FACE VECTORS OF FLAG COMPLEXES

Transcription

1 FACE VECTORS OF FLAG COMPLEXES ANDY FROHMADER Abstact. A conjectue of Kalai and Eckhoff that the face vecto of an abitay flag complex is also the face vecto of some paticula balanced complex is veified. 1. Intoduction We begin by intoducing the main esult. Pecise definitions and statements of some elated theoems ae defeed to late sections. The main object of ou study is the class of flag complexes. A simplicial complex is a flag complex if all of its minimal non-faces ae two element sets. Equivalently, if all of the edges of a potential face of a flag complex ae in the complex, then that face must also be in the complex. Flag complexes ae closely elated to gaphs. Given a gaph G, define its clique complex C = C(G) as the simplicial complex whose vetex set is the vetex set of G, and whose faces ae the cliques of G. The clique complex of any gaph is itself a flag complex, as fo a subset of vetices of a gaph to not fom a clique, two of them must not fom an edge. Convesely, any flag complex is the clique complex of its 1-skeleton. The Kuskal-Katona theoem [6, 5] chaacteizes the face vectos of simplicial complexes as being pecisely the intege vectos whose coodinates satisfy some paticula bounds. The gaphs of the ev-lex complexes which attain these bounds invaiably have a clique on all but one of the vetices of the complex, and sometimes even on all of the vetices. Since the bounds of the Kuskal-Katona theoem hold fo all simplicial complexes, they must in paticula hold fo flag complexes. We might expect that flag complexes which do not have a face on most of the vetices of the complex will not come that close to attaining the bounds of the Kuskal-Katona theoem. One way to foce tighte bounds on face numbes is by equiing the gaph of the complex to have a chomatic numbe much smalle than the numbe of vetices. The face vectos of simplicial complexes of a given chomatic numbe wee classified by Fankl, Füedi, and Kalai [4]. Kalai (unpublished; see [8, p. 100]) and Eckhoff [1] independently conjectued that if the lagest face of a flag complex contains vetices, then it must satisfy the known bounds (see [4]) fo complexes of chomatic numbe, even though the flag complex may have chomatic numbe much lage than. We pove thei conjectue. Theoem 1.1. Fo any flag complex C, thee is a balanced complex C with the same face vecto as C. 1

2 2 ANDY FROHMADER Ou poof is constuctive. The Fankl-Füedi-Kalai [4] theoem states that an intege vecto is the face vecto of a balanced complex if and only if it is the face vecto of a coloed ev-lex complex. This happens if and only if it satisfies cetain bounds on consecutive face numbes. Given a flag complex, fo each i, we constuct a coloed ev-lex complex with the same numbe of i-faces and (i + 1)-faces as the flag complex, thus showing that all the bounds ae satisfied. The stuctue of the pape is as follows. Section 2 contains basic facts and definitions elated to simplicial complexes. In Section 3, we discuss the Kuskal- Katona theoem and the Fankl-Füedi-Kalai theoem, and lay the foundation fo ou poof. Finally, Section 4 gives ou poof of the Kalai-Eckhoff conjectue. 