On Connectivity of the Facet Graphs of

Simplicial Complexes

Y. Rabinovich and I. Newman

REPORT No. 46, 2013/2014, spring

ISSN 1103-467XISRN IML-R- -46-13/14- -SE+spring

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On Connectivity of the Facet Graphs of Simplicial Complexes

Ilan I. Newman∗ Yuri Rabinovich†

February 10, 2015

Abstract

The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorialinterest. In particular, it is shown that the facet graphs ofd-cycles, d-hypertrees andd-hypercuts are, re-spectively,(d+ 1)- , d- and (n− d− 1)-vertex-connected. It is also shown that the facet graph of ad-cyclecannot be split into more thans connected components by removing at mosts vertices. In addition, the paperdiscusses various related issues, as well as an extension tocell-complexes.

1 Introduction

Graphs of convex polytopes have been studied for many decades, starting with the classical Steinitz characteri-zation of the graphs of 3-polytopes [10], experiencing a bust with the advance of the Simplex Method for LinearProgramming, and continuing to draw a research effort in themodern era. See, e.g., the books [10, 23], and thesurvey [14] for many related results and open problems. One of the more famous results in the area is BalinskiTheorem [3] from 1961, claiming that the graph (i.e., the1-skeleton) of ad-polytope is(d+1)-vertex connected.This theorem and its various geometrical, topological and algebraic extensions have received a considerable at-tention, see e.g., [4, 5, 7, 2] for a very partial list of old and new related results. It has been extended to simpled-cycles in simplicial complexes in l [12], where it is shown that the geometric realization of such graphs inRd+1 is generically rigid. See also the very recent [1] for an algebraic treatment of graphs of simpled-cycles.

In this paper we study the connectivity properties offacetgraphs of simplicial complexes and, more gen-erally, of cell complexes. That is, the facet graphGd(K) of a d-complexK, has a vertex for everyd-face ofK, and two such vertices are connected by an edge if the corresponding faces share a(d − 1)-face. Since thefacet graph of a convexd-polytopeP is isomorphic to the graph of its dual polytopeP ∗, the facet graphs ofconvex polytopes do not require a separate study. This is notthe case for simplicial complexes, where graphsand facet graphs differ significantly. For example, it is folklore that if a pured-complexK is strongly connected,i.e., its facet graphGd(K) is connected, then the graph ofK is d-connected. Obviously, this implication cannotbe reversed, and connectivity of the graph ofK implies nothing about the connectivity of its facet graph. Foranother example, observe that while the graph of a simple simplicial cycle is generically rigid by [12], its facetgraph does not have to be such, as demonstrated by the boundary of the cross-polytope.

Motivated on one hand by the classical results about graphs of convex polytopes, and on the other hand bythe emerging combinatorial theory of simplicial complexes(see, e.g., [17] and the references therein), we studythe facet graphs of the basic objects of this theory: simpled-cycles,d-hypertrees andd-hypercuts. These are thehigher dimensional analogues of simple cycles, spanning trees and cuts in the complete graphKn.

The paper proceeds as follows. We start with building our tools, and show that a simplicial complex inducedby at mostd simplices of dimension at mostd, collapses to its(d− 2)-skeleton. This lemma will be generalizedin the last section, and one of its variants will be shown to beequivalent to the Homological Mixed Connectivity

∗Department of Computer Science, University of Haifa, Haifa, Israel. Email:ilan@cs.haifa.ac.il. This Research wassupported by The Israel Science Foundation (grant number 862/10.)

†Department of Computer Science, University of Haifa, Haifa, Israel. Email:yuri@cs.haifa.ac.il. This Research wassupported by The Israel Science Foundation (grant number 862/10.). Part of this research was done while this author visited Mittag-Leffler Institute, Stockholm

1

Theorem [11, 7], an elegant topological generalization of Balinski Theorem to higher-dimensional skeletons. Weshall also discuss the duality of simple cycles and hypercuts in the complete simplicial complex onn vertices.

Next, we address the connectivity of the facet graphs of the basic combinatorial-topological objects. Itcomes, perhaps, as a little surprise that the facet graph of asimpled-cycle is(d+1)-connected. The facet graphof a d-hypertreeT turns out to bed-connected, while the facet graph of ad-hypercutGd(H) is (n − d − 1)-connected. All the results are tight.

In Section 4.3, inspired by [16], we study what happens to thefacet graphGd(Zd) of a simpled-cycleZd

upon removal ofs of its d-simplices. The discussion, containing a study of an extremal problem about the Bettinumbers of smalld-complexes, leads to a somewhat unexpected conclusion thatthe remaining part ofGd(Zd)has at mosts components.

In the last section we study cell complexes with mild topological assumptions about the structure of the cells,and show, among other things, that the facet graphs of a simpled-cycles are still(d+ 1)-connected.

The paper employs only the very basic notions of Algebraic Topology (defined in the body of the paper),and should hopefully be accessible to anyone interested in Combinatorial Topology.

2 Preliminaries

2.1 Basic Standard Algebraic Topology Notions

We mostly use the basics of Homology Theory, beautifully presented in [19]. Throughout the paper we workover a fixed finite set (universe), identified with[n], and an arbitrary fixed fieldF. Many of our results hold ifFis replaced by any Abelian group, however, in this paper we shall not pursue this direction.

A d-dimensional simplex, abbreviated asd-simplex, is an oriented setσ ⊆ [n] with |σ| = d + 1. In thispaper the orientation is expressed by viewingσ as an ordered(d+ 1)-tupleσ = (s1, s2, . . . , sd+1), wheres1 <s2 < . . . < sd+1. A faceof aσ is any (oriented) simplex supported on the subset ofV [σ] = s1, s2, . . . , sd+1.

A simplicial complexK is a collection of simplices over[n] closed under containment, i.e., ifσ ∈ K, thenso are all the faces ofσ. As before,σ ∈ K is called a face ofK. The dimension ofK is the largest dimensionover all its faces. Some of the complexes discussed in this paper arepured-dimensional complexes, i.e., all themaximal faces ofK are all of the same dimension. Such faces are calledfacets.

The completed-dimensional complexKdn = σ ⊂ [n] | |σ| ≤ d + 1 contains all possible simplices over

[n] of dimension at mostd.Chains: LetK be ad-complex. We denote byK(d) the set of alld-faces ofK. A d-chainof K is formal

sumCd =∑

σi∈K(d) ciσi with ci ∈ F. Alternatively,Cd can viewed as a|K(d)|-dimensionalF-valued vector

indexed by members ofK(d). d-chains ofK form a linear space overF.The supportSupp(Cd) is the set of alld-simplices appearing inCd with not-zero coefficients. The pure sim-

plicial simplexK(Cd) associated withCd is the downwards closure ofSupp(Cd) with respect to containment.A complexK is said to havefull d-skeleton ifK(d) contains all

(n

d+1

)d-simplices.

The Boundary Operator: For ad-simplexσ = (s1, s2, . . . , sd+1), its d-boundary is defined as a(d− 1)-chain∂d(σ) =

∑d+1i=1 (−1)i−1(σ \si), where(σ \si) is an oriented facet ofσ obtained by the deletion ofsi.

Taking a linear extension of this definition, one obtains a linear boundary operator∂d from thed-chains over[n]to the(d − 1)-chains over[n]. (Observe that for a specific complexK, thed-chains ofK are mapped by∂d tothe(d− 1)-chains ofK). The key property of the boundary operators is that∂d−1∂d = 0.

Using the vector form ofd-chains,∂d is represented by a(nd

)×( nd+1

)matrixMd whose rows are indexed

by all (d − 1)-simplices, and columns byd-simplices, and∂Cd = MdCd. The entries ofMd are given byMd(τ, σ) = sign(σ, τ), also written as[σ : τ ], wheresign(σ, τ) = 0 if τ is not a facet ofσ, andsign(σ, τ) =(−1)i−1 if τ is a facet ofσ obtained by deletion of thei’th element in the orderedV [σ]. The requirement∂d−1∂d = 0 translates toMd−1Md = 0 for anyd = 1, 2, . . . n. For technical reasons, for any vertexσi ∈ [n],it’s (−1)-boundary∂0(σi) is defined as1. This extends linearly to0-chains. I.e., the setting is that of the reducedhomology.

2

The simplicial complexK(∂dCd) will be often denoted by∆Cd.Cycles and Boundaries: A d-chain inker(∂d) is called ad-cycle. The fact∂d∂d+1 = 0 implies that if

Cd = ∂d+1K thenCd is ad-cycle. Such a cycle is called ad-boundaryof K. The boundary of any(d + 1)-simplex is a simpled-cycle of sized+2, which is the smallest possible size of anyd-cycle. The space ofd-cyclessupported onK is denoted byZd(K), and the space ofd-boundaries supported ofK is denoted byBd(K). Thefactor spaceZd(K) / Bd(K) = Hd(K) is called thed-th (reduced) homology group ofK. The dimension ofHd(K) is thed-th Betti number ofK, denoted byβd(K).

