$k$-shellable simplicial complexes and graphs

0.2 MB·

by  Rahim Rahmati-Asghar

k-SHELLABLE SIMPLICIAL COMPLEXES AND GRAPHS
7 RAHIMRAHMATI-ASGHAR
1
0
2 Abstract. Inthispaperweshowthatak-shellablesimplicialcomplexisthe
expansion of a shellable complex. We prove that the face ring of a pure k-
n
shellable simplicial complex satisfies the Stanley conjecture. In this way, by
a
applyingexpansionfunctortothefaceringofagivenpureshellablecomplex,
J
weconstruct alargeclassofringssatisfyingtheStanleyconjecture.
1 Also,bypresentingsomecharacterizationsofk-shellablegraphs,weextend
1 someresultsduetoCastrill´on-Cruz,Cruz-EstradaandVanTuyl-Villareal.

]
C
A
.
h Introduction
t
a Let ∆ be a simplicial complex on the vertex set X := {x ,...,x }. Denote by
1 n
m
hF ,...,F ithesimplicialcomplex∆withfacetsF ,...,F . ∆iscalledshellable if
1 r 1 r
[ itsfacetscanbegivenalinearorderF ,...,F ,calledashelling order,suchthatfor
1 r
1 all2≤j,thesubcomplexhFji∩hF1,...,Fj−1iispureofdimensiondim(Fj)−1(see
v [3] for probably the earliest definition of this term or as well [2] for a more recent
8 exposition). Studying combinatorial properties of shellable simplicial complexes
6 and algebraic constructions of their face rings and also the edge ideals associated
8
to shellable graphs is a current trend in combinatorics and commutative algebra.
2
See for example [2, 3, 5, 9, 12, 21].
0
. In this paper, we recall from [17] the concept of k-shellability, and extend some
1
results obtained previously by researchers. Actually, k is a positive integer and for
0
7 k =1, 1-shellability coincides with shellability.
1 Richard Stanley [20], in his famous article “Linear Diophantine equations and
: local cohomology”, made a striking conjecture predicting an upper bound for the
v
i depth of a multigraded module. This conjecture is nowadays called the Stanley
X
conjecture and the conjectured upper bound is called the Stanley depth of a mod-
r ule. TheStanleyconjecturehasbecomequite popular,withnumerouspublications
a
dealing with different aspects of the Stanley depth. Although a counterexample
has apparentlyrecently 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 ,...,x ] be the polynomial ring over a field K. Dress proved in
1 n
[9] that the simplicial complex ∆ 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 ⊂S is a monomial ideal, and S/I is a
multigradedpretty cleanring,thenthe Stanley conjectureholdsforS/I. Itfollows
that, for a shellable simplicial complex ∆, the face ring K[∆]= S/I satisfies the
∆
Stanley conjecture where I denotes the Stanley-Reisner ideal of ∆. 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
1
2 RAHIMRAHMATI-ASGHAR
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.
AsimplegraphGiscalledshellableifitsindependencecomplex∆ isashellable
G
simplicial complex. Shellable graphs were studied by several researchers in recent
years. Forexample,VanTuylandVillarrealin[21]classifiedallofshellablebipartite
graphs. Also, Castrill´on 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 generalfacts onsimplicialcomplexes, we referto Stan-
ley’s book [19].
Asimplicialcomplex∆ispure ifallofitsfacets(maximalfaces)areofthesame
dimension. The link and deletion of a face F in ∆ are defined respectively
lk (F)={G∈∆:G∩F =∅ and G∪F ∈∆}
∆
and
dl∆(F)={G∈∆:F *G}.
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 ∆ and F is a face of ∆ if and only if there is no edge of G joining any two
G G
verticesofF. The edge ideal of Gis defineda quadraticsquarefreemonomialideal
I(G)=(x x :x x ∈E(G)). ItisknownthatI(G)=I . WesayGisashellable
i j i j ∆G
graph if ∆ is a shellable simplicial complex.
