$k$-shellable simplicial complexes and graphs

TL;DR

This paper explores the properties of $k$-shellable simplicial complexes and graphs, demonstrating their relation to shellability, face rings, and the Stanley conjecture, and extending existing results in the field.

Contribution

It proves that $k$-shellable complexes are expansions of shellable complexes and shows their face rings satisfy the Stanley conjecture, extending known results to a broader class.

Findings

$k$-shellable complexes are expansions of shellable complexes

Face rings of pure $k$-shellable complexes satisfy Stanley conjecture

Characterizations of $k$-shellable graphs extend previous results

Abstract

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture. Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrill\'{o}n-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

Equations

$${\mathrm{lk}}_{\Delta }(F)=\{G\in \Delta :G\cap F=\varnothing \text{mbox}andG\cup F\in \Delta \}$$

$${\mathrm{dl}}_{\Delta }(F)=\{G\in \Delta :F⊈G\}.$$

$$F^{\alpha }=\{x_{i_{1}1},\dots ,x_{i_1{k}_{i_1}},\dots ,x_{i_{r}1},\dots ,x_{i_r{k}_{i_r}}\}$$

$$x_{11},\dots ,x_{1k_1},\dots ,x_{n1},\dots ,x_{n{k}_n}.$$

$$I^{\alpha }=\sum _{i=1}^r{P}_1^{{\nu }_1(u_i)}\dots P_n^{{\nu }_n(u_i)}\subset S^{\alpha }$$

$$\begin{array}[]{rl}I^{\alpha}=&(x_{11},x_{12})^{2}(x_{21},x_{22})+(x_{11},x_{12})(x_{31})+(x_{21},x_{22})(x_{31})^{2}\\ =&(x^{2}_{11}x_{21},x_{11}x_{12}x_{21},x^{2}_{12}x_{21},x^{2}_{11}x_{22},x_{11}x_{12}x_{22},x^{2}_{12}x_{22},x_{11}x_{31},x_{12}x_{31},x_{21}x^{2}_{31},x_{22}x^{2}_{31})\end{array}$$

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\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array}$$

$$F_1\cup G_1,F_1\cup G_2,\dots ,F_1\cup G_s,\dots ,F_r\cup G_1,F_r\cup G_2,\dots ,F_r\cup G_s$$

$$\begin{array}[]{rl}\Delta^{[k]}_{j}=&\langle F^{[k]}_{j}\rangle\cap\langle F^{[k]}_{1},\ldots,F^{[k]}_{j-1}\rangle\\ =&\langle F^{[k]}_{j}\backslash\{x_{i_{1}1},\ldots,x_{i_{1}k}\},\ldots,F^{[k]}_{j}\backslash\{x_{i_{t}1},\ldots,x_{i_{t}k}\}\rangle.\end{array}$$

$${\Delta }_j^{[k]}=⟨F_j^{[k]}\backslash \{x_{i_1}{\}}^{[k]},\dots ,F_j^{[k]}\backslash \{x_{i_t}{\}}^{[k]}⟩.$$

$$\mathcal{F}:0=M_0\subset M_1\subset \dots \subset M_{r-1}\subset M_r=M$$

$$M_i=\bigcap _{l=1}^{r-i}{P}_{F_l^c}and{M}_i/M_{i-1}\cong \prod _{j=1}^{k^{a_i}}S/P_{F_{r-i+1}^c}(-{\mathbf{a}}_{ij}),$$

$$\bigcap _{j=1}^{i-1}{P}_{F_j^c}+P_{F_i^c}=P_{F_i^c}+P_{{\sigma }_{i1}}\dots P_{{\sigma }_{i{a}_i}}$$

$$\begin{array}[]{ll}(\bigcap^{i-1}_{j=1}P_{F^{c}_{j}})/(\bigcap^{i}_{j=1}P_{F^{c}_{j}})&\cong\bigcap^{i-1}_{j=1}P_{F^{c}_{j}}+P_{F^{c}_{i}}/P_{F^{c}_{i}}\\ &\cong P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}+P_{F^{c}_{i}}/P_{F^{c}_{i}}\\ &\cong P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}/(P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}\cap P_{F^{c}_{i}})\\ &\cong P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}/P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}P_{F^{c}_{i}}\end{array}$$

$$\begin{array}[]{rcc}\theta:\prod^{k^{a_{i}}}S&\rightarrow&P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}/P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}P_{F^{c}_{i}}\\ (r_{1},\ldots,r_{k^{a_{i}}})&\mapsto&\sum^{k^{a_{i}}}_{j}r_{j}f_{ij}+P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}P_{F^{c}_{i}}\end{array}$$

$$P_{{\sigma }_{i1}}\dots P_{{\sigma }_{i{a}_i}}/P_{{\sigma }_{i1}}\dots P_{{\sigma }_{i{a}_i}}{P}_{F_i^c}\cong \prod _{j=1}^{k^{a_i}}S/P_{F_i^c}(-{\mathbf{a}}_{ij}).$$

$$(0)=I_{\Delta }\subset I_{{\Delta }_{r-1}}\subset \dots \subset I_{{\Delta }_1}\subset K[\Delta ]$$

$$(0)=I_{{\Delta }^{[k]}}\subset I_{{\Delta }_{r-1}^{[k]}}\subset \dots \subset I_{{\Delta }_1^{[k]}}\subset K[{\Delta }^{[k]}].$$

$$\begin{array}[]{rl}\sum^{j}_{t=1}(f_{t})+\bigcap^{i}_{t=1}P_{F^{c}_{t}}/\sum^{j-1}_{t=1}(f_{t})+\bigcap^{i}_{t=1}P_{F^{c}_{t}}\cong&(f_{j})/(f_{j})\cap\big{(}(f_{1},\ldots,f_{j-1})+\bigcap^{i}_{t=1}P_{F^{c}_{t}}\big{)}\\ \cong&(f_{j})/f_{j}P_{L_{ij}}.\end{array}$$

$$\mathrm{sdepth}(M)=\max \{\mathrm{sdepth}(\mathcal{D}):\mathcal{D}\text{mbox}isaStanleydecompositionofM\},$$

$$\max \{n_i:i=2,\dots ,k^s\}=n_{k^s}=(k-1)s.$$

$$P_{F_1}\dots P_{F_s}=x_1(f_1,\dots ,f_{k^{s-1}})+\dots +x_k(f_1,\dots ,f_{k^{s-1}}).$$

$$(x_1{f}_1,\dots ,x_1{f}_{k^{s-1}},\dots ,x_k{f}_1,\dots ,x_k{f}_{k^{s-2}}):x_k{f}_{k^{s-1}}=(x_1,\dots ,x_{k-1})+(f_1,\dots ,f_{k^{s-2}}):f_{k^{s-1}}.$$

$$\mathcal{F}:0=M_0\subset M_1\subset \dots \subset M_r=M$$

$$K[{\Delta }^{[k]}]=\underset{i=1}{\overset{r}{⨁}}\underset{j=1}{\overset{k^{a_i}}{⨁}}{u}_{ij}K[Z_{ij}]$$

$$\begin{array}[]{rl}|Z_{ik^{a_{i}}}|&=kn-(|Q_{k^{a_{i}}}|+|F^{c}_{i}|)\\ &=kn-((k-1)a_{i}+k\mathrm{ht}(P_{G^{c}_{i}}))\\ &=(k-1)(n-\mathrm{ht}(P_{G^{c}_{i}})-a_{i})+n-\mathrm{ht}(P_{G^{c}_{i}})\\ &=(k-1)(|G_{i}|-a_{i})+|G_{i}|\\ &\geq|G_{i}|=\dim(K[\Delta]).\end{array}$$

$$E(G)\cup \{x_{i^{\prime }}{x}_j:x_i{x}_j\in E(G)\}.$$

$$V(G^{\prime })=\{x_{ij}:i=1,\dots ,n\text{mbox}andj=1,\dots .k_i\}$$

$$E(G^{\prime })=\{x_{ir}{x}_{js}:x_i{x}_j\in E(G),r=1,\dots ,k_i\text{mbox}andj=1,\dots ,k_i\}.$$