A d-cycleZ is calledsimpleif no other (non-zero)d-cycle is supported onSupp(Z). Sometimes, slightlyabusing the notation, the supports ofd-cycles will also be calledd-cycles.

Cocycles, Coboundaries and Hypercuts: The coboundary operatorδd−1 is a linear operator adjoint to∂p. It is described by the left action ofMd, or, equivalently, by the right action ofMT

d . For historical reasons,both the range and the domain ofδd−1 are calledcochains, and denotedCd andCd−1 respectively. In this paper,while retaining the notation, we shall not make any distinction whatsoever betweend-chains andd-cochains1.

SinceMTd M

Td−1 = (Md−1Md)

T = 0, it holds thatδdδd−1 = 0. The kernel ofδd, ker(δd), is the space ofd-cocycles. Ad-cocycleZ∗ is called ad-hypercutif it is simple, i.e., no other non-zero cocycle is supportedonSupp(Z∗).

Hypertrees: A setA of d-simplices over[n] is calledacyclic if there are nod-cycles supported onA.Equivalently,A is acyclic if the columns vectors ofMd

n corresponding to its elements are linearly independentover F. Thus, it immediately follows that all maximal acyclic setsA ⊆ Kd

n have the same cardinality. Amaximal acyclic set ofd-simplices inKd

n is calledd-hypertree. Hypertrees were first introduced and studied byKalai [13]. The cardinality of everyd-hypertree is

(n−1d

)over any field.

For anyd-simplexσ ∈ Kdn and anyd-hyperteeT , there exists a uniqued-cycle of the formZd = σ −

CapT (σ), whereCapT (σ) =∑

ζi∈T ciζi. Observe that∂dσ = ∂dCapT (σ).

A complexK with full (d− 1)-skeleton hasHd−1(K) = 0 if and only ifK contains ad-hypertree.Relevant Matroidal Notions: Given the definitions above, it is clear thatKd

n defines a linear matroidMd

overF, whose cycles correspond to supports of simpled-cycles as above, and whose maximald-acyclic setscorrespond tod-hypertrees. With slightly more effort one can show that thesupports of the cycles of thedualmatroidof Md, i.e., the cocycles ofMd, correspond tod-hypercuts. Implied by the basic matroid theory is thefact that everyd-hypercut intersect everyd-hypertree (c.f. [21]).

2.2 Facet Graphs

The facet graphG(K) = Gd(K) of K, whereK is ad-complex (or, with a slight abuse of notation, just a set ofd-simplices, or even ad-chain), is a simple graph whose vertices correspond to thed-simplices inK, and twovertices form an edge if the correspondingd-simplices have a common(d − 1)-dimensional face. Thus, eachedge ofG(K) corresponds to a unique(d − 1)-face ofK. However, a(d − 1)-face ofK may correspond tomany or none of the edges ofG(K).

With a slight abuse of notation, we shall speak of facet graphof d-cycles andd-hypercuts, although techni-cally they are not complexes but chains.

3 Tools

The following simple lemma will be at the core of many arguments to come.

Lemma 3.1 LetD be a collection of at mostd simplices of dimension at mostd, and letK(D) be the corre-sponding simplicial complex. Then, every(d − 1)-cycle supported onK(D) is a (d − 1)-boundary ofK(D).That is, every such cycleZ is of a formZ =

∑σ∈D cσ∂σ.

1While ad-chainCd is regarded as a freeF-weighed sum of the elements ofK(d), thed-cochainCp is regarded as a mapping fromK(d) to F.

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The combinatorial proof presented here is based on the following two claims. In Section 4.3 we shall establisha more general version of the lemma, using an algebraic-topological approach.

Call ani-faceζ of a simplicial complexK exposedif it is contained in a unique(i + 1)-faceτ of K, (inparticular, suchτ must be maximal). Anelementaryi-collapseis the operation of elimination (or, alternatively,collapse) of a pair of facesζ, τ as above fromK, resulting in a proper subcomplex ofK. The notion ofi-collapse(due to Wegner [22]) is frequently used in Combinatorial Topology.

Claim 3.1 Letσ be ad-simplex, and letT be a subset of faces ofσ of dimension less thand, with |T | ≤ d− 1 .LetK(T ) be the complex defined byT . Call a face ofσ unmarkedif it is not inK(T ). Then, there is a sequenceof (d − 1) and(d− 2) elementary collapses that eliminate thed-face ofσ, and all the unmarked(d− 1)-facesof σ.

Proof. Since|T | ≤ d−1, there exists an unmarked(d−1)-faceτ of σ. Collapsing it together with thed-faceof σ, we arrive at∆σ\τ.

Consider the facet graphGd−1(∆σ). It is isomorphic toKd+1, the complete graph ond+ 1 vertices, wherethe (d − 1)-faces of∆σ correspond to the vertices, and the(d − 2)-faces correspond (in a 1-1 manner) tothe edges. LetH be the subgraph ofGd−1(∆σ), that is obtained by removing all the vertices and the edgescorresponding to the marked faces. Consider all vertices ofH as being colored white.

We now will consider the following process of elementary(d − 2)-collapses that, in turn, will color thevertices ofH blue once the corresponding(d − 1)-faces are collapsed. We start with a single blue vertex thatcorresponds toτ .

Observe that an edge ofH corresponds to a (currently) exposed(d−2)-face if and only if one of its endpointsis white, and the other is blue. Similarly, it corresponds toan (already) collapsed(d− 2)-face if and only if bothits endpoints are blue. Thus, an operation of an elementary(d− 2)-collapse on∆σ that involves only unmarkedfaces, can be interpreted in the terms ofH as follows: pick a blue vertex with a white neighbour, and make thisneighbour blue. The goal can be equivalently restated as colouring all the vertices ofH blue.

Clearly, this is possible if and only ifH is connected. Indeed, recall thatH is obtained fromKd+1 byremoving at most(d− 1) vertices and edges in total (not counting the edges whose removal was caused by thatof a vertex). Letr be the number of removed vertices, andq be the number of subsequently removed edges.Removingr vertices turnsKd+1 intoKd−r+1. The latter graph is obviously(d− r)-edge-connected. Therefore,removing additionalq edges fromKd−r+1, whereq ≤ (d− 1)− r, results in a connected graph.

Claim 3.2 LetS be a collection of at mostd simplices of dimension at mostd, and letK(S) be the correspond-ing simplicial complex. Then, ford > 1, all d- and (d − 1)-faces ofK(S) can be eliminated by a series ofelementary(d − 1)- and (d − 2)-collapses. Ford = 1 the situation is slightly different: the unique 1-face (ifany) ofK(S) can obviously be eliminated by an elementary0-collapse, however there is no way to eliminatethe surviving 0-face(s).

Proof. The proof is by an induction on the number ofd-simplices inS.If S has nod-simplices, then every(d − 1)-faceτ ∈ K(S) is a(d − 1)-simplex inS. Observe that every

suchτ has a (distinct) exposed(d−2)-facetζ inK(S). Indeed,τ hasd facets, while any simplex inS\τ mayun-expose at most one facet, and|S\τ| < d. Thus, all(d−1)-faces ofK(S) can be eliminated by elementary(d− 2)-collapses that eliminate pairs of facesζ, τ as above.

Otherwise, ifS contains somed-simplices, proceed as follows. Pick anyd-simplexσ ∈ S, setT = σ ∩ξ | ξ ∈ S\σ, and mark the faces ofK(T ) in σ. The assumption,|S| ≤ d implies that|T | ≤ d − 1. Henceby Claim 3.1, thed-face, as well as all the unmarked(d− 1)-faces ofσ can be collapsed by elementary(d− 1)and(d − 2) collapses. Since any elementary(d − 1)- or (d − 2)-collapse inσ that involves only the unmarkededges, can be carried out inK(S) as well, resulting in a complexK ′(S) be the resulting complex.

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Observe that any series of elementary(d − 1)- and(d − 2)-collapses inK(S\σ) can also be performedin K ′(S). Indeed, at any stage of collapse, an exposed(d− 1)- or a(d− 2)-faceτ ∈ K(S\σ) is necessarilyexposed inK ′(S) as well, since the faces inK ′(S)\K(S\σ) are of dimension< d− 1.

Employing this observation, and applying the induction hypothesis toK(S\σ), the conclusion follows.

For those familiar with the properties of the collapse operation, the implication Claim 3.2=⇒ Lemma 3.1 isimmediate. For the sake of completeness, here is a simple self-contained argument:

Proof. (of Lemma 3.1)Let K be a simplicial complex, and assume thatK ′ = K \ζ, τ was obtainedfrom K by an elementary(d − 2)-collapse involving an exposed(d − 2)-face ζ, and the (unique, maximal)(d − 1)-faceτ containing it. Then, any(d − 1)-cycleZ supported onK, is supported onK ′ as well. In otherwords, the coefficientcτ of τ in Z must be0. Indeed, sinceζ is contained only inτ , the coefficient ofζ in ∂Zis sign(τ, ζ) cτ , and thussign(τ, ζ) cτ = 0.

