, *xn* . Then, we have ð Þ f gn *x*1, *x*<sup>2</sup> f g *x*<sup>1</sup> ∪ f g¼ *xn*�<sup>1</sup> f g *x*2, *xn*�<sup>1</sup> and ð Þ f gn *x*1, *x*<sup>2</sup> f g *x*<sup>1</sup> ∪ f g *xn* ¼ f g *x*2, *xn* . Since f g *x*2, *xn*�<sup>1</sup> and f g *x*2, *xn* are not the facets of Δ2ð Þ *Cn* . So Δ2ð Þ *Cn* is not matroid for all *n*≥5.

For the simplicial complexes, one has the following implication:

#### **Matroid** ) **vertex decomposable** ) **shellable** ) **Cohen** � **Macaulay**

Note that these implications are strict, but by the following theorem, for path complexes, the reverse implications are also valid.

**Theorem 3.1.** *Let t*≥3*. Then the following conditions are equivalent*:

i. Δ*t*ð Þ *Cn is matroid;*

ii. Δ*t*ð Þ *Cn is vertex decomposable;*


v. *n* ¼ *t or t* þ 1.

*Vertex Decomposability of Path Complexes and Stanley's Conjectures DOI: http://dx.doi.org/10.5772/intechopen.101083*

*Proof.* (i)) (ii), (ii)) (iii) and (iii)) (iv) is well known. (iv)) (v): Follows from Lemma 3.3 and Proposition 2.1. (v)) (i): If *n* ¼ *t*, then Δ*t*ð Þ *Cn* is a simplex which is a matroid. If *n* ¼ *t* þ 1, then

$$\Delta\_l(\mathcal{C}\_n) = \langle \{\mathbf{x}\_1, \dots, \mathbf{x}\_l\}, \{\mathbf{x}\_2, \dots, \mathbf{x}\_{l+1}\}, \{\mathbf{x}\_3, \dots, \mathbf{x}\_{l+1}, \mathbf{x}\_1\}, \dots, \{\mathbf{x}\_{l+1}, \mathbf{x}\_1, \dots, \mathbf{x}\_{l-1}\} \rangle.$$

For any two facets *F* and *G* of Δ*t*ð Þ *Cn* one has ∣*F* ∩ *G*∣ ¼ *t* � 1. We claim that for any two facets *F* and *G* of Δ*t*ð Þ *Cn* and any *xi* ∈ *F*, there exists a *x <sup>j</sup>* ∈ *G* such that ð Þ *F*nf g *xi* ∪ *x <sup>j</sup>* � � is a facet of <sup>Δ</sup>*t*ð Þ *Cn* . We have to consider two cases. If *xi* <sup>∈</sup>*<sup>F</sup>* and *xi* ∉ *G*, then we choose *x <sup>j</sup>* ∈ *G* such that *x <sup>j</sup>* ∉ *F*. Thus, ð Þ *F*nf g *xi* ∪ *x <sup>j</sup>* � � <sup>¼</sup> *<sup>G</sup>* which is a facet of Δ*t*ð Þ *Cn* .

For other case, if *xi* ∈ *F* and *xi* ∈ *G*, then we choose *x <sup>j</sup>* ∈ *G* such that *x <sup>j</sup>* is the same *xi*. Therefore, ð Þ *F*nf g *xi* ∪ f g *xi* ¼ *F* is a facet of Δ*t*ð Þ *Cn* , which completes the proof.

#### **4. Vertex decomposability path complexes of trees**

As the main result of this section, for all *t*≥2, we characterize all such trees whose Δ*t*ð Þ *G* is vertex decomposable. Let *H p*ð Þ , *n*, *q* denote the double starlike tree obtained by attaching *p* pendant vertices to one pendant vertex of the path *Pn* and *q* pendant vertices to the other pendant vertex of *Pn*. Also, let *H p*ð Þ , *n* be graph obtained by attaching *p* pendant vertices to one pendant vertex of the path *Pn*.

**Remark 4.1.** Let *Pn* ¼ f g *x*1, … , *xn* be a path on vertices f g *x*1, … , *xn* and *H*ð Þ 2, *n* be a graph obtained by attaching two pendant vertices to pendant vertex *xn*. Then, Δ*t*ð Þ *H*ð Þ 2, *n* is vertex decomposable for all *t*≥ 2.