G
In the following we recall the concept of expansion functor in a combinatorial
and an algebraic setting from [16] and [1], respectively.
Let α = (k1,...,kn) be an n-tuple with positive integer entries in Nn. For
F ={x ,...,x }⊆X define
i1 ir
Fα ={x ,...,x ,...,x ,...,x }
i11 i1ki1 ir1 irkir
as a subset of Xα :={x ,...,x ,...,x ,...,x }. Fα is called the expansion
11 1k1 n1 nkn
of F with respect to α.
For a simplicial complex ∆ = hF ,...,F i on X, we define the expansion of ∆
1 r
with respect to α as the simplicial complex ∆α =hFα,...,Fαi (see [16]).
1 r
In[1]BayatiandHerzogdefinedtheexpansionfunctorinthecategoryoffinitely
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
monomialideals and refer the readerto [1]for moregeneralcase in the categoryof
finitely generated multigraded S-modules.
Set Sα a polynomial ring over K in the variables
x ,...,x ,...,x ,...,x .
11 1k1 n1 nkn
k-SHELLABLE SIMPLICIAL COMPLEXES AND GRAPHS 3
Whenever I ⊂ S is a monomial ideal minimally generated by u ,...,u , the ex-
1 r
pansion of I with respect to α is defined
r
Iα =XP1ν1(ui)...Pnνn(ui) ⊂Sα
i=1
whereP =(x ,...,x )is a prime idealofSα andν (u ) is the exponentof x in
j j1 jk j i j
u .
i
Example 1.1. Let I ⊂ K[x ,...,x ] be a monomial ideal minimally generated by
1 3
G(I)={x21x2,x1x3,x2x23} and let α=(2,2,1)∈N3. Then
Iα = (x ,x )2(x ,x )+(x ,x )(x )+(x ,x )(x )2
11 12 21 22 11 12 31 21 22 31
= (x2 x ,x x x ,x2 x ,x2 x ,x x x ,x2 x ,x x ,x x ,x x2 ,x x2 )
11 21 11 12 21 12 21 11 22 11 12 22 12 22 11 31 12 31 21 31 22 31
It was shown in [1] that the expansion functor is exact and so (S/I)α =Sα/Iα.
The following lemma implies that two above concepts of expansion functor are
related.
Lemma 1.2. ([16, Lemma 2.1]) For a simplicial complex ∆ and α ∈ Nn we have
(I )α =I . In particular, K[∆]α =K[∆α].
∆ ∆α
In this paper we just study the functors α=(k1,...,kn)∈Nn with ki =kj for
all i,j. For convenience, we set α = [k] when every component of α is equal to
k ∈N. We call I[k] (resp. ∆[k]) the expansion of I (resp. ∆) with respect to k.
2. Some combinatorial and algebraic properties of k-shellable
complexes
Thenotionofk-shellablesimplicialcomplexeswasfirstintroducedbyEmtander,
Mohammadi and Moradi [11] to provide a natural generalization of shellability. It
was shown in [11, Theorem 6.8] that a simplicial complex ∆ is k-shellable if and
only if the Stanley-Reisner ideal of its Alexander dual has k-quotionts, i.e. there
exists an ordering u1,...,ur of the minimal generators of I∆∨ such that if we for
s = 1,...,t, put I = (u ,...,u ), then for every s there are monomials v ,
s 1 s si
i=1,...,r , deg(v )=k for all i, such that I :u =(v ,...,v ).
s si s s s1 srs
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
colonidealsI :u weregeneratedbyregularsequencesforallsandinthisway,all
s s
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∆bead-dimensionalsimplicialcomplexonX andletk
be anintegerwith 1≤k ≤d+1. ∆is calledk-shellable ifits facets canbe ordered
F ,...,F , called k-shelling order, such that for all j = 2,...,r, the subcomplex
1 r
∆j =hFji∩hF1,...,Fj−1i satisfies the following properties:
(i) ItisgeneratedbyanonemptysetofmaximalproperfacesofhF iofdimension
j
|F |−k−1;
j
(ii) If ∆ has more than one facet then for every two disjoint facets σ,τ ∈hF i∩
j j
hF1,...,Fj−1i we have Fj ⊆σ∪τ.