Next, letK be a simplicial complex, and assume thatK ′′ = K\τ, σ was obtained fromK by an elementary(d − 1)-collapse involving an exposed(d − 1)-faceτ , and the (unique, maximal)d-faceσ containing it. LetZbe a(d− 1) cycle supported onK and letZ ′ = Z − sign(σ, τ) cτ · ∂σ. Then,Z ′ is (d− 1)-cycle supported onK ′′, andZ is of a formZ = Z ′ + ∂T for ad-chainT = cτ · σ.

Combining the two observations, we conclude that ifR is obtained fromK by a series of elementary(d−1)-and(d − 2)-collapses, then any(d − 1)-cycleZ supported onK is of the formZ = Z ′′ + ∂U , whereZ ′′ is(d− 1)-cycle supported onR, andU is ad-chain onK.

By Claim 3.2,K(D) collapses to a complex of dimension< d − 1, lacking, in particular, any non-zero(d− 1)-cycles. Thus, any(d− 1)-cycleZ supported onK(D) must be of the formZ = ∂U , as claimed.

Duality between Cycles and co-Cycles in the Complete Complex Kn−1n

In order to discuss the structure of the facet graphs of hypercuts, it will be useful to establish a duality betweenhypercuts and simple cycles. Such duality exists in MatroidTheory [21], and in a related, but a slightly moresophisticated form in the Algebraic Topology. It is at the core of the important Poincare Duality and AlexanderDuality. For a relevant combinatorial exposition of the latter see [8] and the references therein. In fact, Claim 3.3below is an easy special case of the much more involved main result of that paper.

Let Σ be an(n − 1)-simplex (seen as a complex) on the underlying space[n]. I.e.,Σ = Kn−1n . Define a

correspondence between the(k − 1)-chains and the(r − 1)-cochains ofΣ, wherek + r = n, in the followingway.

For σ = 〈p1, p2, . . . , pk〉 where1 ≤ p1 < p2 < . . . < pk ≤ n, let σ = 〈q1, q2, . . . , qr〉, where1 ≤ q1 ≤q2 ≤ · · · qr ≤ n, andqj appears inσ iff it does not appear inσ. Sets(σ) =

∏pi∈σ(−1)pi−1. The dual (signed)

(r − 1)-simplex ofσ is defined byσ∗ = s(σ) · σ .

Extending this definition to chains and cochains, the dual ofa (k− 1)-chain (or cochain)C =∑cσσ is defined

as a(r − 1)-cochain (respectively, chain)C∗ =∑cσσ

∗. The key fact about this correspondence is:

Claim 3.3 (∂k−1C)∗ = δr−1 C∗ .

The proof appears in Appendix A.This leads to the following lemma, to be used in the Section 4.3, dedicated to hypercuts. Letk be a natural

number in the range[1, n], and letk + r = n.

Lemma 3.2 The operator∗ defines a 1-1 correspondence between simple(k − 1)-cyclesZk and (r − 1)-hypercutsHr−1 of Kn−1

n , given byZk−1 7→ Z∗k−1 = Hr−1. Moreover, the corresponding facet graphs

Gk−1(Zk−1) andGr−1(Hr−1) are isomorphic.

5

Proof. Observe that for any chain or cochainC ofKn−1n , C∗∗ = (−1)(

n+12 )−nC, and hence the duality map

∗ is a 1-1 correspondence between the(k − 1)-chains and the(r − 1)-cochains. Since by Claim 3.3, it mapscycles to co-cycles, and co-cycles to cycles, it yields a 1-1correspondence between(k − 1)-cycles and(r − 1)-co-cycles. Moreover, since it preserves containment, it yields a 1-1 correspondence between the minimal, i.e.,simple,(k − 1)-cycles and the minimal(r − 1)-co-cycles, i.e., the(r − 1)-hypercuts.

The isomorphism between the facet graphs ofZk−1 andHr−1 = Z∗k−1 is given by the mappingvσ 7→ vσ

from V [Gk−1(Zk−1)] to V [Gr−1(Hr−1)]. Since a pair of(k − 1)-simplicesσ, ζ ∈ Kn−1n share an(k − 2)-face

(i.e., are adjacent), if and only if they are both contained in ak-simplexξ ∈ Kn−1n , one concludes thatσ, ζ are

adjacent iffσ, ζ are.

4 Basic Results

4.1 Connectivity of Cycles

We are now ready to present the central results of this paper,starting with the(d+1)-connectivity of the simpled-cycles.

Theorem 4.1 LetZ be a simpled-cycle,d ≥ 1. Then, its facet graphG(Z) = Gd(Z) is (d+ 1)-connected.

Proof. Assume by contradiction thatG(Z) is not (d + 1)-connected. Then, there exists a subsetD of d-simplexes inSupp(Z), |D| ≤ d, such that the removal of the vertices corresponding toD in G(Z) disconnectsthe graph. LetV1, . . . , Vr ⊂ V , r > 1, be the vertex sets of the resulting connected components, and letS1, . . . , Sr be the corresponding sets ofd-simplices inSupp(Z). Finally, given thatZ =

∑cjσj , defined-

chainsZi =∑

σj∈Sicjσj.

By definition ofG(Z), differentSi’s have disjoint(d − 1)-supports. Keeping in mind thatZ is ad-cycle,this implies that the(d − 1)-boundariesCi = ∂Zi are all supported onD. Since every(d − 1)-boundary is a(d− 1)-cycle, Lemma 3.1 applies toCi’s, implying, in particular, that there exists ad-chainB1 supported onDsuch that∂B1 = C1. Consequently, thed-chainZ1 −B1 is ad-cycle, as∂(Z1 −B1) = C1 −C1 = 0. SinceZ1

andB1 have disjoint supports,Z1 − B1 6= 0. Also,Z1 − B1 is supported onK1 ∪D, a strict subset ofd-facesof Z. This contradicts the fact thatZ is simple cycle, concluding the proof.

Remark 4.1 The above argument yields, in fact, a slightly more robust type of connectivity than stated. Recallthat Lemma 3.1 applies not only toD as in the statement of Theorem 4.1, but also to a union ofr d-simplicesand q (d − 1)-simplices, wherer + q ≤ d. Thus, the graphG(Z) remains connected after removal of anyrvertices andq edges (or, more precisely, the edges of anyq cliques induced by(d − 1)-faces ofZ), as long asr + q ≤ d.

Theorem 4.1 is tight, e.g., ford-pseudomanifolds, i.e., simpled-cycles, where every(d− 1)-face is included inexactly twod-faces. In this caseGd is (d+1)-regular, and thus at most(d+1)-connected. Ford = 1, all simplecycles are pseudomatifolds, and thus they are exactly2-connected. Ford > 2, other simple cycles exist, and itnot presently clear to us whether such cycles can be more than(d+ 1)-connected, and if yes, by how much.

Theorem 4.1 has an immediate implication on connectivity ofthe facet graphs ofd-biconnectedsetsS ofd-complexes. This interesting notion originates in MatroidTheory, and generalizes the graph-theoretic 2-(edge)-connectivity.

Let S be a set ofd-simplices. Define the following relation onS:

Definition 4.1 σ ∼ ζ if there is asimplecycleZ supported onS containing bothσ andζ. Treatingσ,−σ asa simple cycle, it is also postulated thatσ ∼ σ.

6

It is known from Matroid Theory [21] that∼ is an equivalence relation. CallS bi-connectedif all itsd-simplices are∼ equivalent.

Corollary 4.1 For biconnectedS as above,Gd(S) is (d+ 1)-connected.

4.2 Connectivity of Hypertrees

Next, we establish the(d− 1)-connectivity ofd-hypertrees.

Theorem 4.2 Let T be ad-hypertree inKdn, d ≥ 1, n ≥ d + 2. Then, its facet graphG(T ) = Gd(T ) is

d-connected.

Proof. As before, it suffices to show that for any subsetX of d-simplices ofT , |X| ≤ d−1, the removal of thevertices corresponding to theX fromG(T ) does not disconnect the graph. Consider suchX, letV1, . . . , Vr ⊂ V ,be the vertex sets of the resulting connected components inG(T ), and letS1, . . . ,Sr be corresponding sets ofd-simplices inT . Let alsoS0 = X. We shall prove thatr must be1, and thusX is non-separating, as required.

For i = 1, . . . , r let us color alld-faces ofSi, by color i. In particular, everyd-face ofT σ /∈ X has a(unique) associated color, whileX is colorless.

The first stepis to extend this colouring ofT to all d-simplices inKdn \X in the following manner. Let

σ ∈ Kdn\T be ad-simplex. As explained in Section 2, there is a (unique)d-chainCapT (σ) supported onT

satisfying∂(Cap(σ)) = ∂(σ), namelyZσ = CapT (σ) − σ is a simpled-cycle. Since any non-emptyd-cycleis of size≥ d+1, and|X| ≤ d− 1, it follows thatSupp(Zσ)\X ∪ σ ⊆ T is not empty, and soCap(σ) mustcontain some colouredd-simplices inT . We claim thatall suchd-simplices must have the same color. Thiscolor will be assigned toσ.