*Proof.* By Lemma 3.1 proof is trivial.

**Proposition 4.1.** *Let Pn* ¼ f g *x*1, … , *xn be a path on vertices x*f g 1, … , *xn and H p*ð Þ , *n be a graph obtained by attaching p pendant vertices to pendant vertex xn. Then* Δ*t*ð Þ *H p*ð Þ , *n is vertex decomposable for all t*≥2*.*

*Proof.* We prove the proposition by induction on *p* the number of pendant vertices to pendant vertex *xn* of *Pn*. If *p* ¼ 0 or 1, then *H p*ð Þ , *n* is a path and by Lemma 2.1 Δ*t*ð Þ *H p*ð Þ , *n* is vertex decomposable. If *p* ¼ 2, then by remark 4.1 Δ*t*ð Þ *H p*ð Þ , *n* is vertex decomposable. Now, let *p*> 2 and *y*1, … , *yp* n o be *<sup>p</sup>* pendant vertices to pendant vertex *xn* of *Pn*, then one has

$$H(p, n) \backslash \{y\_1\} = H(p-1, n) \tag{17}$$

and

$$
\Delta\_l(H(p,n)) \backslash \{\mathcal{y}\_1\} = \Delta\_l(H(p-1,n)).\tag{18}
$$

Therefore by induction hypothesis Δ*t*ð Þ *H p*ð Þ � 1, *n* is vertex decomposable. So Δ*t*ð Þn *H p*ð Þ , *n y*<sup>1</sup> � � is vertex decomposable. If *<sup>t</sup>* <sup>¼</sup> 3, then we have

$$\text{link}\_{\Delta \chi(H(p,n))} \{ \mathbf{y}\_1 \} = \left\langle \{ \mathbf{x}\_{n-1}, \mathbf{x}\_n \}, \{ \mathbf{y}\_2, \mathbf{x}\_n \}, \dots, \left\{ \mathbf{y}\_p, \mathbf{x}\_n \right\} \right\rangle \tag{19}$$

It is easy to see that link<sup>Δ</sup>3ð Þ *H p*ð Þ ,*<sup>n</sup> y*<sup>1</sup> � � is vertex decomposable and *<sup>y</sup>*<sup>1</sup> is a shedding vertex. If *t* ¼ 2 or *t*>3, one has

$$\text{link}\_{\Delta\_{\ell}(H(p,n))} \{ \mathbf{y}\_1 \} = \langle \{ \mathbf{x}\_{n-t+2}, \dots, \mathbf{x}\_n \} \rangle. \tag{20}$$

thus link<sup>Δ</sup>*t*ð Þ *H p*ð Þ ,*<sup>n</sup> y*<sup>1</sup> � � is a simplex, which is not a facet of <sup>Δ</sup>*t*ð Þn *H p*ð Þ , *<sup>n</sup> <sup>y</sup>*<sup>1</sup> � �, therefore Δ*t*ð Þ *H p*ð Þ , *n* is vertex decomposable.

**Lemma 4.1.** *Let p* ¼ 2 *and q*≥ 2*,Then* Δ*t*ð Þ *H*ð Þ 2, *n*, *q is vertex decomposable for all* 2≤ *t*≤*n* þ 2*.*

*Proof.* Let *H*ð Þ 2, *n*, *q* denote the double starlike tree obtained by attaching two pendant vertices *y*1, *y*<sup>2</sup> � � to pendant vertex *x*<sup>1</sup> of path *Pn* and *y*<sup>0</sup> <sup>1</sup>, … , *y*<sup>0</sup> *q* n o be pendant vertices to pendant vertex *xn* of *Pn*. So by proposition 3.2