Remark 2.2. It follows from the definition that two concepts 1-shellability and
shellability coincide.
4 RAHIMRAHMATI-ASGHAR
Remark 2.3. Note that the notions of 1-shellability in our sense and Emtander,
MohammadiandMoradi’scoincide. Although,fork >1,asimplicialcomplexmay
be k-shellable in their concept and not in ours. For example, consider the complex
∆=habc,aef,cdfi on {a,b,...,f}. It is easy to check that ∆ 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 ∆ be a d-dimensional (not necessarily pure) simplicial com-
plex on X and let k be an integer with 1≤k ≤d+1. Suppose that the facets of ∆
can be ordered F ,...,F . Then the following conditions are equivalent:
1 r
(a) F ,...,F is a k-shelling of ∆;
1 r
(b) for every 1≤j ≤r there exist the subsets E ,...,E of X such that the E are
1 t i
mutually disjoint and |E |=k for all i and the set of the minimal elements of
i
hF1,...,Fji\hF1,...,Fj−1i is {{a1,...,at}:ai ∈Ei for all i};
(c) for all i,j, 1 ≤ i < j ≤ r, there exist x ,...,x ∈ F \F and some l ∈
1 k j i
{1,...,j−1} with F \F ={x ,...,x }.
j l 1 k
Proof. (a)⇒(b): Let hFji∩hF1,...,Fj−1i=hFj\σ1,...,Fj\σti where |σi|=k for
all i. Since for all i 6=i′, Fj ⊆(Fj\σi)∪(Fj\σi′), we have σi∩σi′ =∅. Hence the
minimal elements of hF1,...,Fji\hF1,...,Fj−1i are in the form{a1,...,at} where
a ∈σ for all i.
i i
(b)⇒(c): For all i, suppose that E = {x ,...,x }. Let 1 ≤ i < j ≤ r
i i1 ik
and let {x1i1,...,xtit} be a minimal element of hF1,...,Fji\hF1,...,Fj−1i. Be-
cause {x1i1,...,xtit} * Fi we may assume that x1i1 ∈ Fj\Fi. We claim that
x ,...,x ∈F \F . Suppose, on the contrary, that for some s, x 6∈F \F then
11 1k j i 1s j i
x1s ∈Fiandsox1s 6∈hF1,...,Fji\hF1,...,Fj−1i. Itfollowsthat{x1s,x2i2,...,xtit}
isnotaminimalelementofhF1,...,Fji\hF1,...,Fj−1i, acontradiction. Therefore
E ⊆F \F .
1 j i
Now suppose that for all l < j, if E1 is contained in Fj\Fl then E1 $ Fj\Fl.
Then there exists y ∈ F \E such that {y,x ,...,x } is a minimal element
j 1 2i2 tit
of hF1,...,Fji\hF1,...,Fj−1i different from the elements of {{a1,...,at} : ai ∈
E for all i}, a contradiction. Therefore there exists l <j with F \F =E .
i j l 1
(c)⇒(a): Let F ∈ hFji∩hF1,...,Fj−1i. Then F ⊆ Fi for some i < j. By
the condition (c), there exist x ,...,x ∈ F \F and some l ∈ {1,...,j−1} with
1 k j i
Fj\Fl ={x1,...,xk}. ButFj\{x1,...,xk}isaproperfaceofhFji∩hF1,...,Fj−1i,
because F \{x ,...,x } = F ∩F . Moreover, F \{x ,...,x } is a maximal face.
j 1 k j l j 1 k
Finally, since F is contained in F \{x ,...,x }, the assertion is completed. (cid:3)
j 1 k
Example 2.5. The Figure 1 indicates the pure shellable and pure 2-shellable sim-
plicial complexes of dimensions 1, 2 and 3 with 3 facets.