Indeed, by Theorem 3.1, the graphG(Zσ) is (d+1)-connected. Since|X| ≤ d−1, it remains connected afterthe removal ofd vertices corresponding toσ∪X. I.e., for any two colouredd-simplicesζ, τ ∈ Cap(σ)\X ⊂T , there exists a pathξ1, ξ2, . . . , ξs of (coloured)d-simplices inCap(σ)\X ⊂ T whereξ1 = ζ, ξs = τ , andevery two consecutiveξi, ξi+1 share a(d − 1)-dimensional face. By definition ofSi’s, if a pair of colouredd-faces ofT share a(d− 1)-dimensional face, then they have the same color. Hence, allξi’s, and in particularζandτ , have the same color.

Having constructed a consistent extension of the colouringof T\X to the entireKdn\X, it will be convenient

to extend the definition ofSi’s to contain alld-simplices ofKdn colouredi. The setS0 = X remains unaffected.

The second stepis to show that any two adjacent (i.e., sharing a(d− 1)-face) colouredd-simplicesσi, σj ∈Kd

n \X have the same color. While inT \X this is immediately implied by the definition of the color classes,in Kd

n \X a proof is required.Assume by contradiction thatσi andσj have different colours. Letτij be the(d − 1)-face they share, and

let ζi ∈ Cap(σi), ζj ∈ Cap(σj) bed-simplices inT so thatσi ∩ ζi = σj ∩ ζj = τij . Obviously there are suchζi, ζj by the definition of a cap. Now, on one hand,ζi andζj are adjacent, and so, if both are colourful, it mustbe the same color. In addition, this color must be the same as this ofσi, σj by consistency of the color extension.Thus, ifσi andσj differ in color, then at least one ofζi, ζj must belong toX. In particular,τij is a(d− 1)-facetof somed-simplex inX.

Let ψ be the (unique)(d + 1)-simplex containing bothσi andσj, and let∆ψ denote the support of itsboundary. In particular,σi, σj ∈ ∆ψ. ConsiderGd(∆ψ), whose vertices are coloured according to the coloursof the correspondingd-simplices, and the vertices and the edges corresponding respectively tod- and(d − 1)-faces ofK(X), are marked. As explained above, any two colourful verticesof Gd(∆ψ) connected by anunmarked edge must be of the same color. Thus, showing that any two colourful vertices of this graph areconnected by an unmarked path, will imply that there is only one color, contrary to the assumption.

Observe thatGd(∆ψ) is isomorphic toKd+2. Observe also that anyd-simplex ofX may cause the markingof a single vertex, or, alternatively, of a single edge ofGd(∆ψ). Since|X| ≤ d − 1, this amounts to at mostd− 1 vertices and edges altogether. However, by an argument already used in the proof of Claim 3.1, removing

7

all marked vertices and edges is not enough to disconnectKd+2. Thus, any two coloured vertices are indeedconnected by an unmarked path, and thus the adjacentσi andσj must be of same color.

To sum up, we have shown so far that eachd-simplexσ ∈ Kdn\X has a well defined color, and that every two

adjacent colouredd-simplices have the same color. Recall that the goal is to show that there is only one color.Hence, to conclude the proof, it suffices to show that the facet graphGd(K

dn) remains connected after removal

of the vertices corresponding to thed-faces ofX, i.e., thatGd(Kdn) is (d + 1)-connected. One way of doing

it is by observing that thed-skeleton ofKdn is biconnected, and then applying Corollary 4.1. SinceGd(K

dn) is

obviously connected, by transitivity of∼, it suffices to check that any two adjacentσ, ζ are contained in a simpled-cycle. And indeed, they are contained in the boundary of the(unique)(d+ 1)-simplex containing both.

In fact, Gd(Kdn) is familiar in Combinatorics as the graph of the hypersimplex polytope∆d(n), or as the

graph of the(d+ 1)’th slice ofn-hypercube, where two strings are adjacent iff they are at Hamming distance 2.See [9] for a relevant discussion. The results of [2] imply that this graph is(d+ 1)(n − d− 1)-connected. Theproof of [2] involves an intricate geometric argument. For completeness, we attach in Appendix B an alternativesimple combinatorial proof of this fact.

To establish the tightness of Theorem 4.2, consider first thestarT = σ ∈ Kdn | n ∈ σ. This is ad-

hypertree: it obviously spans all thed-simplices inKdn. On the other hand, it is acyclic, as everyσ ∈ T contains

an exposed face, namely(σ\n). Now, consider, e.g., the hypertreeT ′ = T \σ ∪ ζ whereσ = (1, . . . , d, n),andζ = (1, . . . , d+1). It is easy to verify thatT ′ is indeed a hypertree. Observe that the(d−1)-face(1, . . . , d)of ζ is exposed inT ′, while every other(d − 1)-face ofζ is shared with a singled-simplex inT ′. Hence, thevertex corresponding toζ in Gd(T

′) has degreed, implying that this graph is not(d+ 1)-connected.Let us remark that the facet graph of ad-hypertree can be more thand-connected. E.g., whenT is a star as

above,Gd(T ) is obviously isomorphic toGd(Kd−1n−1), which isd(n − d− 2)-connected by Theorem 5.7.

Motivated by the dual definition ofr-edge-connectivity in graphs, namely thatG is r-edge connected ifand only if every cut (of the complete graph) intersectsE[G] in at leastr edges, we introduce the followingdefinition. Ad-complexK will be calledr-connectedif for everyd-hypercutH, |H ∩K(d)| ≥ r.

It immediately follows from Theorem 4.2 that:

Corollary 4.2 For d ≥ 1, if a d-complexK is r-connected then,Gd(K) is (d+ r − 1)-connected.

Proof. Assume by contradiction that there is a set ofd-simplicesD = σ1, . . . , σd+r−2 whose removaldisconnectGd(K). Remove firstD′ = σ1, . . . σr−1 fromK. Since by assumption, every hypercut has size atleastr in K, K \D′ still contains ad-tree. But thenGd(K \D′) is d-connected by Theorem 4.2, and hence itremains connected after the removal of the nextd− 1 simplices inD \D′.

To demonstrate the usefulness of Corollary 4.2, apply it toK = Kdn. Since the mincut in this case is of size

n − d (see, e.g., [20]), one concludes thatGd(Kdn) is at least(n − 1)-connected (which is still far from being

tight, by Theorem 5.7).Another implication of Theorem 4.2 is about the connectivity of complements ofd-hypercuts.

Corollary 4.3 LetH ⊂ Kdn be ad-hypercut,d ≥ 1, and letH contain all d-simplices missed byH. Then,

Gd(H) is (d− 1)-connected.

Proof. Recall thatH, being a hypercut, is critical with respect to hittingd-hypertrees, i.e., it hits every suchT . Moreover, for anyσ ∈ H there exists ad-hypertreeTσ such thatTσ ∩H = σ. Hence, augmentingH byanyσ 6∈ H makes it contain ad-hyperteeTσ. Hence by Theorem 4.2, the corresponding facet graphGd(Tσ) isd-connected. Removing the extra vertex corresponding toσ form this graph, leaves us withGd(H), that mustbe(d− 1)-connected.

While for d = 1, Corollary 4.3 is trivially tight, it appears that ford ≥ 2 it can be significantly strengthened.This is left as an open problem.

8

4.3 Connectivity of Hypercuts and Cocycles

Theorem 4.3 LetH be ad-hypercut,d ≥ 1. Then, its facet graphGd(H) is (n− d− 1)-connected.

Proof. This is an immediate consequence of the duality result of Lemma 3.2, claiming thatH∗ ⊂ Kdn is a

simple(n− d− 2)-cycle withGd(H) = Gn−d−2(H∗), and an application of Theorem 4.1 toH∗.

For tightness, consider the following example. Letτ = (1, 2, . . . , d) ∈ Kdn be a(d − 1)-face. Then, thed-

cochainHτ =∑

p 6∈τ sign(τ ∪ p, τ) · (τ ∪ p) is ad-hypercut. Its graphGd(Hτ ) is an(n − d)-clique, which byconvention is(n− d− 1)-connected.

The facet graphs of cocycles that are not hypercuts, behave very differently. Ford = 1, the cocycles areprecisely the graph-theoretic cuts, and so Theorem 4.3 applies. Ford ≥ 3, the facet graph of cocycle can bedisconnected, as exemplified byHτ +Hτ ′ as above, where the Hamming distance betweenτ andτ ′ as setsis atleast3.

Ford = 2, the answer is given by the following theorem:

Theorem 4.4 The facet graph of a (non-empty)2-cocycleZ∗ ofK2n is 2-connected.

