Download A d-Pseudomanifold With fO Vertices Has At Least df

HOUSTON JOURNAL OF MATHEMATICS, Volume 1, No. 1, 1975.
A d-PSEUDOMANIFOLD
WITHfo VERTICESHASAT LEAST
dfo-(d-1)(d+2)d-SIMPLICES
Victor
Klee
Barnette
wasthefirstto prove
thatif fk isthenumber
ofk-faces
of a simple
(d+l)-polytope
P then(*) fo• dfd'(d-1)(d+2).
He
later extended(*) to a graph-theoretic
settingand wasthereby
enabled to prove the dual inequality for triangulated
d-manifolds.Here his methodsare used to provide a different
graph-theoreticextensionof (*) and thus extend the dual
inequalityto simpliciald-pseudomanifolds.
Introduction. A d-polytope is a d-dimensionalset (in a real vector space)that is
the convex hull of a finite set, and it is simple if each of its vertices is incident to
precisely d edges.The inequality (*) becomesan equality when d C ( 1,2 ), as follows
readily from Euler's theorem, and also when P is obtained from a d-simplex by
successivetruncations of vertices.Though (*) was first stated for d = 3 by BrCickner
[6] in 1909, his "proof" was incorrect and the problem remained open for sixty
years. See Grfinbaum [8] for historical details and an indication of the importance of
the "lower bound conjecture."
The inequality (*) was first proved for d C (3,4)
by Walkup [12] in 1970 and
for arbitrary d by Barnett½ [3] in 1971. (see Barnette [4], Grfinbaum [8], Klce [9],
McMullen and Walkup [10], and Walkup [12] for related inequalities.)In the present
note,Barnette's methods are used to establish(*) in a different graph-theoreticsetting
that makesit possibleto extend the dual inequality to pseudomanifolds.
Preliminaries. The notion of an LB-system (LB for "lower bound") providesa
suitable framework for our inductive arguments.For d • 1 an LB-system of type d,
sometimes called a d-system for brevity, is a nonempty finite collection F of
undirected finite graphs(called facets) such that
(a) each facet is d-valent and d-connected,
The preparationof this paperwassupportedby the Office of Naval Research.
81
82
VICTOR KLEE
(b) in the graph Fu= t3 F, each vertex is d-valent(an externalvertex) or
(d+l)-valent (an internal vertex), and
(c) for eachvertex v and set X of d verticesof Fu adjacentto v, at most one
facet contains (v)
U X.
Note that every nonempty subcollectionof a d-systemis itself a d-system,and
the aboveconditionsimply
(d) any two intersectingfacetshave at least d common vertices,and
(e) each external vertex belongsto a singlefacet, each internal vertex to at least
two and at most d + 1 facets.
The aboveaxiomscapturesoirieimportant aspectsof the boundarycomplexof a
simple (d+ 1)-polytope but they aremuchlessrestrictivethan that notion. Suppose,for
example,that P is a nonempty finite collectionof simpled-polytopesin a real vector
space and the intersection of any two members of P is a face of both. Let
F•--(Fp:PCP•, whereFpisthegraph
formed
bythevertices
andedges
ofP.Then
condition(c) is obviousand (a) followsfrom Balinski'stheorem[2] that the graphof
a d-polytope is d-connected.Thus F is a d-systemif condition (b) is satisfied.In this
setting (b) is equivalent to the conjunction of
(b') each (d-1)-face of a member of P is incidentto at most one additional
member
of P,,,,
and
(b") anytwo intersecting
members
of P havea (d-1)-facein common.
A d-system
•Fissaidto beconnected
if thegraph
•u isconnected.
Notethat•u
is actually (d+l)-connected when F is obtained from the boundary complex of a
simple(d+l)-polytope in the mannerof the precedingparagraph.
The following two lemmasand their proofsare inspiredby observationsof Sallee
[11, p. 470] and Barnette [3, p. 123] respectively.
1. LEMMA. If F is a connectedLB-systemof type d then the graph Fu is
d-connected.
PROOF. By (d) and F's connectedness,
the facetscan be arrangedin a sequence
such that each facet after the first has at least d vertices in common with one of its
predecessors.Then use the fact that a graph is d-connected if it is the union of two
d-connectedsubgraphsthat have at least d verticesin common.This follows readily
from a characterization of d-connectednessthat was establishedby Dirac [7, p. 151]
A d-PSEUDOMANIFOLD
WITHfoVERTICES
83
and is also derived easily frownthe max-flow min-cut theorem.
2. LEMMA. Suppose that F is an LB-system of type d, 0 is a connectedproper
subsystemof b• and C is a subsystemoff
'• 0 such that
(i) thereisa vertext of 0 u that is not in Cu, and
(ii) thereis a vertexcommonto 0 u and Cu. Thenat leastd verticesof Cu are
internal in O LJC but external in C.