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Full text

Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

Introduction

Let $\Delta$ be a simplicial complex on the vertex set $X:=\{x_{1},\ldots,x_{n}\}$. Denote by $\langle F_{1},\ldots,F_{r}\rangle$ the simplicial complex $\Delta$ with facets $F_{1},\ldots,F_{r}$. $\Delta$ is called shellable if its facets can be given a linear order $F_{1},\ldots,F_{r}$, called a shelling order, such that for all $2\leq j$, the subcomplex $\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$ is pure of dimension $\dim(F_{j})-1$ (see [3] for probably the earliest definition of this term or as well [2] for a more recent exposition). Studying combinatorial properties of shellable simplicial complexes and algebraic constructions of their face rings and also the edge ideals associated to shellable graphs is a current trend in combinatorics and commutative algebra. See for example [2, 3, 5, 9, 12, 21].

In this paper, we recall from [17] the concept of $k$-shellability, and extend some results obtained previously by researchers. Actually, $k$ is a positive integer and for $k=1$, $1$-shellability coincides with shellability.

Richard Stanley [20], in his famous article “Linear Diophantine equations and local cohomology”, made a striking conjecture predicting an upper bound for the depth of a multigraded module. This conjecture is nowadays called the Stanley conjecture and the conjectured upper bound is called the Stanley depth of a module. The Stanley conjecture has become quite popular, with numerous publications dealing with different aspects of the Stanley depth. Although a counterexample has apparently recently been found to the Stanley conjecture (see [10]), this makes it perhaps even more interesting to explore the relationship between depth and Stanley depth.

Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial ring over a field $K$. Dress proved in [9] that the simplicial complex $\Delta$ is shellable if and only if its face ring is clean. It is also known that cleanness implies pretty cleanness. Furthermore, Herzog and Popescu [12, Theorem 6.5] proved that, if $I\subset S$ is a monomial ideal, and $S/I$ is a multigraded pretty clean ring, then the Stanley conjecture holds for $S/I$. It follows that, for a shellable simplicial complex $\Delta$, the face ring $K[\Delta]=S/I_{\Delta}$ satisfies the Stanley conjecture where $I_{\Delta}$ denotes the Stanley-Reisner ideal of $\Delta$. We extend this result, in pure case, by showing that the face ring of a $k$-shellable simplicial complex satisfies the Stanley conjecture (see Theorem 3.2). We obtain this result by extending Proposition 8.2 of [12] and presenting a filtration for the face ring of a $k$-shellable simplicial complex in Theorem 2.9.

A simple graph $G$ is called shellable if its independence complex $\Delta_{G}$ is a shellable simplicial complex. Shellable graphs were studied by several researchers in recent years. For example, Van Tuyl and Villarreal in [21] classified all of shellable bipartite graphs. Also, Castrillón and Cruz characterized the shellable graphs and clutters by using the properties of simplicial vertices, shedding vertices and shedding faces ([5]).

Here, we present some characterizations of $k$-shellable graphs and extend some results of [5], [7] and [21] (see Theorems 4.5, 4.7 and 4.10). Our idea is to define a new notion, called a $k$-simplicial set, which is a generalization of the notion of simplicial vertex defined in [8] or [14].

1. Preliminaries

For basic definitions and general facts on simplicial complexes, we refer to Stanley’s book [19].

A simplicial complex $\Delta$ is pure if all of its facets (maximal faces) are of the same dimension. The link and deletion of a face $F$ in $\Delta$ are defined respectively

[TABLE]

and

[TABLE]

Let $G$ be a simple (no loops or multiple edges) undirected graph on the vertex set $V(G)=X$ and the edge set $E(G)$. The independence complex of $G$ is denoted by $\Delta_{G}$ and $F$ is a face of $\Delta_{G}$ if and only if there is no edge of $G$ joining any two vertices of $F$. The edge ideal of $G$ is defined a quadratic squarefree monomial ideal $I(G)=(x_{i}x_{j}:x_{i}x_{j}\in E(G))$. It is known that $I(G)=I_{\Delta_{G}}$. We say $G$ is a shellable graph if $\Delta_{G}$ is a shellable simplicial complex.

In the following we recall the concept of expansion functor in a combinatorial and an algebraic setting from [16] and [1], respectively.

Let $\alpha=(k_{1},\ldots,k_{n})$ be an $n$-tuple with positive integer entries in $\mathbb{N}^{n}$. For $F=\{x_{i_{1}},\ldots,x_{i_{r}}\}\subseteq X$ define

[TABLE]

as a subset of $X^{\alpha}:=\{x_{11},\ldots,x_{1k_{1}},\ldots,x_{n1},\ldots,x_{nk_{n}}\}$. $F^{\alpha}$ is called the expansion of $F$ with respect to $\alpha$.

For a simplicial complex $\Delta=\langle F_{1},\ldots,F_{r}\rangle$ on $X$, we define the expansion of $\Delta$ with respect to $\alpha$ as the simplicial complex $\Delta^{\alpha}=\langle F^{\alpha}_{1},\ldots,F^{\alpha}_{r}\rangle$ (see [16]).

In [1] Bayati and Herzog defined the expansion functor in the category of finitely generated multigraded $S$-modules and studied some homological behaviors of this functor. We recall the expansion functor defined by them only in the category of monomial ideals and refer the reader to [1] for more general case in the category of finitely generated multigraded $S$-modules.

Set $S^{\alpha}$ a polynomial ring over $K$ in the variables

[TABLE]

Whenever $I\subset S$ is a monomial ideal minimally generated by $u_{1},\ldots,u_{r}$, the expansion of $I$ with respect to $\alpha$ is defined

[TABLE]

where $P_{j}=(x_{j1},\ldots,x_{jk})$ is a prime ideal of $S^{\alpha}$ and $\nu_{j}(u_{i})$ is the exponent of $x_{j}$ in $u_{i}$.

Example 1.1*.*

Let $I\subset K[x_{1},\ldots,x_{3}]$ be a monomial ideal minimally generated by $G(I)=\{x^{2}_{1}x_{2},x_{1}x_{3},x_{2}x^{2}_{3}\}$ and let $\alpha=(2,2,1)\in\mathbb{N}^{3}$. Then

[TABLE]

It was shown in [1] that the expansion functor is exact and so $(S/I)^{\alpha}=S^{\alpha}/I^{\alpha}$. The following lemma implies that two above concepts of expansion functor are related.

Lemma 1.2**.**

([16, Lemma 2.1]) For a simplicial complex $\Delta$ and $\alpha\in\mathbb{N}^{n}$ we have $(I_{\Delta})^{\alpha}=I_{\Delta^{\alpha}}$. In particular, $K[\Delta]^{\alpha}=K[\Delta^{\alpha}]$.

In this paper we just study the functors $\alpha=(k_{1},\ldots,k_{n})\in\mathbb{N}^{n}$ with $k_{i}=k_{j}$ for all $i,j$. For convenience, we set $\alpha={[k]}$ when every component of $\alpha$ is equal to $k\in\mathbb{N}$. We call $I^{[k]}$ (resp. $\Delta^{[k]}$) the expansion of $I$ (resp. $\Delta$) with respect to $k$.