Proof. It is immediate to verify the claim forn ≤ 4, and thus we assumen ≥ 5. To simplify the discussion,we use the duality between cocycles and cycles, as stated in Claim 3.3 and Lemma 3.2. LetZ be the dual chainof Z∗. Then,Z is ad-cycle,d = n− 4 ≥ 1, ofKn−4

n = Kdd+4, andG2(Z

∗) = Gd(Z).Now,Z, being a cycle, can be represented asZ =

∑Zi, where eachZi is a simpled-cycle, andSupp(Zi) ⊆

Supp(Z). The key point of the argument is thatKdd+4 is a “narrow” place ford-cycles. We claim that for any

twoZi, Zj as above, there exists a(d− 1)-simplexτ belonging toK(Zi) ∩K(Zj).The proof is by induction ond. No assumption about the simplicity thed-cyclesZi, Zj is made or required.

For d = 1, one needs to show that any two cycles inK15 have a common vertex. This is obvious. For general

d ≥ 2, letZ1, Z2 be twod-cycles. Since each ofZ1, Z2 contains at least(d+2) d-simplices, and since2(d+2) >

d+4, they share a common vertexv. AssumingZ1 =∑cℓσℓ, letC1 = linkv(Zi) = ∂

(∑σℓ∋v cℓσℓ

). C1 is

a nonempty(d−1)-boundary, and hence a nonempty(d−1)-cycle. Moreover, keeping in mind that∂(Z1) = 0,we conclude that the vertexv does not appear in the vertex setV (C1). The same applies to the similarly definedC2. Thus,C1, C2 are(d− 1)-cycles on(d+4)− 1 vertices. By induction hypothesis, they share a(d− 2)-faceτ ′. The desired(d− 1)-faceτ is given byτ = (τ ′ ∪ v).

To conclude the proof of the Theorem, consider twod-simplicesσ, ζ ∈ Supp(Z). If they fall in the sameZi,the(d+1)-connectivity ofGd(Z) implies that there are(d+1) vertex-disjoint paths between the correspondingverticesvσ, vζ in Gd(Z). Else,σ ∈ Zi andζ ∈ Zj . If Zi andZj share a commond-simplexξ, then, using theequivalence relation of Def. 4.1, we conclude thatσ ∼ ξ, ζ ∼ ξ =⇒ σ ∼ ζ, and by Cor. 4.1, we again haveat least(d + 1) vertex-disjoint paths. Finally, ifZi andZj have no commond-simplices, by the above claimthey still have a common(d− 1)-simplex. So, there are 2 vertices inV (Gd(Zi)), and 2 vertices inV (Gd(Zj)),that induce a cliqueK4 in Gd(Z). Hence, there are at least2 vertex-disjoint paths between the verticesvσ, vζ inGd(Z).

Remark 4.2 We have chosen to present this proof, because it provides more information about the structure ofG2(Z

∗). A simpler alternative proof would first reduce the problem to n ≤ 6, by using the following argument.By definition ofd-cocycles, the restriction ofZ∗ to any subsetS ⊂ [n] is a d-cocycle ofKd

n as well. Thus,for any pair of2-simplicesσ, ζ ∈ Z∗, instead of considering the paths between the corresponding vertices inG2(Z

∗), it suffices to consider them inG2(Z∗|S), whereS is the union of the vertex sets ofσ andζ.

9

5 Extensions and Refinements

In this section we further develop the results obtained in the previous section. We shall make a wider use ofthe homology-related notions of Algebraic Topology, whichare luckily well suited for the discussion. This willmake the presentation slightly more advanced, but the benefits will be apparent.

5.1 Facet Graphs of Simple Cycles: Beyond Connectivity

What follows is a direct continuation of Theorem 4.1.Analogously to Klee’s question about vertex graphs of convex polytopes [16]2, we ask what happens to

the facet graphG = Gd(Z) of a simpled-cycle Z, d ≥ 1, after removal of a set of verticesVD ⊆ V [G]corresponding to a set ofd-simplicesD ⊆ Supp(Z). The following localization theorem provides an answer tothis question in terms of the topological structure ofK(D).

Lemma 5.1 LetZ be a simpled-cycle andD ⊂ Supp(Z). Assume that removal ofVD fromGd(Z) createsm > 1 connected components. Then,K(D) containsm (d−1)-cycles so that:(a) any two of them have disjoint(d − 1)-supports; (b) anym − 1 of them are linearly independent modulo the space of(d − 1)-boundariesBd−1(K(D)), while allm of them are dependent.

Proof. Proceeding as in the proof of Theorem 4.1, letV1, . . . , Vm ⊂ V , be the vertex sets of the resultingconnected components, and letS1, . . . , Sm be the corresponding sets ofd-simplices inSupp(Z). Given thatZ =

∑cjσj, define thed-chainsZi =

∑σj∈Si

cjσj, andZD =∑

σj∈D cjσj . Finally, define the(d− 1)-cyclesCi = ∂Zi supported onK(D). This will be the set of the desired cycles.

By definition ofGd(Z), differentK(Si)’s have disjoint(d − 1)-supports, and sinceCi is supported onK(Si), the same applies toCi’s. This establishes(a).

To prove(b), consider, e.g., the firstm − 1 cycles, and assume by contradiction that for somed-chainQd

on D, and for some (not all zero) coefficientski ∈ F, it holds that k0 · ∂Qd +∑m−1

i=1 ki · Ci = 0. LetZ ′ = k0 ·Qd+

∑mi=1 ki ·Zi be ad-chain onK(Z). Using the disjointness ofSupp(Zi)’s, and the disjointness of

∪m−1i=1 Supp(Zi) andD, it is easily verified thatZ ′ is neither0 norZ; the latter sinceSupp(Z ′)∩Supp(Zm) = ∅.

Moreover, by definition ofZ ′, it holds that∂Z ′ = 0. This contradictions the simplicity ofZ.To see thatCimi=1 are linearly dependent overBd−1(K(D)), recall thatCi = ∂Zi, and thatZD +∑m

i=1 Zi = Z. Therefore,∂ZD +∑m

i=1Ci = ∂Z = 0.

As an immediate corollary to the lemma one gets:

Corollary 5.1 Let Z be a simpled-cycle, andD ⊆ Supp(Z). The number of the connected components ofGd(Z) \D is at most1 + dim Hd−1(K(D)) = 1 + βd−1(K(D)).

By Lemma 3.1, when|D| ≤ d, it holds thatHd−1(K(D)) = 0, and henceβd−1(K(D)) = 0, implying that thereis a unique connected component. A natural question is how large canβd−1(D) be as a function of|D| alone,in particular when|D| is large. To prepare the necessary background for the discussion, we cite the followingresult.

Theorem 5.1 [17]3 LetT be a set ofr-simplices,|T | = t. Then, the dimension ofZr(K(T )), the space ofr-cycles overK(T ), is maximized when the set familyT = Supp(τ)τ∈T is compressed, i.e., whenT contains

2 Klee studied the following question: what is the maximum possible number of the connected components in the vertex-graph of aconvexd-polytope, after the removal ofm vertices? The answer: it at most1 for m ≤ d, 2 for m = d + 1, and for a generalm, atmost the maximum possible number of facets in a convexd-polytope onm vertices.

3 Strictly speaking, most of the relevant results in that paper are formulated forrankr(T ), rather than fordimZr(K(T )). However,the two parameters are closely related, asrankr(T ) + dimZr(K(T )) = |T r|.

10

the first t elements in the reverse lexicographic order of the(r + 1)-size subsets of[n]. The correspondingnumerical estimation is that fort =

( xr+1

), x ∈ R, the dimension ofZr(K(T )) never exceeds

(x−1r+1

). I.e.,

dimZr(T ) ≤ t− Ω(t1−

1r+1

).

Moreover, fort =( nr+1

), n ∈ N, the optimaldimZr(K(T )) =

(n−1r+1

)is achieved on ther-skeleton ofKr

n.

This leads to the following crude estimation ofβd−1(D). Since the number of(d − 1)-faces ofK(D) can beupper-bounded by(d+ 1) · |D|, using Theorem 5.1 one gets:

βd−1(K(D)) = dimZd−1(K(D))−dimBd−1(K(D)) ≤ dimZd−1(K(D)) ≤ (d+1) · |D|−Ω(|D|1− 1d ) .(1)

A more accurate answer to our question was provided by Roy Meshulam:

Theorem 5.2 [18] For D as above,βd−1(K(D)) ≤ d · |D| − Ω(|D|1− 1d ).

On the other hand, for any sufficiently large integers of the forms = 1d+1

(nd

), there existsD of sizes with

βd−1(D) =(n−1

d

)− 1

d+1

(nd

), which is best possible for suchs. Thus, the upper bound is asymptotically tight up

to the second-order terms.