Theorem 2.6. Let ∆ be a k-shellable complex and σ a face of ∆. Then lk (σ) is
∆
again k-shellable.
Proof. Since∆isk-shellable,sothereexistsank-shellingorderF ,...,F offacets
1 r
of ∆. Let F ,...,F where i < ... < i be all of facets which contain σ. We
i1 it 1 t
claim that F \σ,...,F \σ is a k-shelling order of lk (σ). To this end we want to
i1 it ∆
show that the condition (c) of Proposition 2.4 holds.
Set G =F \σ. Consider l,m with 1≤l<m≤t. By k-shellability of ∆, there
j ij
are x ,...,x ∈ F \F = (F \σ)\(F \σ) = G \G such that for some s < i
1 k im il im il m l m
k-SHELLABLE SIMPLICIAL COMPLEXES AND GRAPHS 5
shellable 2−shellable
1−dim.
2−dim.
3−dim.
Figure 1.
wehave{x ,...,x }=F \F . Itfollowsfromσ ⊂F andF \F ={x ,...,x }
1 k im s im im s 1 k
that σ ⊂ F . This implies that F is among the list F ,...,F . Let F = F .
Hence Gm\sGm′ ={x1,...,xk} ansd the assertion is comi1pleted.im im′ (cid:3)s
For the simplicial complexes ∆ and ∆ defined on disjoint vertex sets, the join
1 2
of ∆ and ∆ is ∆ ·∆ ={σ∪τ : σ ∈∆ ,τ ∈∆ }.
1 2 1 2 1 2
Theorem 2.7. The simplicial complexes ∆ and ∆ are k-shellable if and only if
1 2
∆ ·∆ is k-shellable.
1 2
Proof. Let ∆ and ∆ be k-shellable. Let F ,...,F and G ,...,G be, respec-
1 2 1 r 1 s
tively, the k-shelling orders of ∆ and ∆ . We claim that
1 2
F ∪G ,F ∪G ,...,F ∪G ,...,F ∪G ,F ∪G ,...,F ∪G
1 1 1 2 1 s r 1 r 2 r s
is a k-shelling order of ∆ ·∆ .
1 2
Let F ∪G be a facet of ∆ ·∆ which comes after F ∪G in the above order.
i j 1 2 p q
We have some cases:
Let p<i. Since ∆ is k-shellable, there exist u ,...,u ∈F \F and some l<i
1 1 k i p
such that F \F = {u ,...,u }. It follows that u ,...,u ∈ (F ∪G )\(F ∪G )
i l 1 k 1 k i j p q
and (F ∪G )\(F ∪G )={u ,...,u }.
i j l j 1 k
Let p=i and q <j. Since ∆ is k-shellable, there exist v ,...,v ∈G \G and
2 1 k j q
some m < j such that G \G = {v ,...,v }. Therefore we obtain v ,...,v ∈
j m 1 k 1 k
(F ∪G )\(F ∪G ) and (F ∪G )\(F ∪G )={v ,...,v }.
i j p q i j i m 1 k
Conversely, suppose that ∆ ·∆ is k-shellable with the k-shelling order F ∪
1 2 i1
G ,...,F ∪G . LetF ,...,F betheorderingobtainedfromF ∪G ,...,F ∪
j1 it jt s1 sr i1 j1 it
G after removing the repeated facets beginning on the left-hand. Then it is easy
jt
to check that F ,...,F is a k-shelling order of ∆ . In a similar way, it is shown
s1 sr 1
that ∆ is k-shellable. (cid:3)
2
The following theorem, relates the expansion of a shellable complex to a k-
shellable complex.
6 RAHIMRAHMATI-ASGHAR
Theorem 2.8. Let ∆ be a simplicial complex and k ∈ N. Then ∆ is shellable if
and only if ∆[k] is k-shellable.