2. Some combinatorial and algebraic properties of $k$-shellable complexes

The notion of $k$-shellable simplicial complexes was first introduced by Emtander, Mohammadi and Moradi [11] to provide a natural generalization of shellability. It was shown in [11, Theorem 6.8] that a simplicial complex $\Delta$ is $k$-shellable if and only if the Stanley-Reisner ideal of its Alexander dual has $k$-quotionts, i.e. there exists an ordering $u_{1},\ldots,u_{r}$ of the minimal generators of $I_{\Delta^{\vee}}$ such that if we for $s=1,\ldots,t$, put $I_{s}=(u_{1},\ldots,u_{s})$, then for every $s$ there are monomials $v_{s_{i}}$ , $i=1,\ldots,r_{s}$, $\deg(v_{s_{i}})=k$ for all $i$, such that $I_{s}:u_{s}=(v_{s_{1}},\ldots,v_{s_{r_{s}}}).$

In [17], we gave another definition of $k$-shellability and having $k$-quotients by adding a condition to Emtander, Mohammadi and Moradi’s. In our definition the colon ideals $I_{s}:u_{s}$ were generated by regular sequences for all $s$ and in this way, all of structural properties of monomial ideals with linear quotients were generalized. The reader is referred to [13] for the definition of monomial ideals with linear quotients.

Definition 2.1**.**

([17]) Let $\Delta$ be a $d$-dimensional simplicial complex on $X$ and let $k$ be an integer with $1\leq k\leq d+1$. $\Delta$ is called $k$-shellable if its facets can be ordered $F_{1},\ldots,F_{r}$, called $k$-shelling order, such that for all $j=2,\ldots,r$, the subcomplex $\Delta_{j}=\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$ satisfies the following properties:

1. (i)

It is generated by a nonempty set of maximal proper faces of $\langle F_{j}\rangle$ of dimension $|F_{j}|-k-1$; 2. (ii)

If $\Delta_{j}$ has more than one facet then for every two disjoint facets $\sigma,\tau\in\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$ we have $F_{j}\subseteq\sigma\cup\tau$.

Remark 2.2*.*

It follows from the definition that two concepts 1-shellability and shellability coincide.

Remark 2.3*.*

Note that the notions of $1$-shellability in our sense and Emtander, Mohammadi and Moradi’s coincide. Although, for $k>1$, a simplicial complex may be $k$-shellable in their concept and not in ours. For example, consider the complex $\Delta=\langle abc,aef,cdf\rangle$ on $\{a,b,\ldots,f\}$. It is easy to check that $\Delta$ is $2$-shellable in the sense of [11] but not in ours.

In the following proposition we describe some the combinatorial properties of $k$-shellable complexes.

Proposition 2.4**.**

Let $\Delta$ be a $d$-dimensional (not necessarily pure) simplicial complex on $X$ and let $k$ be an integer with $1\leq k\leq d+1$. Suppose that the facets of $\Delta$ can be ordered $F_{1},\ldots,F_{r}$. Then the following conditions are equivalent:

1. (a)

$F_{1},\ldots,F_{r}$* is a $k$-shelling of $\Delta$;* 2. (b)

for every $1\leq j\leq r$ there exist the subsets $E_{1},\ldots,E_{t}$ of $X$ such that the $E_{i}$ are mutually disjoint and $|E_{i}|=k$ for all $i$ and the set of the minimal elements of $\langle F_{1},\ldots,F_{j}\rangle\backslash\langle F_{1},\ldots,F_{j-1}\rangle$ is $\{\{a_{1},\ldots,a_{t}\}:a_{i}\in E_{i}\ \mbox{for all}\ i\}$; 3. (c)

for all $i,j$, $1\leq i<j\leq r$, there exist $x_{1},\ldots,x_{k}\in F_{j}\backslash F_{i}$ and some $l\in\{1,\ldots,j-1\}$ with $F_{j}\backslash F_{l}=\{x_{1},\ldots,x_{k}\}$.

Proof.

(a)$\Rightarrow$(b): Let $\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle=\langle F_{j}\backslash\sigma_{1},\ldots,F_{j}\backslash\sigma_{t}\rangle$ where $|\sigma_{i}|=k$ for all $i$. Since for all $i\neq i^{\prime}$, $F_{j}\subseteq(F_{j}\backslash\sigma_{i})\cup(F_{j}\backslash\sigma_{i^{\prime}})$, we have $\sigma_{i}\cap\sigma_{i^{\prime}}=\emptyset$. Hence the minimal elements of $\langle F_{1},\ldots,F_{j}\rangle\backslash\langle F_{1},\ldots,F_{j-1}\rangle$ are in the form $\{a_{1},\ldots,a_{t}\}$ where $a_{i}\in\sigma_{i}$ for all $i$.

(b)$\Rightarrow$(c): For all $i$, suppose that $E_{i}=\{x_{i1},\ldots,x_{ik}\}$. Let $1\leq i<j\leq r$ and let $\{x_{1i_{1}},\ldots,x_{ti_{t}}\}$ be a minimal element of $\langle F_{1},\ldots,F_{j}\rangle\backslash\langle F_{1},\ldots,F_{j-1}\rangle$. Because $\{x_{1i_{1}},\ldots,x_{ti_{t}}\}\nsubseteq F_{i}$ we may assume that $x_{1i_{1}}\in F_{j}\backslash F_{i}$. We claim that $x_{11},\ldots,x_{1k}\in F_{j}\backslash F_{i}$. Suppose, on the contrary, that for some $s$, $x_{1s}\not\in F_{j}\backslash F_{i}$ then $x_{1s}\in F_{i}$ and so $x_{1s}\not\in\langle F_{1},\ldots,F_{j}\rangle\backslash\langle F_{1},\ldots,F_{j-1}\rangle$. It follows that $\{x_{1s},x_{2i_{2}},\ldots,x_{ti_{t}}\}$ is not a minimal element of $\langle F_{1},\ldots,F_{j}\rangle\backslash\langle F_{1},\ldots,F_{j-1}\rangle$, a contradiction. Therefore $E_{1}\subseteq F_{j}\backslash F_{i}$.

Now suppose that for all $l<j$, if $E_{1}$ is contained in $F_{j}\backslash F_{l}$ then $E_{1}\subsetneqq F_{j}\backslash F_{l}$. Then there exists $y\in F_{j}\backslash E_{1}$ such that $\{y,x_{2i_{2}},\ldots,x_{ti_{t}}\}$ is a minimal element of $\langle F_{1},\ldots,F_{j}\rangle\backslash\langle F_{1},\ldots,F_{j-1}\rangle$ different from the elements of $\{\{a_{1},\ldots,a_{t}\}:a_{i}\in E_{i}\ \mbox{for all}\ i\}$, a contradiction. Therefore there exists $l<j$ with $F_{j}\backslash F_{l}=E_{1}$.

(c)$\Rightarrow$(a): Let $F\in\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$. Then $F\subseteq F_{i}$ for some $i<j$. By the condition (c), there exist $x_{1},\ldots,x_{k}\in F_{j}\backslash F_{i}$ and some $l\in\{1,\ldots,j-1\}$ with $F_{j}\backslash F_{l}=\{x_{1},\ldots,x_{k}\}$. But $F_{j}\backslash\{x_{1},\ldots,x_{k}\}$ is a proper face of $\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$, because $F_{j}\backslash\{x_{1},\ldots,x_{k}\}=F_{j}\cap F_{l}$. Moreover, $F_{j}\backslash\{x_{1},\ldots,x_{k}\}$ is a maximal face. Finally, since $F$ is contained in $F_{j}\backslash\{x_{1},\ldots,x_{k}\}$, the assertion is completed. ∎

Example 2.5*.*

The Figure 1 indicates the pure shellable and pure 2-shellable simplicial complexes of dimensions 1, 2 and 3 with 3 facets.