Proof. Remember thatβd−1(K(D)) = dim Hd−1(K(D)) = dimZd−1(K(D))− dimBd−1(K(D)).For the upper bound, one may w.l.o.g., assume thatD is acyclic, i.e.,dimBd−1(K(D)) = |D|. Otherwise,

if somed-simplex inD is spanned by the others, removing it fromD effects neither the space of(r− 1)-cycles,nor the space of(r− 1)-boundaries, but reduces the size ofD. Adding some isolatedd-simplices to compensatethe reduction in the size does not effect, again, the(r− 1)-homology group. Thus, one gets an acyclic setD′ ofd-simplices with|D′| = |D|, andβd−1(K(D)) = βd−1(K(D′)).

For an acyclicD, arguing as in (1), one gets

βd−1(K(D)) = dimZd−1(K(D))− dimBd−1(K(D)) ≤((d+ 1) · |D| − Ω(|D|1− 1

d ))− |D| =

= d · |D| − Ω(|D|1− 1d ) .

For the lower bound, one may use the recent breakthrough result of Keevash [15], implying, in particular, thatfor any d, and for sufficiently largen such that(d + 1) divides

(nd

), there exists a setD∗ of d-simplices that

covers every(d − 1)-simplex inKd−1n exactly once. Clearly,|D∗| = s = 1

d+1

(nd

). The goal is to show that for

this s, βd−1(K(D∗)) is the maximum possible.The setD∗ is acyclic, and by the argument above, so is the optimal setDOPT of the same size. Thus, it

suffices to argue thatdimZd−1(K(D∗)) is the biggest possible. The(d − 1)-skeleton ofD∗ is of size(nd

)=

(d + 1) · s, which is the biggest possible for anyD of sizes. Finally, the(d − 1)-skeleton ofD∗ is Kd−1n ,

which by Theorem 5.1, has the biggest possible dimension ofZd−1(K(T )), namely(n−1

d

), among all setsT

of (d − 1)-simplices with|T | =(nd

). Since maxT, |T |=t dimZd−1(K(T )) is monotone increasing int, the

statement follows.

Corollary 5.1, i.e., part(b) of Lemma 5.1, together with Theorem 5.2 imply that the numberof connectedcomponents obtained by removing at mosts vertices from the facet graph of ad-cycle is at mostds. Somewhatsurprisingly, part(a) of Lemma 5.1 yields a stronger upper bound:

Theorem 5.3 LetGd(Z) be the facet graph of a simpled-cycle. Then, removing fromGd(Z) any s vertices,may create at mosts connected components.

11

Proof. Recall that a(d− 1)-cycle is of size at leastd+1. Since them different(d− 1)-cycles in Lemma 5.1are disjoint, they contain, altogether, at least(d+ 1)m different(d− 1)-simplices. On the other hand, letD bethe set ofd-simplices inSupp(Z) corresponding to the removed vertices. Then, the number of(d − 1) faces ofK(D) is at most(d+ 1) · |D| = (d+ 1)s. Thus,(d+ 1)s ≥ (d+ 1)m, and the conclusion follows.

The inequalitys ≥ m is tight (for somes’s), as shown by the following construction achievings = m. Takea triangulationT of a d-pseudomanifold overF, (i.e., every(d − 1)-face ofK(T ) is contained in exactly twod-simplices ofT , andT supports a unique nonemptyd-cycle), with the property that its facet graphGd(T ) isbipartite.

An example of such a triangulation of the sphere is thed-cross-polytope, also known as thed-cocube. Itsfacet graph is the graph of the cube, namely bipartite with2d+1 vertices, and two color classes each of size2d.Another example ford = 2 is provided by taking a torus obtained by appropriately gluing the opposite sides ofa planark × k square, wherek ≥ 4 is even, and subdividing each1 × 1 square cell in it into two triangles bydrawing the North-East diagonal.

Obviously, for suchT , takingD as alld-simplices in one color class ofGd(T ) results is decomposing theresultingGd(T ) \D into singletons.

The graph-theoretic property stated in the above theorem iscalledtoughness. It has implications. E.g., usingTutte’s criterion for existence of a perfect matching in a graph, one concludes via toughness that if ad-cycleZis of even size, thenG(Z) has a perfect matching. For a survey of toughness see [6].

5.2 Cycles in Cell Complexes

So far, we have discussed structures in simplicial complexes. In this section, we would like to discuss a class ofaxiomatically definedcell complexesthat includes simplicial complexes and convex polytopes (more precisely,the combinatorial abstraction preserving the structure oftheir faces). The methods and results obtained forsimplicial complexes will be re-examined and generalized.We are mostly interested in the generalizations ofLemma 3.1 and Theorem 4.1, in particular we will generalize Balinski Theorem for such complexes.

Replacing simplices bycellswith a specified combinatorial structure, and equipped witha boundary opera-tor ∂, gives rise to cell complexes and their homology groups. Thefollowing axioms describe the structure ofthe cells. Notably, the standard assumption that the boundary of a cell is a pseudomanifold will be replaced hereby a significantly weaker assumption that it is a simple cycle.

Formally,abstract cell complexis a graded poset (partially ordered set)P , whose elements ofP will be calledopen cells. The order represents the cell-subcell relation. A (closed) cellK(Co) ⊆ P corresponding to an opencell Co ∈ P, is defined as the set of all elements that are dominated byCo in P, includingCo.

SinceP is a graded poset, therank or dimensionof its elements is well defined. Define also∆Co ⊂ P, theset offacetsof Co in P, as the set of subcellsK(Co) of co-dimension 1. The elements of dimension0 in P areassociated with the singletons in[n]. Moreover,P is formally extended to contain a unique minimal element ofdimension−1, associated with the empty cell∅.

For a closed cellC = K(Co), its 0-dim subcells are denoted byV (C), and are referred to as itsvertex set.A d-chain is a formal sum of weighted opend-cells with coefficients being non-zero elements inF.

The axioms satisfied byP are as follows:

A1: The restriction ofP to anyK(Co) is a lattice. (I.e., every two elements in it have a unique minimalupper bound, and a unique maximal lower bound).

A2: For every dimensiond ≥ 0, there is a boundary operator∂d mapping every opend-cell Co to a(d−1)-chain supported on∆Co. In particular, ∂0i = ∅. It is required that∂d+1∂d = 0. The operator∂d is linearlyextended to a mapping fromd-chains of cells to(d − 1)-chains. Ad-cycle is ad-chainZ for which∂d(Z) = 0.Z is asimpled-cycle if its support does not properly contain the support of any other cycle.

A3: For every opend-cell Co, its boundary∂Co is a simple cycle. Equivalently, up to a multiplicativeconstant,∂Co is the only(d− 1)-cycle in the closed cellC.

12

As before,d-chains inIm(∂d+1) are calledd-boundaries, and byA2 they are cycles.

Definition 5.1 Call a setT ⊆ P of (open or closed) cellscompatibleif there exists a closed cell inP containingthem all. For suchT , define the cell complexK(T ) ⊆ P as the union of closures of cells inT . Observe that byaxiomA1, for any two cells inK(T ), there exists a unique maximal element contained in both.

The whole purpose and requirement of Definition 5.1 is to ensure the property stated in its last sentence.Finally, (reduced) homology groups are defined by the boundary operator∂ just as in simplicial complexes.

The facet graphGd(K) of d-dimensional complexK has a vertex for eachd-cell inK, and has an edge betweentwo vertices if the correspondingd-cells have a common(d− 1)-cell.

Claim 5.1 For d ≥ 1, any nontriviald-cycleZd with compatible support contains at leastd+ 2 d-cells.

Proof. The proof is by induction ond. Ford = 0, one needs at least two singletons for the sum of coefficientsof ∅ to cancel out. Assume correctness for(d − 1). Let Co be any open cell in the support ofZd. Since∂Co isa (d− 1)-cycle, by inductive assumptionK(Co) hasr ≥ d+ 1 facetsΥ1,Υ2, . . .Υr of dimensiond− 1. Since∂Zd = 0, for everyΥi there exists at least one additionald-cell Co

i ∈ Z besidesCo that containsΥi. ObservethatCo

i may not contain any otherΥj, since otherwiseCo andCoi would have more than one common maximal

subcell, contrary toA1. Thus, allCoi are distinct, and the supportZd contains at leastd + 2 d-cells: Co and

Coi ri=1.

Theorem 5.4 The facet graphGd(Zd) of a simpled-cycleZd with compatible support,d ≥ 1, is (d + 1)-connected (in the robust sense of Remark 4.1).4

Proof. To facilitate the discussion, let us formulate, for everyd ≥ 1, the following two statements, generaliz-ing (slightly weakened versions of) Claim 3.1 and Lemma 3.1,respectively:

(I)d : LetCd be a closedd-cell, andT be a set of at mostd− 1 open subcells ofCd of dimension< d. LetK(T ) ⊂ Cd be the cell complex that corresponds toT , then, Hd−1(Cd,K(T )) = 0. 5

(II)d : LetD be a compatible set of at mostd cells of dimension≤ d, andK(D) the corresponding cellcomplex. Then,Hd−1(K(D)) = 0.

The argument used in deriving Theorem 4.1 from Lemma 3.1, carries over to the present settingas is. Basedon axiomsA1 & A2, it shows that(II)d implies our Theorem.