2. Peliminaies on simplicial complexes In this section, we discuss some basic definitions elated to simplicial complexes. Recall that a simplicial complex on a vetex set V is a collection of subsets of V such that, (i) fo evey v V, {v} and (ii) fo evey B, if A B, then A. The elements of ae called faces. The maximal faces (unde inclusion) ae called facets. Fo a face F of a simpicial complex, the dimension of F is defined as dim F = F 1. The dimension of, dim, is defined as the maximum dimension of the faces of. A complex is pue if all of its facets ae of the same dimension. The i-skeleton of a simplicial complex is the collection of all faces of of dimension i. In paticula, the 1-skeleton of is its undelying gaph. It is sometimes useful in inductive poofs to conside cetain subcomplexes of a given simplicial complex, such as its links. Definition 2.1. Let be a simplicial complex and F. The link of F, lk (F ), is defined as lk (F ) := {G F G =, F G }. The link of a face of a simplicial complex is itself a simplicial complex. It will be convenient to define the notion of a link of a vetex of a gaph. Definition 2.2. The link of a vetex v in a gaph G, denoted lk G (v), is the induced subgaph of G on all vetices adjacent to v. Note that lk G (v) coincides with the 1-skeleton of the link of {v} in the clique complex of G. Next we discuss a special class of simplical complexes known as flag complexes. Definition 2.3. A simplicial complex on a vetex set V is a flag complex if all of its minimal non-faces ae two element sets. A non-face of is a subset A V such that A. A non-face A is minimal if, fo all pope subsets B A, B. In the following, we efe to the chomatic numbe of a simplicial complex as the chomatic numbe of its 1-skeleton in the usual gaph theoetic sense. We also need the notion of a balanced complex, as intoduced and studied in [7]. Definition 2.4. A simplicial complex of dimension d 1 is balanced if it has chomatic numbe d. Note that the chomatic numbe of a simpicial complex of dimension d 1 must be at least d, as it has some face with d vetices, all of which ae adjacent, so

3 FACE VECTORS OF FLAG COMPLEXES 3 coloing that face takes d colos. A balanced complex is then one whose chomatic numbe is no lage than it has to be. Not all simplicial complexes ae balanced complexes. Fo example, a pentagon (five vetices, five edges, and one empty face) is not a balanced complex, because it has chomatic numbe thee but dimension only one. In this pape, we study the face numbes of flag complexes. Definition 2.5. The i-th face numbe of a simplicial complex C, denoted c i (C) is the numbe of faces in C containing i vetices. These ae also called i-faces of C. If dim C = d 1, the face vecto of C is the vecto c(c) = (c 0 (C), c 1 (C),..., c d (C)). In paticula, fo any non-empty complex C, we have c 0 (C) = 1, as thee is a unique empty set of vetices, and it is a face of C. Since flag complexes ae the same as clique complexes of gaphs, it is sometimes convenient to talk about face numbes in the language of gaphs. Definition 2.6. The i-th face numbe of a gaph is the i-th face numbe of its clique complex. Likewise, the clique vecto of a gaph is the face vecto of its clique complex. The face numbes defined hee ae shifted by one fom what is often used fo simplicial complexes. This is done because we ae pimaily concened with flag complexes, o equivalently, clique complexes of gaphs, whee it is moe natual to index i as the numbe of vetices in a clique of the gaph, following Eckhoff [3]. The gaph concept coesponding to the dimension of a simplicial complex is the clique numbe. Definition 2.7. The clique numbe of a gaph is the numbe of vetices in its lagest clique. Note that the clique numbe of a gaph is one lage than the dimension of its clique complex. 3. The Kuskal-Katona and Fankl-Füedi-Kalai theoems Fo the geneal case of simplicial complexes, the question of which face vectos ae possible is answeed by the Kuskal-Katona theoem [6, 5]. Stating the theoem equies the following lemma. Lemma 3.1. Given any positive integes m and k, thee is a unique s and unique n k > n k 1 > > n k s k s > 0 such that ( ) ( ) ( ) m = k k 1 k s The epesentation descibed in the lemma is called the k-canonical epesentation of m. Theoem 3.2 (Kuskal-Katona). Fo a simplicial complex C, let ( ) ( ) ( ) m = c k (C) = k k 1 k s

