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# Vertex Decomposability of Path Complexes and Stanley’s Conjectures

Written By

Seyed Mohammad Ajdani and Francisco Bulnes

Reviewed: October 5th, 2021 Published: February 1st, 2022

DOI: 10.5772/intechopen.101083

From the Edited Volume

## Advanced Topics of Topology [Working Title]

Dr. Francisco Bulnes

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## 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 IΔ 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 Nn-graded ring and M a Zn-graded R-module. Then, Stanley conjectured that depthM≤sdepthM. 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=Kx1xn, where Kis a field. Fix an integer nt2and let Gbe a directed graph. A sequence xi1,,xitof distinct vertices is called a path of length tif there are t1distinct directed edges e1,,et1where ejis a directed edge from xijto xij+1. Then, the path ideal of Gof length tis the monomial ideal ItG=(xi1xit:xi1,,xitis a path of length t in G) in the polynomial ring R=Kx1xn. The distance dxyof two vertices xand yof a graph Gis the length of the shortest path from xto y. The path complex ΔtGis defined by

ΔtG=xi1xit:xi1xitisapathoflengthtinG.

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 , it is shown that I2Cnis sequentially Cohen-Macaulay, if and only if, n=3or n=5. Generalizing this result, in , it is proved that ItCn, (t>2), is sequentially Cohen-Macaulay, if and only if n=tor n=t+1or n=2t+1. Also, the Betti numbers of the ideal ItCnand ItLnis computed explicitly in . In particular, it has been shown that [6, 7]:

Theorem 1.1([1, Corollary 5.15]). Letn, t, panddbe integers such thatnt2, n=t+1p+d, wherep0and0d<t+1. Then,

1. The projective dimension of the path ideal of a graph cycleCnor lineLnis given by,

pdItCn=2p,d02p1,d=0pdItLn=2p1,dt2p,d=tE1

2. The regularity of the path ideal of a graph cycleCnor lineLnis given by,

regItCn=t1p+d+1regItLn=pt1+1,d<tpt1+t,d=tE2

In  it has been shown that ΔtGis a simplicial tree if Gis a rooted tree and t2. One of interesting problems in combinatorial commutative algebra is the Stanley’s conjectures. The Stanley’s conjectures are studied by many researchers. Let Rbe a Nn-graded ring and Ma Zn-graded R-module. Then, Stanley  conjectured that

depthMsdepthME3

He also conjectured in  that each Cohen-Macaulay simplicial complex is partitionable. Herzog, Soleyman Jahan, and Yassemi in [11, 12, 13, 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 ΔtCn. In Section 1, we recall some definitions and results, which will be needed later. In Section 2, for all t>2we show that the following conditions are equivalent:

1. ΔtCnis matroid;

2. ΔtCnis vertex decomposable;

3. Δt(Cn)is shellable;

4. ΔtCnis Cohen-Macaulay;

5. n=tor t+1.

(see Theorem 2.6).

In Section 3, for all t2we show that ΔtGis vertex decomposable if and only if G=Hpnqor G=Hpn. 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=tor t+1then ΔtCnis partitionable and Stanley’s conjecture holds for KΔtCnand KΔtG, where G=Hpnqor G=Hpn.

## 2. Preliminaries

In this section, we recall some definitions and results which will be needed later.

Definition 2.1.A simplicial complexΔover a set of vertices V=x1xn, is a collection of subsets of V, with the property that:

1. xiΔ, for all i;

2. if FΔ, then all subsets of Fare also in Δ(including the empty set).

An element of Δis called a faceof Δand complement of a face Fis V\Fand it is denoted by Fc. Also, the complement of the simplicial complex Δ=F1Fris Δc=F1cFrc. The dimensionof a face Fof Δ, dimF, is F1where Fis the number of elements of Fand dim=1. The faces of dimensions 0 and 1 are called verticesand edges, respectively. A non-faceof Δis a subset Fof Vwith FΔ. We denote by NΔ, the set of all minimal non-faces of Δ. The maximal faces of Δunder inclusion are called facetsof Δ. The dimensionof the simplicial complex Δ, dimΔ, is the maximum of dimensions of its facets. If all facets of Δhave the same dimension, then Δis called pure.