Theorem 2.6**.**

Let $\Delta$ be a $k$-shellable complex and $\sigma$ a face of $\Delta$. Then $\mathrm{lk}_{\Delta}(\sigma)$ is again $k$-shellable.

Proof.

Since $\Delta$ is $k$-shellable, so there exists an $k$-shelling order $F_{1},\ldots,F_{r}$ of facets of $\Delta$. Let $F_{i_{1}},\ldots,F_{i_{t}}$ where $i_{1}<\ldots<i_{t}$ be all of facets which contain $\sigma$. We claim that $F_{i_{1}}\backslash\sigma,\ldots,F_{i_{t}}\backslash\sigma$ is a $k$-shelling order of $\mathrm{lk}_{\Delta}(\sigma)$. To this end we want to show that the condition (c) of Proposition 2.4 holds.

Set $G_{j}=F_{i_{j}}\backslash\sigma$. Consider $l,m$ with $1\leq l<m\leq t$. By $k$-shellability of $\Delta$, there are $x_{1},\ldots,x_{k}\in F_{i_{m}}\backslash F_{i_{l}}=(F_{i_{m}}\backslash\sigma)\backslash(F_{i_{l}}\backslash\sigma)=G_{m}\backslash G_{l}$ such that for some $s<i_{m}$ we have $\{x_{1},\ldots,x_{k}\}=F_{i_{m}}\backslash F_{s}$. It follows from $\sigma\subset F_{i_{m}}$ and $F_{i_{m}}\backslash F_{s}=\{x_{1},\ldots,x_{k}\}$ that $\sigma\subset F_{s}$. This implies that $F_{s}$ is among the list $F_{i_{1}},\ldots,F_{i_{m}}$. Let $F_{i_{m^{\prime}}}=F_{s}$. Hence $G_{m}\backslash G_{m^{\prime}}=\{x_{1},\ldots,x_{k}\}$ and the assertion is completed. ∎

For the simplicial complexes $\Delta_{1}$ and $\Delta_{2}$ defined on disjoint vertex sets, the join of $\Delta_{1}$ and $\Delta_{2}$ is $\Delta_{1}\cdot\Delta_{2}=\{\sigma\cup\tau:\ \sigma\in\Delta_{1},\tau\in\Delta_{2}\}$.

Theorem 2.7**.**

The simplicial complexes $\Delta_{1}$ and $\Delta_{2}$ are $k$-shellable if and only if $\Delta_{1}\cdot\Delta_{2}$ is $k$-shellable.

Proof.

Let $\Delta_{1}$ and $\Delta_{2}$ be $k$-shellable. Let $F_{1},\ldots,F_{r}$ and $G_{1},\ldots,G_{s}$ be, respectively, the $k$-shelling orders of $\Delta_{1}$ and $\Delta_{2}$. We claim that

[TABLE]

is a $k$-shelling order of $\Delta_{1}\cdot\Delta_{2}$.

Let $F_{i}\cup G_{j}$ be a facet of $\Delta_{1}\cdot\Delta_{2}$ which comes after $F_{p}\cup G_{q}$ in the above order. We have some cases:

Let $p<i$. Since $\Delta_{1}$ is $k$-shellable, there exist $u_{1},\ldots,u_{k}\in F_{i}\backslash F_{p}$ and some $l<i$ such that $F_{i}\backslash F_{l}=\{u_{1},\ldots,u_{k}\}$. It follows that $u_{1},\ldots,u_{k}\in(F_{i}\cup G_{j})\backslash(F_{p}\cup G_{q})$ and $(F_{i}\cup G_{j})\backslash(F_{l}\cup G_{j})=\{u_{1},\ldots,u_{k}\}$.

Let $p=i$ and $q<j$. Since $\Delta_{2}$ is $k$-shellable, there exist $v_{1},\ldots,v_{k}\in G_{j}\backslash G_{q}$ and some $m<j$ such that $G_{j}\backslash G_{m}=\{v_{1},\ldots,v_{k}\}$. Therefore we obtain $v_{1},\ldots,v_{k}\in(F_{i}\cup G_{j})\backslash(F_{p}\cup G_{q})$ and $(F_{i}\cup G_{j})\backslash(F_{i}\cup G_{m})=\{v_{1},\ldots,v_{k}\}$.

Conversely, suppose that $\Delta_{1}\cdot\Delta_{2}$ is $k$-shellable with the $k$-shelling order $F_{i_{1}}\cup G_{j_{1}},\ldots,F_{i_{t}}\cup G_{j_{t}}$. Let $F_{s_{1}},\ldots,F_{s_{r}}$ be the ordering obtained from $F_{i_{1}}\cup G_{j_{1}},\ldots,F_{i_{t}}\cup G_{j_{t}}$ after removing the repeated facets beginning on the left-hand. Then it is easy to check that $F_{s_{1}},\ldots,F_{s_{r}}$ is a $k$-shelling order of $\Delta_{1}$. In a similar way, it is shown that $\Delta_{2}$ is $k$-shellable. ∎

The following theorem, relates the expansion of a shellable complex to a $k$-shellable complex.

Theorem 2.8**.**

Let $\Delta$ be a simplicial complex and $k\in\mathbb{N}$. Then $\Delta$ is shellable if and only if $\Delta^{[k]}$ is $k$-shellable.

Proof.

Let $\Delta=\langle F_{1},\ldots,F_{r}\rangle$ and let $\Delta_{j}=\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$ for $j=2,\ldots,r$. Fix an integer $j$. If $\Delta_{j}=\langle F_{j}\backslash x_{i_{1}},\ldots,F_{j}\backslash x_{i_{t}}\rangle$, then

[TABLE]

Now by the Definition 2.1, if $F_{1},\ldots,F_{r}$ is a shelling order of $\Delta$ then $F^{[k]}_{1},\ldots,F^{[k]}_{r}$ is a $k$-shelling order of $\Delta^{[k]}$.

Conversely, suppose that $F^{[k]}_{1},\ldots,F^{[k]}_{r}$ is a $k$-shelling order of $\Delta^{[k]}$ and set $\Delta^{[k]}_{j}=\langle F^{[k]}_{j}\rangle\cap\langle F^{[k]}_{1},\ldots,F^{[k]}_{j-1}\rangle$ for $j=2,\ldots,r$. Fix an index $j$. Hence $\Delta^{[k]}_{j}=\langle F^{[k]}_{j}\backslash\sigma_{1},\ldots,F^{[k]}_{j}\backslash\sigma_{t}\rangle$ with $|\sigma_{i}|=k$ for all $i$. By Proposition 2.4(b), $\sigma_{l}\cap\sigma_{m}=\emptyset$ for all $l\neq m$. We claim that for every $i$, $\sigma_{i}$ is the expansion of a singleton set. Suppose, on the contrary, that for some $\sigma_{s}$ we have $x_{i_{1}l},x_{i_{2}m}\in\sigma_{s}$ with $i_{1}\neq i_{2}$ and let $F^{[k]}_{j}\cap F^{[k]}_{s^{\prime}}=F^{[k]}_{j}\backslash\sigma_{s}$ for some $s^{\prime}$. It follows from $|\sigma_{s}|=k$ that $x_{i_{1}l^{\prime}}\not\in\sigma_{s}$ for some $l^{\prime}$ with $1\leq l^{\prime}\leq k$. In particular, we conclude that $x_{i_{1}l}\not\in F^{[k]}_{s^{\prime}}$ but $x_{i_{1}l^{\prime}}\in F^{[k]}_{s^{\prime}}$. This is a contradiction, because $F^{[k]}_{s^{\prime}}$ is the expansion of $F_{s^{\prime}}$.