Δ*t*ð Þn *H*ð Þ 2, *n*, *q y*<sup>1</sup> � � is vertex decomposable. Now, we prove that link<sup>Δ</sup>*t*ð Þ *<sup>H</sup>*ð Þ 2,*n*,*<sup>q</sup> <sup>y</sup>*<sup>1</sup> � � is vertex decomposable. If *t* ¼ 3, then link<sup>Δ</sup>3ð Þ *<sup>H</sup>*ð Þ 2,*n*,*<sup>q</sup> y*<sup>1</sup> � � <sup>¼</sup> f g *<sup>x</sup>*1, *<sup>x</sup>*<sup>2</sup> , *<sup>x</sup>*1, *<sup>y</sup>*<sup>2</sup> � � � � which is vertex decomposable. If *t* ¼ *n* þ 2, then

$$\text{link}\_{\Delta\_{n+2}(H(2,n,q))} \{ y\_1 \} = \left\langle \left\{ \mathbf{x}\_1, \dots, \mathbf{x}\_n, y\_1' \right\}, \left\{ \mathbf{x}\_1, \dots, \mathbf{x}\_n, y\_2' \right\}, \dots, \left\{ \mathbf{x}\_1, \dots, \mathbf{x}\_n, y\_q' \right\} \right\rangle.$$

It is easy to see that link<sup>Δ</sup>*n*þ2ð Þ *<sup>H</sup>*ð Þ 2,*n*,*<sup>q</sup> y*<sup>1</sup> � � is vertex decomposable. If *<sup>t</sup>* <sup>¼</sup> 2 or 4≤*t* ≤*n* þ 1, then we have link<sup>Δ</sup>*t*ð Þ *<sup>H</sup>*ð Þ 2,*n*,*<sup>q</sup> y*<sup>1</sup> � � <sup>¼</sup> h i f g *<sup>x</sup>*1, … , *xt*�<sup>1</sup> . Thus, link<sup>Δ</sup>*t*ð Þ *<sup>H</sup>*ð Þ 2,*n*,*<sup>q</sup> y*<sup>1</sup> � � is a simplex, which is vertex decomposable. It is clear that *<sup>y</sup>*<sup>1</sup> is a shedding vertex. □

**Proposition 4.2.** *Let Q*1, *Q*<sup>2</sup> *be two paths of maximum length k in tree G and y be a leaf of G such that y*∈ *Q*<sup>1</sup> ∩ *Q*2*,* ∣*Q*<sup>1</sup> ∩ *Q*2∣ ¼ *L. Then* Δ*t*ð Þ *G is not vertex decomposable.*

*Proof.* Suppose *<sup>Q</sup>*<sup>1</sup> <sup>¼</sup> *<sup>y</sup>*1, *<sup>y</sup>*2, … , *yk*�*<sup>L</sup>*, *<sup>x</sup>*1, *<sup>x</sup>*2, … , *xL*�1, *<sup>y</sup>* � � and.

*Q*<sup>2</sup> ¼ *y*<sup>0</sup> <sup>1</sup>, *y*<sup>0</sup> <sup>2</sup>, … , *y*<sup>0</sup> *<sup>k</sup>*�*<sup>L</sup>*, *<sup>x</sup>*1, *<sup>x</sup>*2, … , *xL*�1, *<sup>y</sup>* � � be two paths of length *<sup>k</sup>* in *<sup>G</sup>* such that *Q*<sup>1</sup> ∩ *Q*<sup>2</sup> ¼ f g *x*1, *x*2, … , *xL*�1, *y* and *deg y*ð Þ¼ 1. Since link<sup>Δ</sup>*k*ð Þ *<sup>G</sup>* f g *x*1, … , *xL*�1, *y* is disconnected and pure of positive dimension. By remark 1.11 Δ*k*ð Þ *G* is not Cohen-Macaulay and hence, <sup>Δ</sup>*k*ð Þ *<sup>G</sup>* is not vertex decomposable. □

**Proposition 4.3.** *Let G be a double starlike tree such that is not a path. Then* Δ*t*ð Þ *G is vertex decomposable for all* 2≤ *t*≤*n* þ 2*.*