Let FΔ=F1Fqbe the facet set of Δ. It is clear that FΔdetermines Δcompletely and we write Δ=F1Fq. A simplicial complex with only one facet is called a simplex. A simplicial complex Γis called a subcomplexof Δ, if FΓFΔ.

For vV, the subcomplex of Δobtained by removing all faces FΔwith vFis denoted by Δ\v. That is,

Δ\v=FΔ:vF.E4

The linkof a face FΔ, denoted by linkΔF, is a simplicial complex on Vwith the faces, GΔsuch that, GF=and GFΔ. The link of a vertex vVis simply denoted by linkΔv.

Let Δbe a simplicial complex over nvertices x1xn. For Fx1xn, we set:

xF=xiFxi.E6

We define the facet idealof Δ, denoted by IΔ, to be the ideal of Sgenerated by xF:FFΔ. The non-face idealor the Stanley-Reisner idealof Δ, denoted by IΔ, is the ideal of Sgenerated by square-free monomials xF:FNΔ. Also, we call KΔS/IΔthe Stanley-Reisner ringof Δ.

Definition 2.2.A simplicial complex Δon x1xnis said to be a matroid if, for any two facets Fand Gof Δand any xiF, there exists a xjGsuch that F\xixjis a facet of Δ.

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

1. Both Δ\vand linkΔvare vertex decomposable, and

2. No face of linkΔvis a facet of Δ\v.

A vertex vwhich satisfies in condition (b) is called a shedding vertex.

Definition 2.4.A simplicial complex Δis shellable, if the facets of Δcan be ordered F1,,Fssuch that, for all 1i<js, there exists some vFj\Fiand some l1j1with Fj\Fl=v.

A simplicial complex Δis called disconnected, if the vertex set Vof Δis a disjoint union V=V1V2such that no face of Δhas vertices in both V1and V2. Otherwise, Δis connected. It is well known that

Matroidvertex decomposableshellableCohenMacaulay

Definition 2.5.Given a simplicial complex Δon V, we define Δ, the Alexander dualof Δ, by

Δ=V\F:FΔ.E7

It is known that for the complex Δone has IΔ=IΔc. Let I0be a homogeneous ideal of Sand Nbe the set of non-negative integers. For every iN0, one defines:

tiSI=maxj:βi,jSI0E8

where βi,jSIis the i,j-th graded Betti number of Ias an S-module. The Castelnuovo-Mumford regularityof Iis given by

regI=suptiSIi:iZ.E9

We say that the ideal Ihas a d-linear resolution, if Iis generated by homogeneous polynomials of degree dand βi,jSI=0, for all ji+dand i0. For an ideal that has a d-linear resolution, the Castelnuovo-Mumford regularity would be d. If Iis a graded ideal of S, then we write Idfor the ideal generated by all homogeneous polynomials of degree dbelonging to I.

Definition 2.6.A graded ideal Iis componentwise linear if Idhas a linear resolution for all d.

Also, we write Idfor the ideal generated by the squarefree monomials of degree dbelonging to I.

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

0=M0M1Mr=ME10

such that each Mi/Mi1is Cohen-Macaulay, and the Krull dimensions of the quotients are increasing:

dimM1/M0<dimM2/M1<<dimMr/Mr1.E11

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

Definition 2.8(Alexander duality). For a square-free monomial ideal I=M1MqS=Kx1xn, the Alexander dualof I, denoted by I, is defined to be:

I=PM1PMqE12

where, PMiis prime ideal generated by xj:xjMi.

Theorem 2.1([7, Proposition 8.2.20], [5, Theorem 3]). LetIbe a square-free monomial ideal inS=Kx1xn.