Therefore we conclude that for all $j=2,\ldots,r$, the complex $\Delta^{[k]}_{j}$ is in the form

[TABLE]

Finally, for all $j$, $\Delta_{j}=\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$ will be in the form $\Delta_{j}=\langle F_{j}\backslash x_{i_{1}},\ldots,F_{j}\backslash x_{i_{t}}\rangle$. This implies that $\langle F_{j}\rangle\cap\langle F_{1},\ldots,F_{j-1}\rangle$ is pure of dimension $\dim(F_{j})-1$ for all $j\geq 2$, as desired. ∎

Let $R$ be a Noetherian ring and $M$ be a finitely generated multigraded $R$-module. We call

[TABLE]

a multigraded finite filtration of submodules of $M$ if there exist the positive integers $a_{1},\ldots,a_{r}$ such that $M_{i}/M_{i-1}\cong\prod^{a_{i}}_{j=1}R/P_{i}(-\mathbf{a}_{ij})$ for some $P_{i}\in{\mathrm{Supp}}(M)$. $\mathcal{F}$ is called a multigraded prime filtration if $a_{1}=\ldots=a_{r}=1$. It is well known that every finitely generated multigraded $R$-module $M$ has a multigraded prime filtration (see for example [15, Theorem 6.4]). In the following we present a multigraded finite filtration for the face ring of a $k$-shellable simplicial complex which we need in Section 3.

For $F\subset X$. We set $F^{c}=X\backslash F$ and $P_{F}=(x_{i}:x_{i}\in F)$.

Theorem 2.9**.**

Let $\Delta$ be a simplicial complex and $k$ a positive integer. If $F_{1},\ldots,F_{r}$ is a $k$-shelling order of $\Delta$ then there exists a filtration $0=M_{0}\subset M_{1}\subset\ldots\subset M_{r}=S/I_{\Delta}$ with

[TABLE]

for all $i=1,\ldots,r$. Here $a_{i}=|\mathbf{a}_{ij}|$ for all $j=1,\ldots,k^{a_{i}}$.

Proof.

We set $a_{1}=0$ and for each $i>2$ we denote by $a_{i}$ the number of facets of $\langle F_{i}\rangle\cap\langle F_{1},\ldots,F_{i-1}\rangle$. If $F_{1},\ldots,F_{r}$ is a $k$-shelling of $\Delta$, then for $i=2,\ldots,r$ we have

[TABLE]

where $\sigma_{il}=F_{i}\backslash F_{i_{l}}$ and $|\sigma_{il}|=k$ for $l=1,\ldots,a_{i}$. Actually, $F_{i_{l}}\cap F_{i}$’s are all of facets of $\langle F_{i}\rangle\cap\langle F_{1},\ldots,F_{i-1}\rangle$. Since $\sigma_{il}\cap\sigma_{il^{\prime}}=\emptyset$ for $1\leq l<l^{\prime}\leq a_{i}$, one can suppose that $P_{\sigma_{i1}}\ldots P_{\sigma_{ia_{i}}}=(f_{ij}:j=1,\ldots,k^{a_{i}})$. Set $\mathbf{a}_{ij}=\deg(f_{ij})$ and it is clear that for all $j=1,\ldots,k^{a_{i}}$, $a_{i}=|\mathbf{a}_{ij}|$. We have the following isomorphisms:

[TABLE]

where $a_{i}=|\mathbf{a}_{ij}|$ for $j=1,\ldots,k^{a_{i}}$. Now it is easy to check that the homomorphism

[TABLE]

is an epimorphism. In particular, it follows that

[TABLE]

This completes the proof. ∎

Remark 2.10*.*

In view of Theorem 2.9, let $\Delta=\langle F_{1},\ldots,F_{r}\rangle$ be a shellable simplicial complex and $\Delta_{j}=\langle F_{1},\ldots,F_{j}\rangle$. Then we have the prime filtration

[TABLE]

for $K[\Delta]$. In particular, it follows the following filtration for $K[\Delta^{[k]}]$:

[TABLE]

In other words, Theorem 2.9 gives a filtration for the face ring of the expansion of a shellable simplicial complex with respect to $k$.

Remark 2.11*.*

The filtration described in Theorem 2.9 in the case that $k>1$ is not a prime filtration, i.e. the quotient of any two consecutive modules of the filtration is not cyclic. Consider the same notations of Theorem 2.9, we have the following prime filtration for $K[\Delta]$ when $\Delta$ has a $k$-shelling order:

$\mathcal{F}:0=\bigcap^{r}_{j=1}P_{F^{c}_{j}}\subset\ldots\subset$

$\bigcap^{i}_{j=1}P_{F^{c}_{j}}\subset\ldots\subset\sum^{k^{a_{i}}-1}_{j=1}(f_{j})+\bigcap^{i}_{j=1}P_{F^{c}_{j}}\subset\sum^{k^{a_{i}}}_{j=1}(f_{j})+\bigcap^{i}_{j=1}P_{F^{c}_{j}}=\bigcap^{i-1}_{j=1}P_{F^{c}_{j}}$

$\subset\ldots\subset K[\Delta]$

where $(f_{1},\ldots,f_{j-1}):(f_{j})$ is generated by linear forms for all $j=2,\ldots,k^{a_{i}}$ and all $i=1,\ldots,r$.

For all $2\leq j\leq k^{a_{i}}$, suppose that $(f_{1},\ldots,f_{j-1}):(f_{j})=P_{Q_{j}}$. Set $P_{Q_{1}}=(0)$. We have

[TABLE]

where $L_{ij}=F^{c}_{i}\cup Q_{j}$, for $i=1,\ldots,r$ and $j=1,\ldots,k^{a_{i}}$. Therefore the set of prime ideals which defines the cyclic quotients of $\mathcal{F}$ is ${\mathrm{Supp}}(\mathcal{F})=\{P_{L_{ij}}:i=1,\ldots,r\ \mbox{and}\ j=1,\ldots,k^{a_{i}}\}$.

3. The Stanley conjecture

Consider a field $K$, and let $R$ be a finitely generated $\mathbb{N}^{n}$-graded $K$-algebra, and let $M$ be a finitely generated $\mathbb{Z}^{n}$-graded $R$-module. Stanley [20] conjectured that, in this case, there exist finitely many subalgebras $A_{1},\ldots,A_{r}$ of $R$, each generated by algebraically independent $\mathbb{N}^{n}$-homogeneous elements of $R$, and there exist $\mathbb{Z}^{n}$-homogeneous elements $u_{1},\ldots,u_{r}$ of $M$, such that $M=\bigoplus^{r}_{i=1}u_{i}A_{i}$, where $\dim(A_{i})\geq\mathrm{depth}(M)$ for all $i$ and where $u_{i}A_{i}$ is a free $A_{i}$-module of rank one.

Consider a finitely generated $\mathbb{Z}^{n}$-graded $S$-module $M$, a subset $Z$ of $\{x_{1},\ldots,x_{n}\}$, and a homogeneous element $u\in M$. The $K$-subspace $uK[Z]$ of $M$ is called a Stanley space of dimension $|Z|$ if it is a free $K[Z]$-module of rank 1, i.e., the elements of the form $uv$, where $v$ is a monomial in $K[Z]$, form a $K$-basis of $uK[Z]$. A Stanley decomposition of $M$ is a decomposition $\mathcal{D}$ of $M$ into a finite direct sum of Stanley spaces. The Stanley depth of $\mathcal{D}$, denoted $\mathrm{sdepth}(\mathcal{D})$, is the minimal dimension of a Stanley space in a decomposition $\mathcal{D}$. We set

[TABLE]

and we call this number the Stanley depth of $M$. The Stanley conjecture says that $\mathrm{sdepth}(M)\geq\mathrm{depth}(M)$ always holds.