Statement(II)d will be proven by induction ond, with the base cased = 1, and two-parts induction step(II)d−1 =⇒ (I)d, and(I)d =⇒ (II)d.

The base cased = 1: Axiom A3 immediately implies(II)1.

(II)d−1 =⇒ (I)d: Let Cd, T andK(T ) be as in the premise of(I)d. We need to show that any relative(d−1)-cycle in(Cd,K(T )) is a relative boundary. Consider such a relative(d−1)-cycleXd−1, i.e.,∂d−1Xd−1 ⊆∆(K(T )). If ∂d−1Xd−1 = 0 there is nothing to prove. Assume then thatXd−1 is not a cycle, but rather just arelative cycle with respect toK(T ). By induction hypothesis,(II)d−1 holds, implying thatHd−2(K(T )) = 0.Therefore, there exists a(d− 1)-chainYd−1 supported onK(T ), such that∂Xd−1 = ∂Yd−1 (hereA2 is used toconclude that∂Xd−1 is a (d − 2)-cycle inK(T )). Then,Xd−1 − Yd−1 is a (d − 1)-cycle inCd. However, byA3, the only(d− 1)-cycles inCd are of the formc ∂Co

d , for somec ∈ F, hence,Xd−1 = Yd−1+ c ∂Cod . Keeping

in mind thatYd−1 is supported onK(T ), one concludes thatXd−1 is a relatived-boundary.

4 The assumption about a compatible support is essential. Consider, e.g., the following set of open2-cells (originating from facesof convex2-polytopes):C1 = (1, 2, 3, 4, 5) with boundary∂2C1 = (1, 2) + (2, 3) + (3, 4) + (4, 5) − (1, 5), C2 = (1, 2, 3, ) with∂2(1, 2, 3) = (1, 2) + (2, 3) − (1, 3), C3 = (1, 3, 4) with ∂2C3 = (1, 3) + (3, 4) − (1, 4), andC4 = (1, 4, 5) with ∂C4 = (1, 4) +(4, 5)− (1, 5). Clearly, this set of cells is not compatible. Respectively, the facet graphG2(Z2) of the2-cycleZ2 = C1−C2−C3−C4

is not3-connected.5Meaning that for every relative(d − 1)-cycleXd−1 overCd, i.e., a(d − 1)-chain such that∂Xd−1 = 0 outside ofK(T ), there

exists a(d− 1)-boundaryBd−1 overCd, such thatBd−1 = Xd−1 outside ofK(T ). In fact, more can be said: due to simplicity of∂Cd,the boundaryBd−1 shall always be of the formc · ∂Cd.

13

(I)d =⇒ (II)d: The argument is similar to the proof of Lemma 3.1. LetD be a set of cells as in thepremise of(II)d. The proof is by induction on the number ofd-cells inD. Let Zd−1 be a cycle onK(D). IfD contains nod-cells, thenK(D) contains at most|D| ≤ d different (d − 1)-cells. However, by Claim 5.1, a(d− 1)-cycle has support of size≥ d+ 1. Thus, in this case there are no nontrivial(d− 1)-cycles inK(D).

Hence, there exists ad-cell Cd = K(Cod) ∈ D, whereCo

d is an opend-cell. LetD′ = D \Cod. Let

T = Ψ0 ∧ Cod Ψo∈D′ , whereΨ0 ∧ Co

d denotes the maximal element inP dominated by bothΨ0 andCod, as in

A1. Observe thatT is a compatible set,|T | ≤ d− 1, and that the cells inT are of dimension≤ d− 1.Applying (I)d toCd, T , one concludes thatZd−1 is a relative boundary ofC0

d with respect toK(D′). Namely,there exists a boundaryBd−1 of Cd such thatZ ′

d−1 = Zd−1−Bd−1 is supported onK(D′). However,D′ has onelessd-cell thanD, and by inductionZ ′

d−1 = B′d−1 for some boundaryB′

d−1 inD′. Thus,Zd−1 = Bd−1+B′d−1,

which is a(d− 1)-boundary inK(D).

To conclude this paper, we would like to close the circle and return to where we have started, the Balinski’sTheorem. For this, we need one more variant of(II)d.

Definition 5.2 A cell complexK (in particular, a closed cell) is calledhomologicallyk-connectedif Hi(K) = 0for i = 0, 1, . . . , k.

Theorem 5.5 LetD be a compatible set of at mostd homologically(d− 1)-connected cells ofanydimension.Then, Hd−1(K(D)) = 0.

Proof. The proof is by induction ond, and it is almost identical to the inductive argument of the previoustheorem. Interestingly, the axiomA3 is not needed this time. We are concerned only with the modified (I∗)dand(II∗)d, where the modification consists of dropping any assumptions about the dimension of the cells inTandD, but preserving the conditions|T | ≤ d−1, and|D| ≤ d respectively. Also, in(I∗)d, the cellC is assumedto be of dimension≥ d.

The basis,(II∗)1, follows directly from the assumptions of the Theorem.The implication (II∗)d−1 =⇒ (I∗)d works just like in Theorem 5.4, with the following sole change: the

conclusion that the(d− 1)-cycleXd−1 − Yd−1 onC is a(d− 1)-boundary, is now derived from the assumptionof the Theorem thatHd−1(C) = 0.

The implication (I∗)d =⇒ (II∗)d still works, with induction on the number of cells inD of dimension atleastd.

We would like to show that Theorem 5.5 is in fact equivalent (via Alexander duality) to the following elegantgeneralization of Balinski’s Theorem due to [11], further strengthened by A. Bjorner in [7].

Theorem 5.6 ( Homological Mixed-Connectivity Theorem ) The boundary complexB = ∆P of a convex(d + 1)-polytope remains homologicallyr-connected, with non-emptyr-skeleton, after removal6 of any setF, |F | ≤ d− r of its openfaces, forr = 0, . . . , d− 1.

Clearly,B is homologicallyk-connected if and only if so is its(k + 1)-skeleton, the subcomplex ofB obtainedby retaining only the faces of dimension≤ k+ 1. Thus, the caser = 0 of the the Mixed-Connectivity Theoremis the Balinski’s Theorem.

Claim 5.2 For cells corresponding to convex polytopes,(II∗) ⇐⇒ The Homological Mixed-ConnectivityTheorem.

Proof. Formally, the Homological Mixed Connectivity Theorem is about vanishing of the lower homologygroups of the cell complexB\U(F ), whereU(F ) (not a complex!) is theupperclosure ofF in B with respect

6 Removing open faces means removing the faces themselves andtheir super-faces, but not their subfaces.

14

to containment. LetP ∗ be the dual polytope ofP , and, respectively, letF ∗ be the set faces dual toF in P ∗.Then, by Combinatorial Alexander duality for polytopes (see e.g., [8]7, in particular the discussion towards theend of the Introduction section, and the reference therein),

Hr(B \ U(F )) ∼= Hd−r−1(K(F ∗)) .

Since the cohomology groups are isomorphic to the homology groups, Hd−r−1(K(F ∗)) ∼= Hd−r−1(K(F ∗)).Therefore, the cell complexB \U(F ) has a vanishingr’th homology group if and only ifHd−r−1(K(F ∗)) = 0.As the cells inF ∗ satisfy the assumptions of Theorem 5.5 for anyd, and|F ∗| = |F | ≤ d − r, this is preciselythe statement of Theorem 5.5. Thus, Theorem 5.5 implies Theorem 5.6.

Observing that the argument is completely reversible, provided that the setF is a set of faces of somepolytopeP , and thatF indeed satisfies this condition due to the compatibility assumption of of Theorem 5.5,the reverse implication follows as well.

The fact that ther-skeleton ofB \U(F ) is not empty, can be shown by induction. Clearly, the most ”de-structive” setF is the set ofd − r 0-cells, i.e., points. Consider suchF , and remove its points fromB one byone. The link of the first pointp1 is a nonempty(d − 1)-cycleZd−1 in the cell complexB, and it survives theremoval ofp1. Similarly, the removal of the second point inF either missesZd, or reduces it to a nonempty(d−2)-cycle inB, etc. After the removal of the entireF fromB, a nonempty(d− r)-cycleZd−r survives.

Acknowledgments: We are grateful to Roy Meshulam and Eran Nevo for enlightening discussions.