1. The idealIis componentwise linear ideal if and only ifS/Iis sequentially Cohen-Macaulay.

2. The idealIhas aq-linear resolution if and only ifS/Iis Cohen-Macaulay of dimensionnq.

Remark 2.1.Two special cases, we will be considering in this paper, are when Gis a cycle Cn, or a line graph Lnon vertices x1xnwith edges

ECn=x1x2x2x3xn1xnxnx1;ELn=x1x2x2x3xn1xn

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 ΔtCnis matroid, vertex decomposable, shellable, and Cohen-Macaualay if and only if n=tor n=t+1. For the proof, we shall need the following lemmas and propositions.

Lemma 3.1.LetΔtPnbe a simplicial complex on the pathPn=x1xnand2tn. ThenΔtPnis vertex decomposable.

Proof.If t=n, then ΔnPnis a simplex which is vertex decomposable. Let 2t<nthen one has

ΔtPn=x1xtx2xt+1xnt+1xn.E13

So ΔtPn\xn=x1xtx2xt+1xntxn1. Now, we use induction on the number of vertices of Pnand by induction hypothesis ΔtPn\xnis vertex decomposable. On the other hand, it is clear that linkΔtPnxn=xnt+1xn1. Thus, linkΔtPnxnis a simplex which is not a facet of ΔtPn\xn. Therefore, ΔtPnis vertex decomposable.

Lemma 3.2.LetΔ2Cnbe a simplicial complex onx1xn. ThenΔ2Cnis vertex decomposable.

Proof.Since Δ2Cn=x1x2x2x3xn1xnxnx1then we have

Δ2Cn\xn=x1x2x2x3xn2xn1.E14

By Lemma 2.1 Δ2Cn\xnis vertex decomposable. Also, it is trivial that linkΔ2Cnxn=xn1x1is vertex decomposable and no face of linkΔ2Cnxnis a facet of Δ2Cn\xn. Therefore, Δ2Cnis vertex decomposable.□

Lemma 3.3.LetΔtCnbe a simplicial complex onx1xnand3tn2. ThenΔtCnis not Cohen-Macaulay.

Proof.It suffices to show that IΔtCnhas not a linear resolution. Since IΔtCn=IΔtCnc, then one can easily check that IΔtCn=IntCn. By Theorem 1.1 we have

regIΔtCn=nt1p+d+1.E15

Since 3tn2then one has regIΔtCnntand by Theorem 2.1 ΔtCnis not Cohen-Macaulay.

Proposition 3.1.LetΔtCnbe a simplicial complex onx1xnandt3. ThenΔtCnis vertex decomposable if and only ifn=tort+1.

Proof.By Lemma 3.3, it suffices to show that if n=tor t+1, then ΔtCnis vertex decomposable. If n=t, then ΔnCnis a simplex which is vertex decomposable.

If t=n1, then we have

Δn1Cn=x1xn1x2xnx3xnx1xnx1xn2.

Now, we use induction on the number of vertices of Cnand show that Δn1Cnis vertex decomposable. It is clear that Δn1Cn\xn=x1xn1is a simplex, which is vertex decomposable.

On the other hand,

By induction hypothesis linkΔn1Cnxnis vertex decomposable. It is easy to see that no face of linkΔn1Cnxnis a facet of Δn1Cn\xn. Therefore, Δn1Cnis vertex decomposable.

Proposition 3.2.Δ2Cnis a matroid if and only ifn=3or4.

Proof.If n=3or 4, then it is easy to see that Δ2Cnis a matroid. Now, we prove the converse. It suffices to show that Δ2Cnis not a matroid for all n5. We consider two facets x1x2and xn1xn. Then, we have x1x2\x1xn1=x2xn1and x1x2\x1xn=x2xn. Since x2xn1and x2xnare not the facets of Δ2Cn. So Δ2Cnis not matroid for all n5.

For the simplicial complexes, one has the following implication:

Matroidvertex decomposableshellableCohenMacaulay

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

Theorem 3.1.Lett3. Then the following conditions are equivalent:

1. ΔtCnis matroid;

2. ΔtCnis vertex decomposable;

3. ΔtCnis shellable;

4. ΔtCnis Cohen-Macaulay;

5. n=tort+1.