The following lemma is needed in the proof of the main theorem of this section.

Lemma 3.1**.**

Let $F_{1},\ldots,F_{s}\subset X$ with $F_{i}\cap F_{j}=\emptyset$ for all $i\neq j$ and $|F_{i}|=k>1$. Let $f_{1},\ldots,f_{k^{s}}$ be a sequence of minimal generators of $P_{F_{1}}\ldots P_{F_{s}}$ ordered with respect to lexicographical ordering $x_{1}>x_{2}>\ldots>x_{n}$. Suppose that $n_{i}$ is the minimal number of homogeneous generators of $(f_{1},\ldots,f_{i-1}):(f_{i})$ for $i=2,\ldots,k^{s}$. Then

[TABLE]

Moreover, for all $i$, the colon ideal $(f_{1},\ldots,f_{i-1}):(f_{i})$ is generated by linear forms.

Proof.

By [6, Corollary 1.5.], $P_{F_{1}}\ldots P_{F_{s}}$ has linear quotients. To show equality, we use induction on $s$. If $s=1$, the assertion is clear. Assume that $s>1$. Let $P_{F_{1}}\ldots P_{F_{s-1}}=(f_{1},\ldots,f_{k^{s-1}})$ and $F_{s}=\{x_{1},\ldots,x_{k}\}$. Then

[TABLE]

Moreover,

[TABLE]

Now, by the induction hypothesis, we have $n_{k^{s}}=(k-1)+(k-1)(s-1)=(k-1)s$, as desired. ∎

Let

[TABLE]

be a prime filtration of $M$ with $M_{i}/M_{i-1}\cong(S/P_{i})(-\mathbf{a}_{i})$. Then this filtration decomposes $M$ as a multigraded $K$-vector space, that is, we have $M=\bigoplus^{r}_{i=1}u_{i}K[Z_{i}]$ and this is a Stanley decomposition of $M$ where $u_{i}\in M_{i}$ is a homogeneous element of degree $\mathbf{a}_{i}$ and $Z_{i}=\{x_{j}:x_{j}\not\in P_{i}\}$.

Now suppose that $\Delta=\langle G_{1},\ldots,G_{r}\rangle$ is a pure shellable simplicial complex on $X$ and $k$ is a positive integer. By Theorem 2.8, $\Delta^{[k]}=\langle G^{[k]}_{1},\ldots,G^{[k]}_{r}\rangle$ is pure $k$-shellable. For all $i$, set $F_{i}=G^{[k]}_{i}$. Consider the prime filtration $\mathcal{F}$ of $K[\Delta^{[k]}]$ described in Section 2. Then we have the Stanley decomposition

[TABLE]

where $Z_{ij}=\{x_{l}:l\not\in L_{ij}\}$, $\deg(u_{ij})=\mathbf{a}_{ij}$ and $|\mathbf{a}_{ij}|=a_{i}$ for all $i,j$. We claim that for all $i,j$, $|Z_{ij}|\geq\mathrm{depth}(K[\Delta^{[k]}])$.

By Corollaries 4.1 and 2.1 of [17], we have $\mathrm{depth}(K[\Delta^{[k]}])=\dim(K[\Delta])=|G_{i}|$. On the other hand, $|G_{i}|\geq a_{i}$. Now by combining all of these results with Lemma 3.1, we have

[TABLE]

Thus we have shown the main result of this section:

Theorem 3.2**.**

The expansion of the face ring of a pure shellable simplicial complex with respect to $k>0$ satisfies the Stanley conjecture. In particular, the face ring of a pure $k$-shellable complex satisfies the Stanley conjecture.

4. $k$-shellable graphs

Let $G$ be a simple graph and let $\Delta_{G}$ the independence complex of $G$. We say that $G$ is $k$-shellable if $\Delta_{G}$ has this property. The purpose of this section is to characterize $k$-shellable graphs.

Following Schrijver [18], the duplication of a vertex $x_{i}$ of a graph $G$ means extending its vertex set $X$ by a new vertex $x_{i^{\prime}}$ and replacing $E(G)$ by

[TABLE]

In other words, if $V(G)=\{x_{1},\ldots,x_{n}\}$ then the graph $G^{\prime}$ obtained from $G$ by duplicating $k_{i}-1$ times the vertex $x_{i}$ has the vertex set

[TABLE]

and the edge set

[TABLE]

Example 4.1*.*

Let $G$ be a simple graph on the vertex set $V(G)=\{x_{1},\ldots,x_{5}\}$ and $E(G)=\{x_{1}x_{3},x_{1}x_{4},x_{2}x_{4},x_{2}x_{5},x_{3}x_{5},x_{4}x_{5}\}$. Let $G^{\prime}$ be obtained from $G$ by duplicating $1$ times the vertices $x_{1}$ and $x_{4}$ and [math] times the other vertices. Then $G$ and $G^{\prime}$ are in the form

[TABLE]

Also, the independence complexes of $G$ and $G^{\prime}$ are, respectively, $\Delta_{G}=\langle x_{1}x_{2},x_{1}x_{5},x_{2}x_{3},x_{3}x_{4}\rangle$ and $\Delta_{G^{\prime}}=\langle x_{11}x_{12}x_{21},x_{11}x_{12}x_{51},x_{21}x_{31},x_{31}x_{41}x_{42}\rangle$. Note that $\Delta_{G^{\prime}}=\Delta^{\alpha}_{G}$ where $\alpha=(2,1,1,2,1)$.

In the following theorem we show that the simple graph obtained from duplicating $k-1$ times any vertex of a shellable graph is $k$-shellable.

Theorem 4.2**.**

Let $G$ be a simple graph on $X$ and let $G^{\prime}$ be a new graph obtained from $G$ by duplicating $k-1$ times any vertex of $G$. Then $G$ is shellable if and only if $G^{\prime}$ is $k$-shellable.

Proof.

It suffices to show that $\Delta_{G^{\prime}}=\Delta^{[k]}_{G}$. Then Theorem 2.8 completes the assertion.

After relabeling of the vertices of $G^{\prime}$ one can assume that $G^{\prime}$ is a graph with the vertex set $X^{[k]}=\{x_{ij}:i=1,\ldots,n,\ j=1,\ldots,k\}$ and the edge set

[TABLE]

Let $F$ be an independent set of $G^{\prime}$ and let $\bar{F}=\{x_{i}:x_{ir}\in F\ \mbox{for some}\ r\}$. If $|\bar{F}|=1$ then $\bar{F}$ is an independent set in $G$. So assume that $|\bar{F}|>1$. Suppose, on the contrary, that $x_{i},x_{j}\in\bar{F}$ and $x_{i}x_{j}\in G$. By the construction of $G^{\prime}$, for all $x_{ir}$ and $x_{js}$ of $V(G^{\prime})$, $x_{ir}x_{js}\in G^{\prime}$. Therefore $F$ contains an edge $x_{ir}x_{js}$ of $G^{\prime}$, a contradiction. This implies that $\bar{F}$ is an independent set in $G$. In particular, since $F\subset(\bar{F})^{[k]}$ we have $F\in\Delta^{[k]}_{G}$.