Proof. Let ∆ = hF1,...,Fri and let ∆j = hFji∩hF1,...,Fj−1i for j = 2,...,r.
Fix an integer j. If ∆ =hF \x ,...,F \x i, then
j j i1 j it
∆[k] = hF[k]i∩hF[k],...,F[k] i
j j 1 j−1
= hF[k]\{x ,...,x },...,F[k]\{x ,...,x }i.
j i11 i1k j it1 itk
[k] [k]
Now by the Definition 2.1, if F ,...,F is a shelling order of ∆ then F ,...,F
1 r 1 r
is a k-shelling order of ∆[k].
Conversely, suppose that F[k],...,F[k] is a k-shelling order of ∆[k] and set
1 r
∆[k] = hF[k]i∩hF[k],...,F[k] i for j = 2,...,r. Fix an index j. Hence ∆[k] =
j j 1 j−1 j
[k] [k]
hF \σ ,...,F \σ i with |σ | = k for all i. By Proposition 2.4(b), σ ∩σ = ∅
j 1 j t i l m
for all l 6= m. We claim that for every i, σ is the expansion of a singleton set.
i
Suppose,onthe contrary,thatforsomeσ wehavex ,x ∈σ withi 6=i and
s i1l i2m s 1 2
let Fj[k] ∩Fs[k′] = Fj[k]\σs for some s′. It follows from |σs| = k that xi1l′ 6∈ σs for
somel′ with1≤l′ ≤k. Inparticular,weconclude thatxi1l 6∈Fs[′k] butxi1l′ ∈Fs[′k].
This is a contradiction, because Fs[′k] is the expansion of Fs′.
[k]
Therefore we conclude that for all j =2,...,r, the complex ∆ is in the form
j
∆[k] =hF[k]\{x }[k],...,F[k]\{x }[k]i.
j j i1 j it
Finally,forallj,∆j =hFji∩hF1,...,Fj−1iwillbeintheform∆j =hFj\xi1,...,Fj\xiti.
ThisimpliesthathFji∩hF1,...,Fj−1iispureofdimensiondim(Fj)−1forallj ≥2,
as desired. (cid:3)
LetRbeaNoetherianringandM beafinitelygeneratedmultigradedR-module.
We call
F :0=M0 ⊂M1 ⊂...⊂Mr−1 ⊂Mr =M
amultigraded finitefiltration ofsubmodulesofM ifthereexistthepositiveintegers
a1,...,ar such that Mi/Mi−1 ∼= Qaj=i1R/Pi(−aij) for some Pi ∈ Supp(M). F is
calledamultigradedprimefiltration ifa =...=a =1. Itiswellknownthatevery
1 r
finitelygeneratedmultigradedR-moduleM hasamultigradedprimefiltration(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 ⊂X. We set Fc =X\F and P =(x :x ∈F).
F i i
Theorem2.9. Let∆beasimplicial complex andk apositiveinteger. IfF ,...,F
1 r
is a k-shelling order of ∆ then there exists a filtration 0=M ⊂M ⊂...⊂M =
0 1 r
S/I with
∆
r−i kai
Mi = \PFlc and Mi/Mi−1 ∼= YS/PFrc−i+1(−aij),
l=1 j=1
for all i=1,...,r. Here ai =|aij| for all j =1,...,kai.