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=t, then ΔtCnis a simplex which is a matroid.

If n=t+1, then

ΔtCn=x1xtx2xt+1x3xt+1x1xt+1x1xt1.

For any two facets Fand Gof ΔtCnone has FG=t1. We claim that for any two facets Fand Gof ΔtCnand any xiF, there exists a xjGsuch that F\xixjis a facet of ΔtCn. We have to consider two cases. If xiFand xiG, then we choose xjGsuch that xjF. Thus, F\xixj=Gwhich is a facet of ΔtCn.

For other case, if xiFand xiG, then we choose xjGsuch that xjis the same xi. Therefore, F\xixi=Fis a facet of ΔtCn, which completes the proof.

## 4. Vertex decomposability path complexes of trees

As the main result of this section, for all t2, we characterize all such trees whose ΔtGis vertex decomposable. Let Hpnqdenote the double starlike tree obtained by attaching ppendant vertices to one pendant vertex of the path Pnand qpendant vertices to the other pendant vertex of Pn. Also, let Hpnbe graph obtained by attaching ppendant vertices to one pendant vertex of the path Pn.

Remark 4.1.Let Pn=x1xnbe a path on vertices x1xnand H2nbe a graph obtained by attaching two pendant vertices to pendant vertex xn. Then, ΔtH2nis vertex decomposable for all t2.

Proof.By Lemma 3.1 proof is trivial.

Proposition 4.1.LetPn=x1xnbe a path on verticesx1xnandHpnbe a graph obtained by attachingppendant vertices to pendant vertexxn. ThenΔtHpnis vertex decomposable for allt2.

Proof.We prove the proposition by induction on pthe number of pendant vertices to pendant vertex xnof Pn. If p=0or 1, then Hpnis a path and by Lemma 2.1 ΔtHpnis vertex decomposable. If p=2, then by remark 4.1 ΔtHpnis vertex decomposable. Now, let p>2and y1ypbe ppendant vertices to pendant vertex xnof Pn, then one has

Hpn\y1=Hp1nE17

and

ΔtHpn\y1=ΔtHp1n.E18

Therefore by induction hypothesis ΔtHp1nis vertex decomposable. So ΔtHpn\y1is vertex decomposable. If t=3, then we have

It is easy to see that linkΔ3Hpny1is vertex decomposable and y1is a shedding vertex. If t=2or t>3, one has

thus linkΔtHpny1is a simplex, which is not a facet of ΔtHpn\y1, therefore ΔtHpnis vertex decomposable.

Lemma 4.1.Letp=2andq2, ThenΔtH2nqis vertex decomposable for all2tn+2.

Proof.Let H2nqdenote the double starlike tree obtained by attaching two pendant vertices y1y2to pendant vertex x1of path Pnand y1yqbe pendant vertices to pendant vertex xnof Pn. So by proposition 3.2 ΔtH2nq\y1is vertex decomposable. Now, we prove that linkΔtH2nqy1is vertex decomposable. If t=3, then linkΔ3H2nqy1=x1x2x1y2which is vertex decomposable. If t=n+2, then

It is easy to see that linkΔn+2H2nqy1is vertex decomposable. If t=2or 4tn+1, then we have linkΔtH2nqy1=x1xt1. Thus, linkΔtH2nqy1is a simplex, which is vertex decomposable. It is clear that y1is a shedding vertex. □

Proposition 4.2.LetQ1,Q2be two paths of maximum lengthkin treeGandybe a leaf ofGsuch thatyQ1Q2, Q1Q2=L. ThenΔtGis not vertex decomposable.

Proof.Suppose Q1=y1y2ykLx1x2xL1yand.

Q2=y1y2ykLx1x2xL1ybe two paths of length kin Gsuch that Q1Q2=x1x2xL1yand degy=1. Since linkΔkGx1xL1yis disconnected and pure of positive dimension. By remark 1.11 ΔkGis not Cohen-Macaulay and hence, ΔkGis not vertex decomposable. □

Proposition 4.3.LetGbe a double starlike tree such that is not a path. ThenΔtGis vertex decomposable for all2tn+2.