PROOF. By (d) thereare at leastd verticescommonto QU andCu, andby (e)
they are all internalin Q •JC. Suppose
that somevertexs of QU(hCu is internalin C.
By the preceding
lemmathereare d independent
pathsin QU that join s to t, andon
eachpath the lastvertexin Cu is internalin Q u C but externalin C.
Main result.
3. THEOREM.
The number of internal vertices of a connected LB-system F of
typed isat leastdl•FId whenthereisan external
vertexandat leastn = d[F...I
- (d-I)
(dq-2) when all vertices are internal.
PROOF.For an artibraryfixedd >• 1, wefirstemployinduction
on IFilto prove
the theorem with n replacedby n - 1. The assertionsare obviouswhen IFI = 1. For the
inductivestep,let v be a vertexof Fu, externalif possible,
let Q be the collectionof all
facets
incident
tov,andletc.C..
1,---,C•r
n betheconnected
components
ofF•'--...Q.
Bythe
precedinglemma,eachsubgraph
C.uof Fu hasat leastd verticesthat are internalin
Q•J..C.i
butexternal
inCi, whence
bytheinductive
hypothesis
C.
u also
hasdl.•.ild
vertices
thatareinternal
in Ci. Thusatleastdl•ilvertices
ofC-uareinternal
inF, and
since
the,.•i'sarepairwise
disjoint
thetotalnumber
of these
vertices
isatleast
dZ•al•iI = d(IF•I-I•QI).
That is the desired conclusion when v is external in F, for then IQI = 1, and it is also
the stated conclusion •vhen all vertices are internal, because IQi •< d + 1 by (e) and
hence
d(IF•I-IQI)
+ 1 >•dlF•l-d(d+l) + 1 = n- 1.
It remainsonlyto showthereisno connected
d-system
F•with precisely
k =dlF_.J[d2-d+ 1
vertices.
Suppose
thereis suchan F...,
let v and•Qbe asin thepreceding
paragraph,
let
5' beanisomorph
of•Fdisjoint
fromF•,andletv'and...Q'
bethecorrespondents
in•
of v and•Q.By thedualof a construction
usedby Barnette
[4, p. 351] fora similar
84
VICTOR KLEE
purpose,it is possibleto eliminatev and v' and meld each facet in Q with its
correspondent
in •Q'(simply
combine
corresponding
edges)
soasto obtainfrom
•FUF•'aconnected
d-system
withprecisely
2k-2vertices
and2IFil-I•QI
facets.
From
the inequality of the precedingparagraphit follows that
2dlF•l
- 2d2-2d=2k-2•>d(21F•I-I•Q[)
- d2-d+ 1,
whence
dl•Ql>
d2+dand
(e)iscontradicted.
For the case in which there are external vertices, the theorem's inequality
becomes an equality if the members of F are the graphs of simple d-polytopes
P1,'-',Pm whichare pairwisedisjointexceptthat for 1 <j •<m the intersection
Pj-191Pjis a (d-1)-simplex.
Whenall vertices
areinternal,
equality
arises
if the
members of F are the graphs of the various d-faces of one of the truncation
(d+ 1)-polytopes mentioned earlier.
The dual inequality for pseudomanifolds.As the term is usedhere a (simplicial)
d-complex is a nonempty finite collection C of (d+l)-sets, the d-simplices of C,
together with all subsetsof these d-simplices.The dual graph DG(C) has as its vertices
thed-simplices
of •C,twosuch
vertices
being
joinedbyanedgeof DG(C•)
if andonlyif
the two d-simplicesintersect in a (d-1)-simplex. For each k-simplex K of C, star (K,C)
is the d-complex
generated
by alld-simplices
of •CthatcontainK andlink(K,•C)isthe
(d-k-1)-complex
consisting
of allmembers
of star(K,C•)thataredisjoint
fromK. Note
thatthedualgraphs
DG(star(K,C•))
andDG(link(K,•C))
areisomorphic.
Asthetermis usedhere,a d-pseudomanifold
isa d-complex
•Msuchthat(i) each
(d-1)-simplex of M is contained in precisely two d-simplicesand (ii) the dual graph
DG(M•is connected.
Notethatif a d-complex
C satisfies
(i) theneachsubcomplex
of
C that is generatedby a connectedcomponentof DG(C) is a d-pseudomanifold;sucha
complex is called a strong component of C. Note also that if (i) holds for C and K is a
k-simplex of C then (i) holds for link(K,C)--that is, each (d-k-2)-simplex of link (K,C)
is containedin preciselytwo (d-k-1)-simplicesof the link.
The following lemma and proof were presentedby Barnette [5, p. 65] for his
graph manifolds and by Adler, Dantzig and Murty [1]
for a special class of
pseudomanifoldscalled abstract polytopes.
