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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. REFERENCES 1. I. Adler,G. B. DantzigandK. G. Murty,Existence of A-avoiding pathsin abstractpolytopes, 2. Math. Programming Studies1(1974), 4142. M. Balinski,On the graphstructureof convexpolyhedrain n-space,PacificJ. Math. 11(1961), 3. D. Barnette,The minimumnumberof verticesof a simplepolytope,IsraelJ. Math. 10(1971), 4. D. Barnette,A proof of the lower bound conjecturefor convexpolytopes,PacificJ. Math. 46(1973), 349-354. D. Barnette,Graphtheoremsfor manifolds,IsraelJ. Math. 16(1973), 62-72. 431-434. 121-125. 5. 6. M. BrtiLkner, Uber die Ableitung der allgemeinen Polytope und die nach Isomorphisms verschiedenen Typenderallgemeinenachtzelle(Oktatope), Verh. Nedfl. Akad.Wetensch. (Sect.I) 10(1909), 3-27. 7. G.A. Dirac,Extensions of Menger's theorem, J. LondonMath.Soc.38(1963),148-161. 8. B. Grtinbaum, Convex Polytopes, Wiley-Interscience (London-New York-Sidney), 1967. 9. V. Klee, Polyotopepairsand their relationship to linearprogramming, Acta Math. 133(1974), 1-25. 10. P. McMullenand D. W. Walkup,A generalizedlower-boundconjecturefor simplicialpolytopes, Mathematika18(1971), 264-273. 11. G. T. Sallee,Incidencegraphsofpolytopes,J. CombinatorialTheory2(1967), 466-506. 12. D. Walkup,The lowerboundconjecture for 3- and4-manifolds, ActaMath. 125(1970),75-107. Universityof Washington Seattle,Washington ReceivedJuly 10, 1975