k-SHELLABLE SIMPLICIAL COMPLEXES AND GRAPHS 7
Proof. We set a = 0 and for each i > 2 we denote by a the number of facets of
1 i
hFii∩hF1,...,Fi−1i. IfF1,...,Fr isak-shellingof∆,thenfori=2,...,r wehave
i−1
(1) \PFjc +PFic =PFic +Pσi1...Pσiai
j=1
where σ = F \F and |σ | = k for l = 1,...,a . Actually, F ∩F ’s are all of
il i il il i il i
facets of hFii∩hF1,...,Fi−1i. Since σil ∩σil′ = ∅ for 1 ≤ l < l′ ≤ ai, one can
supposethatPσi1...Pσiai =(fij :j =1,...,kai). Setaij =deg(fij)anditis clear
that for all j =1,...,kai, ai =|aij|. We have the following isomorphisms:
(Tij−=11PFjc)/(Tij=1PFjc) ∼=Tij−=11PFjc +PFic/PFic
∼=∼=∼=PPPσσσiii111.........PPPσσσiiiaaaiii//+(PPσPσi1Fi1ic../...P.PFPσicσiaiiaiP∩FicPFic)
whereai =|aij| forj =1,...,kai. Nowit iseasyto checkthatthe homomorphism
(r1θ,.:.Q.,rkakiaiS) →7→ PPσijk1a.i.r.jPfσijia+i/PPσσii11......PPσσiiaaiiPPFFicic
is an epimorphism. In particular, it follows that
kai
Pσi1...Pσiai/Pσi1...PσiaiPFic ∼= YS/PFic(−aij).
j=1
This completes the proof. (cid:3)
Remark 2.10. InviewofTheorem2.9,let∆=hF ,...,F ibeashellablesimplicial
1 r
complex and ∆ =hF ,...,F i. Then we have the prime filtration
j 1 j
(0)=I ⊂I ⊂...⊂I ⊂K[∆]
∆ ∆r−1 ∆1
for K[∆]. In particular, it follows the following filtration for K[∆[k]]:
(0)=I ⊂I ⊂...⊂I ⊂K[∆[k]].
∆[k] ∆r[k−]1 ∆1[k]
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 filtrationdescribedinTheorem2.9inthe casethat k >1is not
a prime filtration,i.e. the quotientof anytwo consecutivemodules ofthe filtration
is not cyclic. Consider the same notations of Theorem 2.9, we have the following
prime filtration for K[∆] when ∆ has a k-shelling order:
r
F :0=Tj=1PFjc ⊂...⊂
Tij=1PFjc ⊂...⊂Pkj=ai1−1(fj)+Tij=1PFjc ⊂Pjk=ai1(fj)+Tij=1PFjc =Tij−=11PFjc
⊂...⊂K[∆]
where (f1,...,fj−1):(fj) is generatedby linear forms for all j =2,...,kai and all
i=1,...,r.
Forall2≤j ≤kai, supposethat(f1,...,fj−1):(fj)=PQj. SetPQ1 =(0). We
have
Pjt=1(ft)+Tit=1PFtc/Ptj=−11(ft)+Tit=1PFtc ∼=∼= ((ffjj))//f(fjjP)L∩ij.(cid:0)(f1,...,fj−1)+Tit=1PFtc(cid:1)
8 RAHIMRAHMATI-ASGHAR
where Lij = Fic ∪Qj, for i = 1,...,r and j = 1,...,kai. Therefore the set of
prime ideals which defines the cyclic quotients of F is Supp(F) = {P : i =
Lij
1,...,r and j =1,...,kai}.
3. The Stanley conjecture
Consider a field K, and let R be a finitely generated Nn-graded K-algebra,and
let M be a finitely generated Zn-graded R-module. Stanley [20] conjectured that,
in this case, there exist finitely many subalgebras A ,...,A of R, each gener-
1 r
ated by algebraically independent Nn-homogeneous elements of R, and there ex-
ist Zn-homogeneous elements u1,...,ur of M, such that M = Lri=1uiAi, where
dim(A )≥depth(M) for all i and where u A is a free A -module of rank one.
i i i i
ConsiderafinitelygeneratedZn-gradedS-moduleM,asubsetZ of{x1,...,xn},
andahomogeneouselementu∈M. TheK-subspaceuK[Z]ofM iscalledaStanley
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 D of M into a finite direct sum of Stanley
spaces. The Stanley depth of D, denoted sdepth(D), is the minimal dimension of a
Stanley space in a decomposition D. We set
sdepth(M)=max{sdepth(D):D is a Stanley decomposition of M},
and we call this number the Stanley depth of M. The Stanley conjecture says that
sdepth(M)≥depth(M) always holds.