4 4 ANDY FROHMADER be the k-canonical epesentation of m. Then ( ) ( c k+1 (C) + k + 1 k ) + + ( ). k s + 1 Futhemoe, given a vecto (1, c 1, c 2,..., c t ) which satisfies this bound fo all 1 k < t, thee is some complex that has this vecto as its face vecto. To constuct the complexes which demonstate that the bound of the Kuskal- Katona theoem is attained, we need the evese-lexicogaphic ( ev-lex ) ode. To define the ev-lex ode of i-faces of a simplicial complex on n vetices, we stat by labelling the vetices 1, 2,.... Let N be the natual numbes, let A and B be distinct subsets of N with A = B = i, and let A B be the symmetic diffeence of A and B. Definition 3.3. Fo A, B N with A = B, we say that A pecedes B in the ev-lex ode if max(a B) B, and B pecedes A othewise. Fo example, {2, 3, 5} pecedes {1, 4, 5}, as 3 is less than 4, and {3, 4, 5} pecedes {1, 2, 6}. Definition 3.4. The ev-lex complex on m i-faces is the pue complex whose facets ae the fist m i-sets possible in ev-lex ode. This complex is denoted C i (m). We can also specify moe than one numbe in the face vecto. Fo two sequences i 1 < < i and (m 1,..., m ), let C = C i1 (m 1 ) C i2 (m 2 ) C i (m ). A standad way to pove the Kuskal-Katona theoem involves showing that if the numbes m 1,..., m satisfy the bounds of the theoem, then the complex C has exactly m j i j -faces fo all j and no moe. In this case, we efe to C as the ev-lex complex on m 1 i 1 -faces,..., m i -faces. Fo example, if the complex C has ( ( 9 3) + 6 2) = 99 3-faces, then the Kuskal- Katona theoem says that it can have at most ( ( 9 4) + 6 3) = faces. The ev-lex complex on 99 3-faces and faces gives an example showing that this bound is attained. The 1-skeleton of the ev-lex complex that gives the example fo the existence pat of the Kuskal-Katona theoem always has as lage of a clique as is possible without exceeding the numbe of edges allowed. It also has a chomatic numbe of eithe the numbe of non-isolated vetices o one less than this, as the edges fom a clique on all of the non-isolated vetices except possibly fo the last one. It tuns out that if we equie a much smalle chomatic numbe, we can get a much smalle bound. To take an exteme example, if c 3 (C) = 1140, then the Kuskal- Katona theoem equies that c 4 (C) But if we equie the complex C to be 3-coloable, then we tivially cannot have any faces on fou vetices, and c 4 (C) = 0. We could ask what face vectos occu fo -coloable complexes fo a given. This was solved by Fankl, Füedi, and Kalai [4]. In ode to explain thei esult, we need the concept of a Tuán gaph. Definition 3.5. The Tuán gaph T n, is the gaph obtained by patitioning n vetices into pats as evenly as possible, and making two vetices adjacent exactly if they ae not in the same pat. Define ( ) n k to be the numbe of k-cliques of the gaph T n,.