References

[1] K. A. Adiprasito, A. Goodarzi, M. Varbaro,Connectivity of Pseudomanifold Graphs From an AlgebraicPoint of View, arXiv:1409.6657 [math.CO]

[2] Ch. A. Athanasiadis,On the Graph Connectivity of Skeleta of Convex Polytopes, Discrete& ComputationalGeometry Vol. 42 No. 2 (2009), pp. 155-165

[3] M. L. Balinski, On the Graph Structure of Convex Polyhedra in n-Space, Pacific Journal of Mathematics11 (2) (1961), pp. 431-434,

[4] David Barnette,Graph Theorems for Manifolds, Israel Journal of Mathematics Vol. 16, Issue 1 (1973), pp.62-72

[5] David Barnette,Decompositions of Homology Manifolds and Their Graphs, Israel Journal of MathematicsVol. 41, Issue 3 (1982), pp. 203-212

[6] D. Bauer, H. Broersma, E.Schmeichel,Toughness in Graphs - A Survey, Graphs and Combinatorics Vol.22 (2006), pp. 1-35

[7] A. Bjorner,Mixed Connectivity, Polytope Boundaries, and Matroid Basis Graphs, a talk at Billerafest 2008,http://emis.matem.unam.mx/journals/SLC/wpapers/s61vortrag/bjoerner3.pdf

[8] A. Bjorner, M. Tancer,Combinatorial Alexander Duality - a Short and Elementary Proof, Discrete andComputational Geometry 42(4) (2009), pp. 586-593

7 The point of [8] is to provide a self-contained exposition based on the first principles. Alternatively, the Combinatorial Alexanderduality for polytopes can be derived from the topological Alexander duality, see, e.g., [19], after providing a geometric argument that|B \ U(F )|, the geometric realization ofB \ U(F ), is homotopy equivalent to|B∗\K(F ∗)|.

15

[9] B. Bollobas, I. Leader,Isoperimetric Problems forr-sets, Combinatorics, Probability and Computing Vol.13/2 (2004), pp. 277-279

[10] Branko Gruenbaum,Convex Polytopes, 2nd ed., Springer (2003)

[11] G. Floystad,Cohen-Macaulay cell complexes, Algebraic and geometric combinatorics, Contemp. Math.423, Amer. Math. Soc., Providence, RI, 2006, p.205-220.

[12] A. Fogelsanger,The Generic Rigidity of Minimal Cycles, Ph.D. Thesis, Cornell University, 1988.

[13] G. Kalai, Enumeration of Q-acyclic Simplicial Complexes, Israel Journal of Mathematics 45 (1983),337351.

[14] Gil Kalai, Polytope Skeletons And Paths, Handbook of Discrete and Computational Geometry (2nd Edi-tion), Chapter 20, CRC Press, 2004.

[15] P. Keevash,The Existence of Designs, arXiv:1401.3665 [math.CO]

[16] V. Klee, A property of polyhedral graphs, J. Math. Mech. 13 (1964), pp. 1039-1042.

[17] N. Linial, I. Newman, Y. Peled, Y. RabinovichExtremal Problems on Shadows and Hypercuts in SimplicialComplexes, arXiv:1408.0602 [math.CO]

[18] R. Meshulam,Personal Communication

[19] J. R. Munkres,Elements of Algebraic Topology, Addison-Wesley, 1984.

[20] I. Newman and Y. Rabinovich,On Multiplicativeλ-Approximations and Some Geometric Applications,SIAM J. Comput. 42(3), 855-883 (2013).

[21] J. G. Oxley,Matroid Theory, Oxford University Press, 2006.

[22] G. Wegner,d-Collapsing and Nerves of Families of Convex SetsArchives of Mathematics 26 (1975),pp. 317321

[23] G. M. Ziegler,Lecture on Polytopes, Springer-Verlag, 1995.

Appendix A: Proof of Claim 3.3

The statement is that(∂k−1C)∗ = δr−1 C∗ .

Proof. By linearity of all the involved operators, it suffices to verify the claim for (k − 1)-simplices. Thebasic identity behind the Claim is:

sign(σ, σ\p) = sign(σ ∪ p, σ) · (−1)p−1 , (2)

wherep ∈ σ, and bothσ\p andσ ∪ p denote, with some abuse of notation, signed simplices ordered in theincreasing order.

To verify (2), assume thatp is thei’th element inσ. Then, by definition,sign(σ, σ\p) = (−1)i−1. On theother hand, the order ofp in σ ∪ p must bep− i+ 1, thussign(σ ∪ p, σ) = (−1)p−i, and (2) follows.

The next identity is an immediate consequence of (2):

s(σ) · sign(σ, σ\p) · s(σ\p) = sign(σ ∪ p, σ) . (3)

16

We can now establish the Claim.

(∂σ)∗ =

(∑

p∈σsign(σ, σ\p) · (σ\p)

)∗

=∑

p 6∈σsign(σ, σ\p) · [s(σ\p) · (σ ∪ p)] =

= s(σ)·∑

p 6∈σ[s(σ)·sign(σ, σ\p)·s(σ\p)]·(σ∪p) = s(σ)·

∑

p 6∈σsign((σ∪p), σ)·(σ∪p) = s(σ)·δ(σ) = δ(σ∗) ,

where the fourth equality follows from (3).

Appendix B: Connectivity of Gd(Kdn)

The facet graph ofGd(Kdn), to be denoted byG(n, d + 1), is the graph whose vertices correspond to(d + 1)-

subsets of[n], and a a pair of vertices forms an edge if the symmetric difference between the corresponding setsis of size2.

Theorem 5.7 For all pairs n, d, wheren > d + 1, andd > 0, the graphG(n, d + 1) is (d + 1)(n − d − 1)connected.

Proof. The proof is by induction on the pairs(n, d). Menger’s Theorem will be used throughout.For the base cased = 1, G(n, d + 1) is isomorphic to the line graph ofKn, which is easily verified to be

2(n− 2) connected. Observe also that the statement is correct forn ≤ d+3: Forn = d+2, Gd(d+2, d+1) isjust ad+2 clique. The casen = d+3 reduces to the cased = 1, since two(d+1)-sets in[d+3] are adjacent ifand only if their complements, i.e., sets of size2, are adjacent. Hence we assume in what follows thatn ≥ d+4.

Separating the vertex setV of G(n, d + 1) to V0, corresponding to(d + 1)-sets not containingn, andV1,the rest, we observe thatG0 = G(n, d+ 1)|V0 is isomorphic toG(n − 1, d+ 1), whileG1 = G(n, d+ 1)|V1 isisomorphic toG(n − 1, d).

LetX be a subset of vertices ofV , with |X| < (d+1)(n−d−1). We show thatG(n, d+1)\X is connected.

Case 1: |X ∩ V0| < (d+ 1)(n − d− 2).By induction assumption, in this caseG0\X is connected. Thus, eitherV1\X = ∅, in which case we are done,or V1\X 6= ∅. In the latter case, it suffices to show that for everyσ ∈ V1\X, there exists a path inG \X fromσ to a member ofV0\X.

For any subsetS ⊂ V1 let Ni(S), i = 0, 1 contain all neighbours of ofS in Vi respectively. Note that foranyτ ∈ V1, |N0(τ)| = (n− d− 1) and|N1(τ)| = d(n− d− 1).

Considerσ ∈ V1 \X, and assume w.l.o.g., thatσ\n = [d]. WriteN1(σ) = ∪dj=1N

j1 (σ), whereN j

1 (σ) =τ ∈ N1(σ)| j /∈ τ. Write alsoA0(τ) = N0(τ) \N0(σ). The following is a a simple observation that we willuse.

Claim 5.3 Let τ ∈ N i1(σ), τ

′ ∈ N j1 (σ). Then,|A0(τ)| = n− d− 2.

Furthermore, ifi = j, then|A0(τ) ∩A0(τ′)| = 1, and if i 6= j, then|A0(τ) ∩A0(τ

′)| = 0.

Let ri = |N i1(σ)\X|, for i = 1, . . . d.

If N0(σ) or any ofN0(Ni1(σ) \X) contains a member inV0\X, we are done. Otherwise,

|X ∩ V1| ≥ | ∪di=1 N

i1(σ) ∩X| = d(n − d− 1) −

d∑

1

ri .

17

On the other hand, in view of the Claim above,

|X ∩ V0| ≥ |N0(σ) ∪ ∪di=1 N0(N

i1(σ) \X) ∩X | = |N0(σ)|+

d∑

i=1

| ∪τ∈ N i1(σ)\X A0(τ) | ≥

(n− d− 1) +d∑

1

ri(n− d− 2) −d∑

1

(ri2

).

Combining the two estimations, one gets

|X| ≥ (d+ 1)(n − d− 1) +∑

i

(ri(n− d− 3)−

(ri2

)). (4)

Since forn ≥ d + 4, it holds that (n − d − 3) ≥ n−d−22 ≥ ri−1

2 , the last term in (4) is nonnegative, and thus|X| ≥ (d+ 1)(n − d− 1), contradicting the assumption of Case 1.

Case 2: |X ∩ V0| ≥ (d+ 1)(n − d− 2), or, equivalently,|X ∩ V1| < d+ 1 .Since d+ 1 < d(n− d− 1), the graphG1\X is connected by the induction hypothesis. Thus, to establish theconnectivity ofG(n, d+ 1)\X it suffices to show that everyσ ∈ V0\X has a neighbour inV1\X. Sinceσ hasd+ 1 neighbours inV1, and|X ∩ V1| < d+ 1 by the assumption of Case 2, the implication follows.

This completes the proof of the statement.

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