4. LEMMA. If M is a d-pseudomanifold,
x is a vertex(O-simplex)of M, and S
andT ared-simplices
of•Mnotcontaining
x, thenS andT arejoinedinDG(M)bya
Ad-PSEUDOMANIFOLD
WITHfoVERTICES
85
path that doesnot useany d-simplexcontainingx.
PROOF. The assertionis obviouswhen d = 1, for the 1-pseudomanifolds
and
their dual graphsare simplecircuits.Now supposed > 1 and let M, x, S and T be as
described.
By the connectedness
of DG(M) thereis in M a sequence
of successively
adjacent
d-simplices
S= So,S1,--.,Sœ
= T. LetSi'=Si whenx ½Si. Whenx CSi let
Ri = Si '" {x} andlet S[15e
theunique
d-simplex
of•Mthatcontains
Ri butnotx.
Since
So=S andS•=T it suffices
toshow
for1•<i•<œ
thatSi_
1=Si orDG(•M)
admits
a pathfromSi_1 to Si thatdoesnotuseanyd-simplex
incident
to x.
The simplices
Si_1 and Si are adjacent(resp.identical)whenneither(resp.
precisely)
oneof Si_
1andSi contains
x.Fortheremaining
case,
letKi = Ri_1 • Ri and
note that the sequence
Si_l,Si_l, Si, Si describes
a pathfromSi_
1 to Si in
DG(star(Ki,M•)),
whence
S•_
1'--Ki andS•'--Ki belong
tothesame
strong
component
•Qiof link(Ki,•M).
Thedesired
conclusion
thenfollows
fromthefactthat•Qiis a
2-pseudomanifold.
5. THEOREM. The dualgraphof a d-pseudomanifold
is (d-/-1)-connected.
PROOF. The assertionis obviouswhen d = 1 and we proceedby inductionto
showthat if S and T are distinctd-simplices
of M and X is a setof d d-simplices
in
M--- {S,T}
then DG(M) admits a path from S to T that usesno member of X.
Supposing
thecontrary,
let S= So,S
1,'",Sœ= T bea pathfromS to T thatuses
the
minimum
number
ofmembers
ofX•,letj bethesmallest
indexfor
which
Sj+1CX•,and
withm=min{j+d,œ}
letx GU•=jS
i.
If all membersof X contain x let S ' and T' be d-simplicesof M that do not
containx and are identicalwith or adjacentto S and T respectively.
By applyingthe
preceding
lemma
to S'andT' weseethatSandT arejoined
inDG(•M)
byapathnot
using any member of X.
If, on the other hand, somememberof X fails to contain x, then X includesat
mostd - 1 d-simplices
of star(x,•M)and thereexistsk suchthatj + 2 •<k •<m and
Sk ½•X.Byapplying
theinductive
hypothesis
totheappropriate
strong
component
of
link(x,•M)
weseethatDG(star(x,•M))
admits
a pathSj=T1,.-.Tr --Sk thatuses
no
member
of X•,whence
thepath
So,"',Sj,T2,'' ',Tr-1,Sk,'' -,Sœ
contradicts
theminimizing
property
forwhichtheSi'swerechosen.
86
VICTOR KLEE
6. THEOREM. Supposethat M is a d-pseudomanifoldand .for each vertex x of
•Mlet Qxdenote
thesetofallstrong
components
ofstar(x,•M).
LetQ(•M)
= UxOx and
•F= ( DG(Q?
.'•O•Q(•M)}.ThenF• is a connected
LB-system
of typed with
•Fu=DG(M•).
Thenumbers
of vertices
andfacets
of•Farerespectively
fd(M•)
and
ZxlOx[.
PROOF. It follows from the precedingtheorem that F satisfiesaxiom(a), and (b)
is obvious.
To verify(c), notethat if SO is a vertexof DG(M) andS1,.",Sd are
vertices
ofDG(M•)
adjacent
toSOthen
theintersection
Ch•d=0S
i consists
ofaunique
vertexx of M. Tkereis a strong
component
Q of star(x,M) thatincludes
alltheSi's,
andDG(Q•)
istheunique
member
of F•thatcontains
allthevertices
So,S
1," ',Sd'
Plainly
fo(F•
u)= fd(•M).
To seethatIFil= Zx[Qx[,
notethatif x1 andx2 are
distinct
vertices
of •Mand•Qiis a strongcomponent
of star(xi,•M),then•Q14:•Q2'
Indeed,
let S•be a d-simplex
in •Q1,let T = S if y •S, andif y • S let T bethe
d-simplex
of •Mthatcontains
S"- (y } butnoty. ThenT • Q•I'" •Q2'
7. COROLLARY.
A d-pseudomanifold
withfo vertices
hasat leastdfo -(d-l)
(d+2) d-simplices.
PROOF.
Use Theorems
3 and 7.
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Universityof Washington
Seattle,Washington
ReceivedJuly 10, 1975