5 FACE VECTORS OF FLAG COMPLEXES 5 The stuctue of the Fankl-Füedi-Kalai theoem [4] is simila to that of the Kuskal-Katona theoem, beginning with a canonical epesentation of the numbe of faces. Lemma 3.6. Given positive integes m, k, and with k, thee ae unique s, n k, n k 1,..., n k s such that ( ) ( ) ( ) m = + + +, k k 1 1 k s s n k i n k i i > nk i 1 fo all 0 i < s, and n k s k s > 0. This expession is called the (k, )-canonical epesentation of m. Theoem 3.7 (Fankl-Füedi-Kalai). Fo an -coloable complex C, let ( ) ( ) ( ) m = c k (C) = k k 1 k s 1 be the (k, )-canonical epesentation of m. Then ( ) ( ) c k+1 (C) k + 1 k 1 s ( ). k s + 1 s Futhemoe, given a vecto (1, c 1, c 2,... c t ) which satisfies this bound fo all 1 k < t, thee is some -coloable complex that has this vecto as its face vecto. The examples which show that this bound is shap come fom a coloed equivalent of the ev-lex complexes of the Kuskal-Katona theoem: Definition 3.8. A subset A N is -pemissible if, fo any two a, b A, does not divide a b. The -coloed ev-lex complex on m i-faces is the pue complex whose facets ae the fist m -pemissible i-sets in ev-lex ode. This complex is denoted C i (m). The complex Ci (n) is -coloable because we can colo all vetices which ae i modulo with colo i. As with the uncoloed case, we can define a ev-lex complex with specified face numbes of moe than one dimension. Fo two sequences i 1 < < i s and (m 1,..., m s ), let C = Ci 1 (m 1 ) Ci 2 (m 2 ) Ci s (m s ). The poof of Theoem 3.7 involves showing that if the numbes m 1,..., m satisfy the bounds of the theoem, then the complex C has exactly m j i j -faces and no moe. In this case, we efe to C as the -coloed ev-lex complex on m 1 i 1 -faces,..., m i -faces. This complex is likewise -coloable with one colo fo each value modulo. In the case of flag complexes, the face numbes of the complex must still follow the bounds imposed by the chomatic numbe by Theoem 3.7. Still, thee ae gaphs whose clique numbe is fa smalle than the chomatic numbe, and having no lage cliques seems to foce tighte estictions on the clique vecto than the chomatic numbe alone. In paticula, given a gaph G of clique numbe n, we must have c i (G) = 0 fo all i > n, while the bound fom the chomatic numbe and Theoem 3.7 may be athe lage. Note that the chomatic numbe must be at least the size of the lagest clique, as any two vetices in a maximum size clique must have diffeent colos. It has been conjectued by Kalai (unpublished) and Eckhoff [1] that, given a gaph G with clique numbe, thee is an -coloable complex with exactly the

6 6 ANDY FROHMADER same face numbes as the clique complex of the gaph. Thei conjectue genealizes the classical Tuán theoem fom gaph theoy, which states that among all tianglefee gaphs on n vetices, the Tuán gaph T n,2 has the most edges [9]. The goal of the following section is to veify Theoem 1.1, poving thei conjectue. The evese inclusion does not hold. Fo example, the vecto (1, 4, 5, 1) in the case of = 3 satisfies the bounds of the Fankl-Füedi-Kalai theoem, so it is the face vecto of a balanced complex. Howeve, it is not the face vecto of a flag complex: a gaph with fou vetices and five edges must be one edge shot of a clique on fou vetices, which has two tiangles, not one. 4. Poof of the Kalai-Eckhoff conjectue Fix a gaph G with c +1 (G) = 0 and fix k 0. We stat by showing that thee is an -coloable complex C with c k (G) = c k (C) and c k+1 (G) = c k+1 (C) (see Lemma 4.1 below). The case k = 1 of the lemma is given by Tuán s theoem [9]. It was genealized by Zykov [10] to state that if G is a gaph on n vetices of chomatic numbe, then c i (G) ( ) n i. The case k = 2 was poven by Eckhoff [2]. A subsequent pape of Eckhoff [3] established a bound on c i (G) in tems of c 2 (G) fo all 2 i. All of these esults ae special cases of ou Theoem 4.2 and poven independently