Proof.Let G=Hpnqdenote the double starlike tree obtained by attaching ppendant vertices to one pendant vertex of the path Pnand qpendant vertices to the other pendant vertex of Pn. We prove the theorem by induction on pthe number of pendant vertices to pendant vertex x1of Pn. If p=0or p=1, then by proposition 3.2 ΔtGis vertex decomposable. If p=2, then by Lemma 4.3 ΔtGis vertex decomposable. Now, let p>2and y1ypbe ppendant vertices to pendant vertex x1of Pn. Since G\y1is again double starlike tree on p1pendant vertices. Therefore, by induction hypothesis, ΔtG\y1is vertex decomposable. So ΔtG\y1=ΔtG\y1is vertex decomposable. Let t=2, then linkΔ2Gy1=x1is simplex and vertex decomposable. Let t=3, then linkΔ3Gy1=x2x1y2x1ypx1is vertex decomposable. Let 3<tn+1, then linkΔtGy1=x1x2xt1is simplex and vertex decomposable. Let t=n+2, then linkΔtGy1=x1xny1,x1xny2,,{x1,,xn,ypis a path complex of a starlike tree which is vertex decomposable. It is easy to see that no face of linkΔtGy1is a facet of ΔtG\y1. So ΔtGis vertex decomposable.

Now, we are ready that prove the main result of this section.

Theorem 4.1.LetGbe a tree such that is not a path. ThenΔtGis vertex decomposable for allt2if and only ifG=HpnqorHpn.

Proof.We prove by contradiction. Suppose GHpnqand GHpn. So there exists two paths of maximum length kin Gwhich contain Lcommon vertices such that one of these vertices is a leaf. Therefore, by proposition 4.2 ΔkGis not vertex decomposable, which is a contradiction. By proposition 4.1 and Proposition 4.3, the proof is completed.

## 5. Vertex decomposability path complexes of dynkin graphs

Dynkin diagrams first appeared  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,E6,E7, and E8. Dynkin diagrams are closely related to coxter graphs that appeared in geometry (see ).

After that, Dynkin diagrams appeared in many branches of mathematics and beyond, in particular in representation theory. In , 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 Qbe a finite quiver and Q¯the undirected graph obtained from Qby deleting the orientation of all arrows.

Theorem 5.1.(Gabriel theorem). A connected quiverQis of Finite type if and only if the graphQ¯is one of the following simply laced Dynkin diagrams:An,Dn,E6,E7orE8.

Let Lnbe a line graph on vertices x1xnand xjyjbe whisker of Lnat xjwith 3jn1.

We obtain some condition that ΔtLnWxjis vertex decomposable, as an application of our results, vertex decomposability path complexes of Dynkin graphs are shown. Throughout this section, we assume LnWxjbe an undirected graph. By Lemma 3.1, we have the following corollary:

Corollary 5.1.LetAnbe a Dynkin graph and2tn. ThenΔtAnis vertex decomposable.

Proposition 5.1.LetLnbe a line graph on verticesx1xnandxn1yn1be whisker ofLnatxn1.

Then, ΔtLnWxn1is vertex decomposable for all 2tn.

Proof. Then by proposition 4.1 proof is trivial.

Corollary 5.2.LetDnbe a Dynkin graph andn4. ThenΔtDnis vertex decomposable for all2tn.

Proof.We know that Dn=LnWxn1. So by proposition 4.3 ΔtDnis vertex decomposable. □

Theorem 5.2.LetLnbe a line graph on verticesx1xnandxjyjbe whisker ofLnatxjwith3jn2.

Then, ΔtLnWxjis vertex decomposable if and only if 2t3or nt>α, where α=mindyjx1dyjxn.

Proof.We first show that ΔtLnWxjis not vertex decomposable for all 4tα. It is well known that if Δis a Cohen-Macaulay simplicial complex, then linkΔFis Cohen-Macaulay for each face Fof Δ.