Conversely, suppose $H$ is an independent set of $G$. Choose $x_{ir},x_{js}\in H^{[k]}$. If $x_{ir}x_{js}\in G^{\prime}$ then $x_{i}x_{j}\in G$, which is false since $x_{i},x_{j}\in H$. Therefore $H^{[k]}$ is an independent set of $G^{\prime}$ and $H^{[k]}\in\Delta_{G^{\prime}}$. ∎

In the following we want to extend some results from [5, 7, 21]. Firstly, we present a generalization of the concept of simplicial vertex.

Let $G$ be a simple graph. For $U\subset V(G)$ we define the induced subgraph of $G$ on $U$ to be the subgraph $G_{U}$ on $U$ consisting of those edges $x_{i}x_{j}\in E(G)$ with $x_{i},x_{j}\in U$. For $x\in V(G)$, let $N_{G}(x)$ denote the open neighborhood of $x$, that is, all of vertices adjacent to $x$. We also denote by $N_{G}[x]$ the closed neighborhood of $x$, which is $N_{G}(x)$ together with $x$ itself, so that $N_{G}[x]=N_{G}(x)\cup\{x\}$. Set $N_{G}(U)=\bigcup_{x\in U}N_{G}(x)$ and $N_{G}[U]=\bigcup_{x\in U}N_{G}[x]$.

Recall from [8] or [14] that a vertex $x\in V(G)$ is simplicial if the induced subgraph $G_{N_{G}[x]}$ is complete.

The simple graph $G$ is a complete $r$-partite graph if there is a partition $V(G)=V_{1}\cup\ldots\cup V_{r}$ of the vertex set, such that $uv\in E(G)$ if and only if $u$ and $v$ are in different parts of the partition. If $|V_{i}|=n_{i}$, then $G$ is denoted by $K_{n_{1}.\ldots,n_{r}}$.

Definition 4.3**.**

The set $S$ of pairwise non-adjacent vertices of $G$ is a $k$-simplicial set if $G_{N_{G}[S]}$ is a $r$-partite complete graph with $k$-element parts $S_{1},\ldots,S_{r}$ having the following property:

for every $S_{l}$ and every two vertices $x_{i},x_{j}\in S_{l}$, $N_{G}(x_{i})=N_{G}(x_{j})$.

Note that every $1$-simplicial set is a simplicial vertex.

Lemma 4.4**.**

Let $S\subset V(G)$ of pairwise non-adjacent vertices of $G$. Set $G^{\prime}=G\backslash N_{G}[S]$ and $G^{\prime\prime}=G\backslash S$. Then

(i) $\Delta_{G^{\prime}}=\mathrm{lk}_{\Delta_{G}}(S)$ and (ii) $\Delta_{G^{\prime\prime}}=\bigcap_{x\in S}\mathrm{dl}_{\Delta_{G}}(x)$.

Proof.

(i) “$\subseteq$”: Let $F\in\Delta_{G^{\prime}}$. Since $F\subset V(G)\backslash N_{G}[S]$ we have $F\in\Delta_{G}$ and $F\cap S=\emptyset$. It remains to show that $F\cup S\in\Delta_{G}$. Let $F\cup S$ contain an edge $x_{i}x_{j}\in E(G)$. Then it should be $x_{i}\in S$ and $x_{j}\in F$. In particular, since $x_{i}$ and $x_{j}$ are adjacent it follows that $x_{j}\in N_{G}[S]$. This is impossible because $F\subset V(G)\backslash N_{G}[S]$. Therefore $F\cup S\in\Delta_{G}$. This implies that $F\in\mathrm{lk}_{\Delta_{G}}(S)$.

“$\supseteq$”: Let $F\in\mathrm{lk}_{\Delta_{G}}(S)$. Then $F\cup S\in\Delta_{G}$ and $F\cap S=\emptyset$. In order to prove that $F\in\Delta_{G^{\prime}}$ it suffices to show that $F\cap N_{G}(S)=\emptyset$ and no two vertices of $F$ are adjacent in $G^{\prime}$. If $x\in F\cap N_{G}(S)$ then $S\cup\{x\}$ contains an edge of $G$ and this contradicts $F\cup S\in\Delta_{G}$. Also, if $x_{i},x_{j}\in F$ and $x_{i}x_{j}\in G^{\prime}$ then $x_{i}x_{j}\in G$, which is again a contradiction.

(ii) “$\subseteq$”: Let $F\in\Delta_{G^{\prime\prime}}$. If for $x_{i},x_{j}\in F$, $x_{i}x_{j}$ is an edge of $G$, since $\{x_{i},x_{j}\}\cap S=\emptyset$ we obtain that $x_{i}x_{j}\in G^{\prime\prime}$, which is not true. Hence $F\in\Delta_{G}$. In particular, it follows from $F\subset V(G)\backslash S$ that $F\cap S=\emptyset$ and so $F\in\bigcap_{x\in S}\mathrm{dl}_{\Delta_{G}}(x)$.

“$\supseteq$”: Let $F\in\bigcap_{x\in S}\mathrm{dl}_{\Delta_{G}}(x)$. Then for all $x\in S$ we have $F\in\mathrm{dl}_{\Delta_{G}}(x)$. Thus $F\in\Delta_{G}$ and $F\cap S=\emptyset$. Since $F$ contains no edge of $G$ it follows that $F$ contains no edge of $G^{\prime\prime}$, either. This implies that $F\in\Delta_{G^{\prime\prime}}$. ∎

Combining Theorem 2.6 with Lemma 4.4 we obtain the following corollary as an extension of Theorem 2.6. of [21].

Corollary 4.5**.**

Let $S\subset V(G)$ be a set of pairwise non-adjacent vertices. If $G$ is $k$-shellable, then $G^{\prime}=G\backslash N_{G}[S]$ is $k$-shellable, too.

Theorem 4.6**.**

If $S$ is a $k$-simplicial set of $G$ such that both $G\backslash S$ and $G\backslash N_{G}[S]$ are $k$-shellable, then $G$ is $k$-shellable.

Proof.

Let $G^{\prime}=G\backslash N_{G}[S]$ and $G^{\prime\prime}=G\backslash S$. Let $F_{1},\ldots,F_{r}$ and $H_{r+1},\ldots,H_{s}$ be, respectively, the $k$-shelling orders of $\Delta_{G^{\prime\prime}}$ and $\Delta_{G^{\prime}}$. Set $F_{i}=H_{i}\cup S$, for $r+1\leq i\leq s$. We show that $F_{1},\ldots,F_{s}$ is a $k$-shelling order of $\Delta_{G}$.

Note that $\Delta_{G^{\prime\prime}}=\mathrm{dl}_{\Delta_{G}}(S)$. This follows from the fact that $S$ is a $k$-simplicial set. Hence $F_{1},\ldots,F_{s}$ contains all of facets of $\Delta_{G}$. Now let $1\leq j<i\leq s$. If $i\leq r$ or $j>r$, by $k$-shellability of $\Delta_{G^{\prime}}$ and $\Delta_{G^{\prime\prime}}$, we are done. So suppose that $i>r$ and $j\leq r$. Clearly, $S\subseteq F_{i}\backslash F_{j}$. On the other hand, $\Delta_{G^{\prime}}\subset\Delta_{G^{\prime\prime}}$ and so there exists $l\leq r$ such that $H_{i}\subset F_{l}$. This implies that $S=F_{i}\backslash F_{l}$ and therefore the assertion is completed. ∎

The following theorem extends Theorem 2.1.13 of [7].