below. Lemma 4.1. If G is a gaph with c +1 (G) = 0 and k is a nonnegative intege, then thee is some -coloable complex C with c k (C) = c k (G) and c k+1 (C) = c k+1 (G). Poof. We use induction on k. Fo the base case, if k = 0, take C to be a complex with the same numbe of vetices as G and no edges, so that all vetices can be the same colo. Othewise, assume that the lemma holds fo k 1, and we need to pove it fo k. The appoach fo this is to use induction on c k+1 (G). Fo the base case, if c k+1 (G) = 0, then take C to be a disjoint union of c k (G) k-faces. Fo the inductive step, suppose that c k+1 (G) > 0. Let v 0 be the vetex of G contained in the most cliques of k + 1 vetices; in case of a tie, abitaily pick some vetex tied fo the most to label v 0. Let the vetices of G not adjacent to v 0 be v 1, v 2,... v s. Given a gaph G and a vetex v, thee is a bijection between k-cliques of lk G (v) and (k + 1)-cliques of G containing v, whee a k-clique of lk G (v) coesponds to the (k + 1)-clique of G containing the k vetices of the k-clique of lk G (v) togethe with v. Then the numbe of (k +1)-cliques of G containing v is c k (lk G (v)). In paticula, the choice of v 0 gives c k (lk G (v 0 )) c k (lk G (v )) fo evey vetex v G. Define gaphs G 0, G 1,..., G s+1 by setting G i+1 = G {v 0, v 1,... v i } fo 0 i s and G 0 = G. Clealy, G = G 0 G 1 G s+1. Futhe, G s+1 is the induced subgaph on the vetices adjacent to v 0, which is lk G (v 0 ). Since c +1 (G) = 0, c (lk G (v 0 )) = 0, fo othewise, the vetices of an -clique of lk G (v 0 ) togethe with v 0 would fom an ( + 1)-clique of G. Then c (G s+1 ) = 0. Futhe, since c k+1 (G) > 0, and v 0 is contained in the most (k + 1)-cliques of any vetex of G, v 0 is contained in at least one (k + 1)-clique, and so c k (lk G (v 0 )) > 0. Since v is contained in at least one (k + 1)-clique of G, we have c k+1 (G s+1 ) < c k+1 (G). Then by the second inductive hypothesis, thee is some ( 1)-coloable complex C s+1 such that c k (C s+1 ) = c k (G s+1 ) and c k+1 (C s+1 ) = c k+1 (G s+1 ). Since given

7 FACE VECTORS OF FLAG COMPLEXES 7 any ( 1)-coloable complex, thee is an ( 1)-coloable ev-lex complex with the same face numbes, we can take C s+1 to be a ev-lex complex. Futhe, since c k+1 (C s+1 ) and c k (C s+1 ) only foce a lowe bound on c k 1 (C s+1 ), but not an uppe bound, we can take c k 1 (C s+1 ) c k 1 (G). Let c k (lk Gi (v i )) = a i and c k 1 (lk Gi (v i )) = b i. Since G i+1 = G i v i, c k+1 (G i ) c k+1 (G i+1 ) = a i and c k (G i ) c k (G i+1 ) = b i. We have c k (lk G (v 0 )) c k (lk G (v i )) by the choice of v 0. We also have c k (lk G (v i )) c k (lk Gi (v i )) since G i G. Thus c k (C s+1 ) = c k (G s+1 ) = c k (lk G (v 0 )) c k (lk G (v i )) c k (lk Gi (v i )) = a i. Also, since lk Gi (v i ) G i G, we have b i = c k 1 (lk Gi (v i )) c k 1 (G i ) c k 1 (G) c k 1 (C s+1 ). Given an -coloed complex C i+1 such that c k+1 (C i+1 ) = c k+1 (G i+1 ), c k (C i+1 ) = c k (G i+1 ), and the induced subcomplex of C i+1 on the vetices of the fist 1 colos is isomophic to C s+1, we want to constuct a complex C i such that c k+1 (C i ) = c k+1 (G i ), c k (C i ) = c k (G i ), and the induced subcomplex of C i on the vetices of the fist 1 colos is isomophic to C s+1. Constuct C i fom C i+1 by adding a new vetex v i of colo. Let the (k+1)-faces containing v i consist of each of the fist a i k-faces in ev-lex ode of C s+1 togethe with v i, and let the k-faces containing v i consist of each of the fist b i (k 1)-faces in ev-lex ode of C s+1 togethe with v i. If this