Also, we know that all Cohen-Macaulay complexes of positive dimension are connected. It is easy to see that linkΔtLnWxjxjyjis disconnected and pure of positive dimension.

This implies that ΔtLnWxjis not Cohen- Macaulay and hence ΔtLnWxjis not vertex decomposable without loss of generality we can assume that α=dyjx1if t=2or nt>α, one has:

ΔtLnWxj=x1xt,x2xt+1,,xj1xjxj+t2,yjxjxj+1xj+t2,,xnt+1xn

So ΔtLnWxj\yj=ΔtLnand by Lemma [4, 6, 7, 9, 10, 18–20] ΔtLnWxj\yjis vertex decomposable. On the other hand, we have linkΔtLnWxjyj=xjxj+1xj+t2that is a simplex and vertex decomposable.

If t=3, then

Δ3LnWxj=x1x2x3,x2x3x4,,xj1xjyj,yjxjxj+1,,xn2xn1xn

and Δ3LnWxj\yj=Δ3Lnwhich is vertex decomposable.

It is easy to see that linkΔtLnWxjyj=xj1xjxjxj+1is vertex decomposable and yjis a shedding vertex.

Corollary 5.3.LetE6be a Dynkin graph. ThenΔtE6is vertex decomposable if and only if2t3ort=5.

Proof.Since E6=L5Wx3, so by Theorem 5.2 the proof is completed.

Corollary 5.4.LetE7be a Dynkin graph. ThenΔtE7is vertex decomposable if and only if2t3or5t6.

Proof.We know that E7=L6Wx3and the proof follow from Theorem 5.2.

Corollary 5.5.LetE8be a Dynkin graph. ThenΔtE8is vertex decomposable if and only if2t3or5t7.

## 6. Stanley decompositions

Let Rbe any standard graded K-algebra over an infinite field K, i.e, Ris a finitely generated graded algebra R=i0Risuch that R0=Kand Ris generated by R1. There are several characterizations of the depth of such an algebra. We use the one that depthRis the maximal length of a regular R-sequence consisting of linear forms. Let xF=iFxibe a squarefree monomial for some Fnand Zx1xn. The K-subspace xFKZof S=Kx1xnis the subspace generated by monomials xFu, where uis a monomial in the polynomial ring KZ. It is called a squarefree Stanley space if xi:iFZ. The dimension of this Stanley space is Z. Let Δbe a simplicial complex on x1xn. A squarefree Stanley decomposition Dof KΔis a finite direct sum iuiKZiof squarefree Stanley spaces, which is isomorphic as a Zn-graded K-vector space to KΔ, i.e.

KΔiuiKZiE21

We denote by sdepthDthe minimal dimension of a Stanley space in Dand we define sdepthKΔ=maxsdepthD, where Dis a Stanley decomposition of KΔ. Stanley conjectured in  the upper bound for the depth of KΔas the following:

depthKΔsdepthKΔE22

Also, we recall another conjecture of Stanley. Let Δbe again a simplicial complex on x1xnwith facets G1,,Gt. The complex Δis called partitionable if there exists a partition Δ=i=1tFiGiwhere FiGiare suitable faces of Δ. Here, the interval FiGiis the set of faces HΔ:FiHGi. In [10, 18] respectively Stanley conjectured each Cohen-Macaulay simplicial complex is partitionable [16, 17, 18, 19, 20]. This conjecture is a special case of the previous conjecture. Indeed, Herzog, Soleyman Jahan, and Yassemi  proved that for Cohen-Macaulay simplicial complex Δon x1xnwe have that depthKΔsdepthKΔif and only if Δis 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.ifn=tort+1thenΔtCnis partitionable and Stanley’s conjecture holds forKΔtCn.

Corollary 6.2.LetGbe a tree such that is not a path. ifG=HpnqorG=HpnthenΔtGis vertex decomposable for allt2and Stanley’s conjecture holds forKΔtG.