Theorem 4.7**.**

Let $S_{1}$ be a $k$-simplicial set of $G$ and let $S_{1},\ldots,S_{r}$ be the parts of $G_{N_{G}[S_{1}]}$ and $G^{\prime}_{i}=G\backslash N_{G}[S_{i}]$ for all $i=1,\ldots,r$. Then $G$ is $k$-shellable if and only if $G^{\prime}_{i}$ is $k$-shellable for all $i=1,\ldots,r$.

Proof.

Only if part follows from Theorem 4.5. Conversely, let $G^{\prime}_{i}$ be $k$-shellable for all $i=1,\ldots,r$. Hence for every $i$, there exists a $k$-shelling order $F_{i1},\ldots,F_{it_{i}}$ for $\Delta_{G^{\prime}_{i}}=\mathrm{lk}_{\Delta_{G}}(S_{i})$. We claim that

[TABLE]

is a $k$-shelling order for $\Delta_{G}$.

We first show that the above list is the complete list of facets of $\Delta_{G}$. Let $F\in\Delta_{G}$. If for some $i\neq 1$, $S_{i}\subset F$ then $F$ is in the above list. Otherwise, suppose that for all $i\neq 1$, $S_{i}\cap F=\emptyset$. Since the elements of $S_{1}$ are only adjacent to the elements of $S_{i}$’s, for $i\neq 1$, it follows that $S_{1}\subset F$. On the other hand, it is not possible that $F$ contains only some of elements of a part, as $S_{i}$, but not all of elements of $S_{i}$. Because in this case, there are $x_{j}\in S_{i}\cap F$ and $x_{j^{\prime}}\in S_{i}\backslash F$. This means that $N_{G}(x_{j})\cap F=\emptyset$ but $N_{G}(x_{j^{\prime}})\cap F\neq\emptyset$ and so $N_{G}(x_{j})\neq N_{G}(x_{j^{\prime}})$, a contradiction.

Now consider $F_{ij}\cup S_{i}$ and $F_{lm}\cup S_{l}$. We have the following cases:

(i) $i<l$: Then $S_{l}\subset(F_{lm}\cup S_{l})\backslash(F_{ij}\cup S_{i})$. Since $F_{lm}\cup S_{1}$ is an independence set in $G$, there exists a facet $F\in\Delta_{G}$ with $F_{lm}\cup S_{1}\subset F$. In particular, $F$ is in the form $F=F_{1p}\cup S_{1}$ for some $1\leq p\leq t_{1}$. Thus $(F_{lm}\cup S_{l})\backslash(F_{1p}\cup S_{1})=S_{l}$.

(ii) $i=l$ and $j<m$: The assertion follows from $k$-shellability of $\Delta_{G^{\prime}_{i}}$.

Therefore $G$ is $k$-shellable. ∎

Lemma 4.8**.**

Let $S,T\subset V(G)$ and let $T$ be an independent set of $G$. If $N_{G}[T]\subseteq N_{G}[S]$ then independent sets of $G\backslash N_{G}[S]$ are not maximal independent sets of $G\backslash S$.

Proof.

If $F$ is an independent set of $G\backslash N_{G}[S]$ then $F\cup T$ will be a larger independent set of $G\backslash S$. ∎

Corollary 4.9**.**

Let $S$ be a $k$-simplicial set of $G$ and let $S_{1},\ldots,S_{r}$ be the parts of $G_{N_{G}[S]}$. Then for every $S_{i}$, independent sets of $G\backslash N_{G}[S_{i}]$ are not maximal independent sets of $G\backslash S_{i}$.

Proof.

Since $N_{G}[S]\subseteq N_{G}[S_{i}]$ for all $i$, the assertion follows from Lemma 4.8. ∎

In [5, Theorem 2] the authors proved that if $x$ is a simplicial vertex of $G$ and $y$ adjacent to $x$, then $G$ is shellable if and only if $G\backslash N_{G}[y]$ and $G\backslash y$ are shellable. The following theorem extends this result to $k$-shellable graphs.

Theorem 4.10**.**

Let $S_{1}$ be a $k$-simplicial set of $G$ and let $S_{1},\ldots,S_{r}$ be the parts of $G_{N_{G}[S_{1}]}$. Then $G$ is $k$-shellable if and only if for each $i=2,\ldots,r$ the graphs $G^{\prime}_{i}=G\backslash N_{G}[S_{i}]$ and $G^{\prime\prime}_{i}=G\backslash S_{i}$ are $k$-shellable.

Proof.

“Only if part”: Let $G$ be $k$-shellable. It follows from Theorem 4.7 that $G^{\prime}_{i}$ is $k$-shellable for all $i$. Fix an integer $i$. We want to show that $G^{\prime\prime}_{i}$ is $k$-shellable. By again relabeling $S_{i}$’s we can consider $i=r$.

Since each $G^{\prime}_{i}$ is $k$-shellable, so for every $i$, there exists a $k$-shelling order $F_{i1},\ldots,F_{it_{i}}$ for $\Delta_{G^{\prime}_{i}}=\mathrm{lk}_{\Delta_{G}}(S_{i})$. Moreover, by the proof of Theorem 4.7,

[TABLE]

is a $k$-shelling order for $\Delta_{G}$. By the fact that $N_{G}(S_{1})\subseteq N_{G}(S_{r})$ we conclude that for every $F_{rj}$ where $1\leq j\leq t_{r}$, there exists $1\leq l\leq t_{1}$ such that $F_{rj}\subseteq F_{1l}$. Therefore

[TABLE]

will be a list of facets of $G\backslash S_{r}$. Furthermore, it is a $k$-shelling order of $G^{\prime\prime}_{r}$.

“If part”: Let for all $i=2,\ldots,r$ the graphs $G^{\prime}_{i}$ and $G^{\prime\prime}_{i}$ are $k$-shellable. Fix an $i$ and set $G^{\prime}=G^{\prime}_{i},G^{\prime\prime}=G^{\prime\prime}_{i},S=S_{i}$. Let $F_{1},\ldots,F_{r}$ and $H_{1},\ldots,H_{s}$ be, respectively, the $k$-shelling orders of $\Delta_{G^{\prime\prime}}$ and $\Delta_{G^{\prime}}$. We first show that

[TABLE]

is a list of facets of $\Delta_{G}$, and furthermore, this list is a $k$-shelling order of $\Delta_{G}$.

Let $F$ be a facet of $\Delta_{G}$. If $F\cap S=\emptyset$, then since $F$ contains no edge of $G$, it contains no edge of $G^{\prime\prime}$ and so $F\in\Delta_{G^{\prime\prime}}$. Suppose $F\cap S\neq\emptyset$. Let $x\in F\cap S$ and let $y\in S$ with $y\neq x$. Since $N_{G}(y)=N_{G}(x)$, we have $y\in F$. Thus $S\subset F$. On the other $F$ contains no edge of $G^{\prime}$. Therefore $F\backslash S$ is a facet of $\Delta_{G^{\prime}}$.

Now we show that above list is a $k$-shelling order. Set $F_{i+r}=H_{i}\cup S$ for all $i=1,\ldots,s$. Suppose $F_{i}$ and $F_{j}$ with $i<j$. If $j\leq r$ or $i\geq r+1$ then by the $k$-shellability of $\Delta_{G^{\prime\prime}}$ and $\Delta_{G^{\prime}}$, respectively, the assertion is completed. Let $i\leq r$ and $r<j$. Then $S\subset F_{j}\backslash F_{i}$. Since $H_{j-r}\in\Delta_{G^{\prime\prime}}$, there is $F_{l}$ with $l\leq r$ such that $H_{j-r}\subset F_{l}$. Therefore $F_{j}\backslash F_{l}=S$. This completes the assertion. ∎

Acknowledgments: The author would like to thank the referee for careful reading of the paper. The work was supported by the research council of the University of Maragheh.