*Proof.* Let *G* ¼ *H p*ð Þ , *n*, *q* denote the double starlike tree obtained by attaching *p* pendant vertices to one pendant vertex of the path *Pn* and *q* pendant vertices to the other pendant vertex of *Pn*. We prove the theorem by induction on *p* the number of pendant vertices to pendant vertex *x*<sup>1</sup> of *Pn*. If *p* ¼ 0 or *p* ¼ 1, then by proposition 3.2 Δ*t*ð Þ *G* is vertex decomposable. If *p* ¼ 2, then by Lemma 4.3 Δ*t*ð Þ *G* is vertex decomposable. Now, let *p* >2 and *y*1, … , *yp* n o be *<sup>p</sup>* pendant vertices to pendant vertex *x*<sup>1</sup> of *Pn*. Since *G*n *y*<sup>1</sup> � � is again double starlike tree on *<sup>p</sup>* � 1 pendant vertices. Therefore, by induction hypothesis, Δ*<sup>t</sup> G*n *y*<sup>1</sup> � � � � is vertex decomposable. So Δ*<sup>t</sup> G*n *y*<sup>1</sup> � � � � <sup>¼</sup> <sup>Δ</sup>*t*ð Þn *<sup>G</sup> <sup>y</sup>*<sup>1</sup> � � is vertex decomposable. Let *<sup>t</sup>* <sup>¼</sup> 2, then link<sup>Δ</sup>2ð Þ *<sup>G</sup> <sup>y</sup>*<sup>1</sup> � � <sup>¼</sup> h i f g *x*<sup>1</sup> is simplex and vertex decomposable. Let *t* ¼ 3, then link<sup>Δ</sup>3ð Þ *<sup>G</sup> y*<sup>1</sup> � � <sup>¼</sup> f g *x*2, *x*<sup>1</sup> , *y*2, *x*<sup>1</sup> � �, … , *yp*, *<sup>x</sup>*<sup>1</sup> D E n o is vertex decomposable. Let 3<sup>&</sup>lt; *<sup>t</sup>*≤*<sup>n</sup>* <sup>þ</sup> 1, then link<sup>Δ</sup>*t*ð Þ *<sup>G</sup> y*<sup>1</sup> � � <sup>¼</sup> h i f g *<sup>x</sup>*1, *<sup>x</sup>*2, … , *xt*�<sup>1</sup> is simplex and vertex decomposable. Let *<sup>t</sup>* <sup>¼</sup> *n* þ 2, then link<sup>Δ</sup>*t*ð Þ *<sup>G</sup> y*<sup>1</sup> � � ¼ h *<sup>x</sup>*1, … , *xn*, *<sup>y</sup>*<sup>1</sup> � �, *<sup>x</sup>*1, … , *xn*, *<sup>y</sup>*<sup>2</sup> � �, … , <sup>f</sup>*x*1, … , *xn*, *yp*<sup>i</sup> is a

path complex of a starlike tree which is vertex decomposable. It is easy to see that no face of link<sup>Δ</sup>*t*ð Þ *<sup>G</sup> y*<sup>1</sup> � � is a facet of <sup>Δ</sup>*t*ð Þn *<sup>G</sup> <sup>y</sup>*<sup>1</sup> � �. So <sup>Δ</sup>*t*ð Þ *<sup>G</sup>* is vertex decomposable. Now, we are ready that prove the main result of this section.

**Theorem 4.1.** *Let G be a tree such that is not a path. Then* Δ*t*ð Þ *G is vertex decomposable for all t*≥2 *if and only if G* ¼ *H p*ð Þ , *n*, *q or H p*ð Þ , *n .*

*Proof.* ð Þ ) We prove by contradiction. Suppose *G* 6¼ *H p*ð Þ , *n*, *q* and *G* 6¼ *H p*ð Þ , *n* . So there exists two paths of maximum length *k* in *G* which contain *L* common

vertices such that one of these vertices is a leaf. Therefore, by proposition 4.2 Δ*k*ð Þ *G* is not vertex decomposable, which is a contradiction. ð Þ ( By proposition 4.1 and Proposition 4.3, the proof is completed.

### **5. Vertex decomposability path complexes of dynkin graphs**

Dynkin diagrams first appeared [15] in the connection with classification of simple Lie groups. Among Dynkin diagrams a special role is played by the simply laced Dynkin diagrams *An*, *Dn*, *E*6, *E*7, and *E*8. Dynkin diagrams are closely related to coxter graphs that appeared in geometry (see [16]).

After that, Dynkin diagrams appeared in many branches of mathematics and beyond, in particular in representation theory. In [17], Gabriel introduced a notion of a quiver (directed graph) and its representations. He proved the famous Gabriel's theorem on representation of quivers over algebraic closed field. Let *Q* be a finite quiver and *Q* the undirected graph obtained from *Q* by deleting the orientation of all arrows.