The following lemma is needed in the proof of the main theorem of this section.
Lemma3.1. LetF ,...,F ⊂X withF ∩F =∅foralli6=j and|F |=k >1. Let
1 s i j i
f ,...,f be a sequence of minimal generators of P ...P ordered with respect
1 ks F1 Fs
to lexicographical ordering x > x > ... > x . Suppose that n is the minimal
1 2 n i
number of homogeneous generators of (f1,...,fi−1):(fi) for i=2,...,ks. Then
max{ni :i=2,...,ks}=nks =(k−1)s.
Moreover, for all i, the colon ideal (f1,...,fi−1):(fi) is generated by linear forms.
Proof. By [6, Corollary 1.5.], P ...P has linear quotients. To show equality,
F1 Fs
we use induction on s. If s = 1, the assertion is clear. Assume that s > 1. Let
PF1...PFs−1 =(f1,...,fks−1) and Fs ={x1,...,xk}. Then
PF1...PFs =x1(f1,...,fks−1)+...+xk(f1,...,fks−1).
Moreover,
(x1f1,...,x1fks−1,...,xkf1,...,xkfks−2):xkfks−1 =(x1,...,xk−1)+(f1,...,fks−2):fks−1.
Now,bytheinductionhypothesis,wehavenks =(k−1)+(k−1)(s−1)=(k−1)s,
as desired. (cid:3)
Let
F :0=M ⊂M ⊂...⊂M =M
0 1 r
be a prime filtration of M with Mi/Mi−1 ∼= (S/Pi)(−ai). Then this filtration de-
r
composesM asamultigradedK-vectorspace,thatis,we haveM = u K[Z ]
Li=1 i i
andthis is a StanleydecompositionofM whereu ∈M is ahomogeneouselement
i i
of degree a and Z ={x :x 6∈P }.
i i j j i
Now suppose that ∆ = hG ,...,G i is a pure shellable simplicial complex on
1 r
X and k is a positive integer. By Theorem 2.8, ∆[k] = hG[k],...,G[k]i is pure
1 r
k-SHELLABLE SIMPLICIAL COMPLEXES AND GRAPHS 9
k-shellable. For all i, set F = G[k]. Consider the prime filtration F of K[∆[k]]
i i
described in Section 2. Then we have the Stanley decomposition
r kai
K[∆[k]]=MMuijK[Zij]
i=1 j=1
where Z ={x :l 6∈L }, deg(u )=a and |a |=a for all i,j. We claim that
ij l ij ij ij ij i
for all i,j, |Z |≥depth(K[∆[k]]).
ij
By Corollaries 4.1 and 2.1 of [17], we have depth(K[∆[k]])=dim(K[∆])=|G |.
i
On the other hand, |G | ≥ a . Now by combining all of these results with Lemma
i i
3.1, we have
|Zikai| =kn−(|Qkai|+|Fic|)
=kn−((k−1)ai+kht(PGc))
i
=(k−1)(n−ht(PGc)−ai)+n−ht(PGc)
i i
=(k−1)(|G |−a )+|G |
i i i
≥|G |=dim(K[∆]).
i
Thus we have shown the main result of this section:
Theorem3.2. Theexpansionofthefaceringofapureshellable 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 ∆ the independence complex of G. We say
G
that G is k-shellable if ∆ has this property. The purpose of this section is to
G
characterize k-shellable graphs.
Following Schrijver [18], the duplication of a vertex x of a graph G means
i
extending its vertex set X by a new vertex xi′ and replacing E(G) by
E(G)∪{xi′xj :xixj ∈E(G)}.