constuction can be done, then c k+1 (C i ) is the numbe of (k + 1)-faces of C i containing v i plus the numbe of (k + 1)-faces of C i not containing v i, which ae a i and c k+1 (C i+1 ), espectively. Then Likewise, we have c k+1 (C i ) = c k+1 (C i+1 ) + a i = c k+1 (G i+1 ) + a i = c k+1 (G i ). c k (C i ) = c k (C i+1 ) + b i = c k (G i+1 ) + b i = c k (G i ). Futhe, it is clea fom the constuction that the induced subcomplex on vetices of the fist 1 colos is unchanged fom C i+1, and hence is isomophic to C s+1. In ode to show that the constuction is possible, we must show that c k (C s+1 ) a i and c k 1 (C s+1 ) b i, and that it is possible fo an ( 1)-coloed complex C to have exactly c k (C) = a i and c k 1 (C) = b i. We have aleady shown the fist two of these. Fo the thid, since G i G, we have c +1 (G i ) c +1 (G) = 0, and so c +1 (G i ) = 0. Then c (lk Gi (v i )) = 0. We also have c k (lk Gi (v i )) = a i and c k 1 (lk Gi (v i )) = b i by the definitions of a i and b i. Then by the fist inductive hypothesis, thee is some ( 1)-coloed complex C i such that c k(c i ) = a i and c k 1 (C i ) = b i. Then we can take C i to be the ( 1)-coloed ev-lex complex with c k(c i ) = a i and c k 1 (C i ) = b i. Since C s+1 is an ( 1)-coloed ev-lex complex with c k (C s+1 ) a i and c k 1 (C s+1 ) b i, C i C s+1, and we can choose the link of v i in C i to be C i. We can epeat this constuction fo each 0 i s to stat with C s+1, then constuct C s, then C s 1, and so foth, until we have an -coloed complex C 0 such that c k (C 0 ) = c k (G) and c k+1 (C 0 ) = c k+1 (G). This completes the inductive step fo the induction on c k+1 (G), which in tun completes the inductive step fo the induction on k. We ae now eady to pove the esult which immediately implies Theoem 1.1, and hence establish the Kalai-Eckhoff conjectue, by taking to be the clique numbe of G.

8 8 ANDY FROHMADER Theoem 4.2. Fo evey gaph G with c +1 (G) = 0, thee is an -coloable complex C such that c i (C) = c i (G) fo all i. Poof. By Lemma 4.1, we can pick an -coloed complex C i such that c i (C i ) = c i (G) and c i+1 (C i ) = c i+1 (G) fo all i 1. By Theoem 3.7, we can take C i to be the ev-lex complex on c i (G) i-faces and c i+1 (G) (i + 1)-faces, and then i=1 C i will have the desied face numbes. Acknowledgements. I would like to thank my thesis adviso Isabella Novik fo suggesting this poblem, and fo he many useful discussions on solving the poblem and witing an aticle. Refeences [1] J. Eckhoff, Intesection popeties of boxes. I. An uppe-bound theoem, Isael J. Math. 62 (1988) no [2] J. Eckhoff, The maximum numbe of tiangles in a K 4 -fee gaph, Discete Math. 194 (1999) no [3] J. Eckhoff, A new Tuán-type theoem fo cliques in gaphs, Discete Math. 282 (2004) [4] P. Fankl, Z. Füedi, and G. Kalai, Shadows of coloed complexes, Math. Scand. 63 (1988) [5] G. Katona, A theoem of finite sets, in: Theoy of Gaphs, Academic Pess, New Yok, 1968, pp [6] J.B. Kuskal, The numbe of simplices in a complex, in: Mathematical Optimization Techniques, Univesity fo Califonia Pess, Bekeley, Califonia, 1963, pp [7] R. Stanley, Balanced Cohen-Macaulay complexes, Tans. Ame. Math. Soc. 249 (1979) [8] R. Stanley, Combinatoics and Commutative Algeba, Second Edition, Bikhause Boston, Inc., Boston, Massachusetts, 1996, [9] P. Tuán, Eine Extemalaufgabe aus de Gaphentheoie Mat. Fiz. Lapok 48 (1941) [10] A.A. Zykov, On some popeties of linea complexes, Ame. Math. Soc. Tansl. (1952) no. 79. Depatment of Mathematics, Univesity of Washington, Seattle, WA addess: fohmade@math.washington.edu