**Theorem 5.1.** *(Gabriel theorem). A connected quiver Q is of Finite type if and only if the graph Q is one of the following simply laced Dynkin diagrams: An*, *Dn*, *E*6, *E*<sup>7</sup> *or E*8*.*

Let *Ln* be a line graph on vertices f g *x*1, … , *xn* and *x <sup>j</sup>*, *y <sup>j</sup>* n o be whisker of *Ln* at *<sup>x</sup> <sup>j</sup>* with 3≤*j*≤ *n* � 1.

We obtain some condition that Δ*<sup>t</sup> Ln* ∪*W x <sup>j</sup>* � � � � is vertex decomposable, as an application of our results, vertex decomposability path complexes of Dynkin graphs are shown. Throughout this section, we assume *Ln* ∪ *W x <sup>j</sup>* � � be an undirected graph. By Lemma 3.1, we have the following corollary:

**Corollary 5.1.** *Let An be a Dynkin graph and* 2 ≤*t*≤*n. Then* Δ*t*ð Þ *An is vertex decomposable.*

**Proposition 5.1.** *Let Ln be a line graph on vertices x*f g 1, … , *xn and xn*�1, *yn*�<sup>1</sup> � � *be whisker of Ln at xn*�<sup>1</sup>*.*

Then, Δ*t*ð Þ *Ln* ∪*W x*ð Þ *<sup>n</sup>*�<sup>1</sup> is vertex decomposable for all 2≤ *t*≤*n*.

Proof. Then by proposition 4.1 proof is trivial.

**Corollary 5.2.** *Let Dn be a Dynkin graph and n* ≥4*. Then* Δ*t*ð Þ *Dn is vertex decomposable for all* 2≤*t* ≤*n.*

*Proof.* We know that *Dn* ¼ *Ln* ∪*W x*ð Þ *<sup>n</sup>*�<sup>1</sup> . So by proposition 4.3 Δ*t*ð Þ *Dn* is vertex decomposable. □

**Theorem 5.2.** *Let Ln be a line graph on vertices x*f g 1, … , *xn and x <sup>j</sup>*, *y <sup>j</sup>* n o *be whisker of Ln at x <sup>j</sup> with* 3≤*j*≤*n* � 2*.*

Then, Δ*<sup>t</sup> Ln* ∪*W x <sup>j</sup>* � � � � is vertex decomposable if and only if 2 ≤*t*≤ 3 or *n* ≥*t*>*α*, where *α* ¼ min *d y <sup>j</sup>* , *x*<sup>1</sup> � �, *d y <sup>j</sup>* , *xn* n o � � .

*Proof.* We first show that Δ*<sup>t</sup> Ln* ∪ *W x <sup>j</sup>* � � � � is not vertex decomposable for all 4≤*t* ≤*α*. It is well known that if Δ is a Cohen-Macaulay simplicial complex, then *link*Δf g*F* is Cohen-Macaulay for each face *F* of Δ.

Also, we know that all Cohen-Macaulay complexes of positive dimension are connected. It is easy to see that *link*<sup>Δ</sup>*t*ð Þ *Ln* <sup>∪</sup>*W x*ð Þ*<sup>j</sup> x <sup>j</sup>*, *y <sup>j</sup>* n o is disconnected and pure of positive dimension.

This implies that Δ*<sup>t</sup> Ln* ∪*W x <sup>j</sup>* � � � � is not Cohen- Macaulay and hence Δ*<sup>t</sup> Ln* ∪ *W x <sup>j</sup>* � � � � is not vertex decomposable without loss of generality we can assume that *α* ¼ *d y <sup>j</sup>* , *x*<sup>1</sup> � � if *<sup>t</sup>* <sup>¼</sup> 2 or *<sup>n</sup>* <sup>≥</sup>*t*>*α*, one has:

$$\Delta\_t(L\_\pi \cup W(\mathfrak{x}\_j)) = \langle \{\mathfrak{x}\_1, \dots, \mathfrak{x}\_t\}, \{\mathfrak{x}\_2, \dots, \mathfrak{x}\_{t+1}\}, \dots,$$

$$\{\mathfrak{x}\_{j-1}, \mathfrak{x}\_j, \dots, \mathfrak{x}\_{j+t-2}\}, \{\mathfrak{y}\_j, \mathfrak{x}\_j, \mathfrak{x}\_{j+1}, \dots, \mathfrak{x}\_{j+t-2}\}, \dots, \{\mathfrak{x}\_{n-t+1}, \dots, \mathfrak{x}\_n\})$$