In other words, if V(G) = {x ,...,x } then the graph G′ obtained from G by
1 n
duplicating k −1 times the vertex x has the vertex set
i i
′
V(G)={x :i=1,...,n and j =1,....k }
ij i
and the edge set
′
E(G)={x x :x x ∈E(G), r =1,...,k and j =1,...,k }.
ir js i j i i
Example 4.1. Let G be a simple graph on the vertex set V(G) = {x ,...,x }
1 5
and E(G) = {x x ,x x ,x x ,x x ,x x ,x x }. Let G′ be obtained from G by
1 3 1 4 2 4 2 5 3 5 4 5
duplicating 1 times the vertices x and x and 0 times the other vertices. Then G
1 4
and G′ are in the form
x x
2 21
x
12
x x x x
1 3 11 31
x
42
x x x x
5 4 51 41
G Gα
10 RAHIMRAHMATI-ASGHAR
Also,theindependencecomplexesofGandG′are,respectively,∆ =hx x ,x x ,x x ,x x i
G 1 2 1 5 2 3 3 4
and ∆G′ = hx11x12x21,x11x12x51,x21x31,x31x41x42i. Note that ∆G′ = ∆αG where
α=(2,1,1,2,1).
Inthe followingtheoremwe showthatthe simple graphobtainedfromduplicat-
ing 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′ 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′ is k-shellable.
Proof. It suffices to show that ∆G′ = ∆[Gk]. Then Theorem 2.8 completes the
assertion.
After relabelingof the verticesof G′ one canassume thatG′ is a graphwith the
vertex set X[k] ={x :i=1,...,n, j =1,...,k} and the edge set
ij
{x x :x x ∈G and 1≤r,s≤k}.
ir js i j
Let F be an independent set of G′ and let F¯ = {x : x ∈ F for some r}. If
i ir
|F¯| = 1 then F¯ is an independent set in G. So assume that |F¯| > 1. Suppose,
on the contrary, that x ,x ∈ F¯ and x x ∈ G. By the construction of G′, for all
i j i j
x and x of V(G′), x x ∈ G′. Therefore F contains an edge x x of G′, a
ir js ir js ir js
contradiction. This implies that F¯ is an independent set in G. In particular, since
F ⊂(F¯)[k] we have F ∈∆[k].
G
Conversely, suppose H is an independent set of G. Choose x ,x ∈ H[k]. If
ir js
x x ∈ G′ then x x ∈ G, which is false since x ,x ∈ H. Therefore H[k] is an
ir js i j i j
independent set of G′ and H[k] ∈∆G′. (cid:3)
In the following we want to extend some results from [5, 7, 21]. Firstly, we
present a generalizationof the concept of simplicial vertex.
Let G be a simple graph. For U ⊂ V(G) we define the induced subgraph of G
on U to be the subgraph G on U consisting of those edges x x ∈ E(G) with
U i j
x ,x ∈ U. For x ∈ V(G), let N (x) denote the open neighborhood of x, that
i j G
is, all of vertices adjacent to x. We also denote by N [x] the closed neighborhood
G
of x, which is N (x) together with x itself, so that N [x] = N (x)∪{x}. Set
G G G
N (U)= N (x) and N [U]= N [x].
G Sx∈U G G Sx∈U G
Recall from [8] or [14] that a vertex x ∈ V(G) is simplicial if the induced sub-
graph G is complete.
NG[x]
The simple graphG is a complete r-partite graph if there is a partition V(G)=
V ∪...∪V of the vertex set, such that uv ∈ E(G) if and only if u and v are in
1 r
different parts of the partition. If |V |=n , then G is denoted by K .
i i n1....,nr
Definition 4.3. The set S of pairwise non-adjacent vertices of G is a k-simplicial
set if G is a r-partite complete graph with k-element parts S ,...,S having
NG[S] 1 r
the following property:
for every S and every two vertices x ,x ∈S , N (x )=N (x ).
l i j l G i G j
Note that every 1-simplicial set is a simplicial vertex.
Lemma 4.4. Let S ⊂ V(G) of pairwise non-adjacent vertices of G. Set G′ =
G\N [S] and G′′ =G\S. Then
G
(i) ∆G′ =lk∆G(S) and (ii) ∆G′′ =Tx∈Sdl∆G(x).