So Δ*<sup>t</sup> Ln* ∪*W x <sup>j</sup>* � � � � <sup>n</sup>*<sup>y</sup> <sup>j</sup>* <sup>¼</sup> <sup>Δ</sup>*t*ð Þ *Ln* and by Lemma [4, 6, 7, 9, 10, 18–20] Δ*<sup>t</sup> Ln* ∪ *W x <sup>j</sup>* � � � � <sup>n</sup>*<sup>y</sup> <sup>j</sup>* is vertex decomposable. On the other hand, we have *link*<sup>Δ</sup>*t*ð Þ *Ln* <sup>∪</sup>*W x*ð Þ*<sup>j</sup> <sup>y</sup> <sup>j</sup>* n o <sup>¼</sup> *<sup>x</sup> <sup>j</sup>*, *<sup>x</sup> <sup>j</sup>*þ1, … , *<sup>x</sup> <sup>j</sup>*þ*t*�<sup>2</sup> � � � � that is a simplex and vertex decomposable.

If *t* ¼ 3, then

$$\Delta\_3(L\_n \cup W(\mathfrak{x}\_j)) = \langle \{\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3\}, \{\mathfrak{x}\_2, \mathfrak{x}\_3, \mathfrak{x}\_4\}, \dots, \mathfrak{x}\_n$$

$$\{\mathfrak{x}\_{j-1}, \mathfrak{x}\_j, \mathfrak{y}\_j\}, \{\mathfrak{y}\_j, \mathfrak{x}\_j, \mathfrak{x}\_{j+1}\}, \dots, \{\mathfrak{x}\_{n-2}, \mathfrak{x}\_{n-1}, \mathfrak{x}\_n\}\rangle$$

and Δ<sup>3</sup> *Ln* ∪*W x <sup>j</sup>* � � � � <sup>n</sup>*<sup>y</sup> <sup>j</sup>* <sup>¼</sup> <sup>Δ</sup>3ð Þ *Ln* which is vertex decomposable.

It is easy to see that *link*Δ*t*ð Þ *Ln* <sup>∪</sup>*W x*ð Þ*<sup>j</sup> <sup>y</sup> <sup>j</sup>* n o <sup>¼</sup> *<sup>x</sup> <sup>j</sup>*�1, *<sup>x</sup> <sup>j</sup>* � �, *<sup>x</sup> <sup>j</sup>*, *<sup>x</sup> <sup>j</sup>*þ<sup>1</sup> � � � � is vertex decomposable and *y <sup>j</sup>* is a shedding vertex.

**Corollary 5.3.** *Let E*<sup>6</sup> *be a Dynkin graph. Then* Δ*t*ð Þ *E*<sup>6</sup> *is vertex decomposable if and only if* 2≤*t* ≤3 *or t* ¼ 5*.*

*Proof.* Since *E*<sup>6</sup> ¼ *L*<sup>5</sup> ∪ *W x*ð Þ<sup>3</sup> , so by Theorem 5.2 the proof is completed.

**Corollary 5.4.** *Let E*<sup>7</sup> *be a Dynkin graph. Then* Δ*t*ð Þ *E*<sup>7</sup> *is vertex decomposable if and only if* 2≤*t* ≤3 *or* 5≤ *t*≤6*.*

*Proof.* We know that *E*<sup>7</sup> ¼ *L*<sup>6</sup> ∪*W x*ð Þ<sup>3</sup> and the proof follow from Theorem 5.2. **Corollary 5.5.** *Let E*<sup>8</sup> *be a Dynkin graph. Then* Δ*t*ð Þ *E*<sup>8</sup> *is vertex decomposable if and only if* 2≤*t* ≤3 *or* 5≤ *t*≤7.

#### **6. Stanley decompositions**

Let *R* be any standard graded *K*-algebra over an infinite field *K*, *i:e*, *R* is a finitely generated graded algebra *R* ¼ ⊕*<sup>i</sup>*≥<sup>0</sup>*Ri* such that *R*<sup>0</sup> ¼ *K* and *R* is generated by *R*1. There are several characterizations of the depth of such an algebra. We use the one that depthð Þ *R* is the maximal length of a regular *R*-sequence consisting of linear forms. Let *xF* ¼ ⊓*<sup>i</sup>* <sup>∈</sup>*Fxi* be a squarefree monomial for some *F* ⊆ ½ � *n* and *Z* ⊆f g *x*1, … , *xn* . The *K*-subspace *xFK Z*½ � of *S* ¼ *K x*½ � 1, … , *xn* is the subspace generated by monomials *xFu*, where *u* is a monomial in the polynomial ring *K Z*½ �. It is called a squarefree Stanley space if *xi* f g : *i*∈ *F* ⊆*Z*. The dimension of this Stanley space is ∣*Z*∣. Let Δ be a simplicial complex on f g *x*1, … , *xn* . A squarefree Stanley decomposition D of *K*½ � Δ is a finite direct sum ⊕*iuiK Zi* ½ � of squarefree Stanley spaces, which is isomorphic as a *<sup>n</sup>*-graded *<sup>K</sup>*-vector space to *<sup>K</sup>*½ � <sup>Δ</sup> , *<sup>i</sup>:e*.

$$K[\Delta] \cong \bigoplus\_{i} \mu\_{i} K[Z\_{i}] \tag{21}$$

We denote by sdepthð Þ D the minimal dimension of a Stanley space in D and we define sdepthð Þ¼ *K*½ � Δ max sdepth f g ð Þ D , where D is a Stanley decomposition of *K*½ � Δ . Stanley conjectured in [9] the upper bound for the depth of *K*½ � Δ as the following:

$$\text{depth}(K[\Delta]) \le \text{sdepth}(K[\Delta]) \tag{22}$$

*Vertex Decomposability of Path Complexes and Stanley's Conjectures DOI: http://dx.doi.org/10.5772/intechopen.101083*

Also, we recall another conjecture of Stanley. Let Δ be again a simplicial complex on f g *x*1, … , *xn* with facets *G*1, … , *Gt*. The complex Δ is called partitionable if there exists a partition <sup>Δ</sup> <sup>¼</sup> <sup>∪</sup> *<sup>t</sup> <sup>i</sup>*¼<sup>1</sup> *Fi*, *Gi* ½ � where *Fi* ⊆ *Gi* are suitable faces of Δ. Here, the interval *Fi*, *Gi* ½ � is the set of faces f g *H* ∈ Δ : *Fi* ⊆ *H* ⊆ *Gi* . In [10, 18] respectively Stanley conjectured each Cohen-Macaulay simplicial complex is partitionable [16–20]. This conjecture is a special case of the previous conjecture. Indeed, Herzog, Soleyman Jahan, and Yassemi [14] proved that for Cohen-Macaulay simplicial complex Δ on f g *x*1, … , *xn* we have that depthð Þ *K*½ � Δ ≤sdepthð Þ *K*½ � Δ if and only if Δ is partitionable. Since each vertex decomposable simplicial complex is shellable and each shellable complex is partitionable [4, 6, 7, 19, 20]. Then as a consequence of our results, we obtain:

**Corollary 6.1.** *if n* ¼ *t or t* þ 1 *then* Δ*t*ð Þ *Cn is partitionable and Stanley's conjecture holds for K*½ � Δ*t*ð Þ *Cn .*

**Corollary 6.2.** *Let G be a tree such that is not a path. if G* ¼ *H p*ð Þ , *n*, *q or G* ¼ *H p*ð Þ , *n then* Δ*t*ð Þ *G is vertex decomposable for all t*≥2 *and Stanley's conjecture holds for K*½ � Δ*t*ð Þ *G .*

#### **Classification**

2010 Mathematics Subject Classification, 13F20; 05E40; 13F55.

#### **Author details**

Seyed Mohammad Ajdani and Francisco Bulnes\*

1 Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

2 IINAMEI, Investigación Internacional Avanzada en Matemáticas e Ingeniería, Chalco, Mexico

\*Address all correspondence to: francisco.bulnes@tesch.edu.mx

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## Section 5
