FACE VECTORS of FLAG COMPLEXES 1. Introduction We
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FACE VECTORS OF FLAG COMPLEXES ANDY FROHMADER Abstract. A conjecture of Kalai and Eckhoff that the face vector of an arbi- trary flag complex is also the face vector of some particular balanced complex is verified. 1. Introduction We begin by introducing the main result. Precise definitions and statements of some related theorems are deferred to later sections. The main object of our study is the class of flag complexes. A simplicial complex is a flag complex if all of its minimal non-faces are two element sets. Equivalently, if all of the edges of a potential face of a flag complex are in the complex, then that face must also be in the complex. Flag complexes are closely related to graphs. Given a graph G, define its clique complex C = C(G) as the simplicial complex whose vertex set is the vertex set of G, and whose faces are the cliques of G. The clique complex of any graph is itself a flag complex, as for a subset of vertices of a graph to not form a clique, two of them must not form an edge. Conversely, any flag complex is the clique complex of its 1-skeleton. The Kruskal-Katona theorem [6, 5] characterizes the face vectors of simplicial complexes as being precisely the integer vectors whose coordinates satisfy some par- ticular bounds. The graphs of the \rev-lex" complexes which attain these bounds invariably have a clique on all but one of the vertices of the complex, and sometimes even on all of the vertices. Since the bounds of the Kruskal-Katona theorem hold for all simplicial com- plexes, they must in particular hold for flag complexes. We might expect that flag complexes which do not have a face on most of the vertices of the complex will not come that close to attaining the bounds of the Kruskal-Katona theorem. One way to force tighter bounds on face numbers is by requiring the graph of the complex to have a chromatic number much smaller than the number of vertices. The face vectors of simplicial complexes of a given chromatic number were classified by Frankl, F¨uredi,and Kalai [4]. Kalai (unpublished; see [8, p. 100]) and Eckhoff [1] independently conjectured that if the largest face of a flag complex contains r vertices, then it must satisfy the known bounds (see [4]) for complexes of chromatic number r, even though the flag complex may have chromatic number much larger than r. We prove their conjecture. Theorem 1.1. For any flag complex C, there is a balanced complex C0 with the same face vector as C. 1 2 ANDY FROHMADER Our proof is constructive. The Frankl-F¨uredi-Kalai[4] theorem states that an integer vector is the face vector of a balanced complex if and only if it is the face vector of a colored \rev-lex" complex. This happens if and only if it satisfies certain bounds on consecutive face numbers. Given a flag complex, for each i, we construct a colored \rev-lex" complex with the same number of i-faces and (i + 1)-faces as the flag complex, thus showing that all the bounds are satisfied. The structure of the paper is as follows. Section 2 contains basic facts and definitions related to simplicial complexes. In Section 3, we discuss the Kruskal- Katona theorem and the Frankl-F¨uredi-Kalaitheorem, and lay the foundation for our proof. Finally, Section 4 gives our proof of the Kalai-Eckhoff conjecture. 2. Preliminaries on simplicial complexes In this section, we discuss some basic definitions related to simplicial complexes. Recall that a simplicial complex ∆ on a vertex set V is a collection of subsets of V such that, (i) for every v 2 V , fvg 2 ∆ and (ii) for every B 2 ∆, if A ⊂ B, then A 2 ∆. The elements of ∆ are called faces. The maximal faces (under inclusion) are called facets. For a face F of a simpicial complex ∆, the dimension of F is defined as dim F = jF j − 1. The dimension of ∆, dim ∆, is defined as the maximum dimension of the faces of ∆. A complex ∆ is pure if all of its facets are of the same dimension. The i-skeleton of a simplicial complex ∆ is the collection of all faces of ∆ of dimension ≤ i. In particular, the 1-skeleton of ∆ is its underlying graph. It is sometimes useful in inductive proofs to consider certain subcomplexes of a given simplicial complex, such as its links. Definition 2.1. Let ∆ be a simplicial complex and F 2 ∆. The link of F , lk∆(F ), is defined as lk∆(F ) := fG 2 ∆ j F \ G = ;;F [ G 2 ∆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 vertex of a graph. Definition 2.2. The link of a vertex v in a graph G, denoted lkG(v), is the induced subgraph of G on all vertices adjacent to v. Note that lkG(v) coincides with the 1-skeleton of the link of fvg 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 vertex set V is a flag complex if all of its minimal non-faces are two element sets. A non-face of ∆ is a subset A ⊆ V such that A 62 ∆. A non-face A is minimal if, for all proper subsets B ⊂ A, B 2 ∆. In the following, we refer to the chromatic number of a simplicial complex as the chromatic number of its 1-skeleton in the usual graph theoretic sense. We also need the notion of a balanced complex, as introduced and studied in [7]. Definition 2.4. A simplicial complex ∆ of dimension d − 1 is balanced if it has chromatic number d. Note that the chromatic number of a simpicial complex of dimension d − 1 must be at least d, as it has some face with d vertices, all of which are adjacent, so FACE VECTORS OF FLAG COMPLEXES 3 coloring that face takes d colors. A balanced complex is then one whose chromatic number is no larger than it has to be. Not all simplicial complexes are balanced complexes. For example, a pentagon (five vertices, five edges, and one empty face) is not a balanced complex, because it has chromatic number three but dimension only one. In this paper, we study the face numbers of flag complexes. Definition 2.5. The i-th face number of a simplicial complex C, denoted ci(C) is the number of faces in C containing i vertices. These are also called i-faces of C. If dim C = d − 1, the face vector of C is the vector c(C) = (c0(C); c1(C); : : : ; cd(C)): In particular, for any non-empty complex C, we have c0(C) = 1, as there is a unique empty set of vertices, and it is a face of C. Since flag complexes are the same as clique complexes of graphs, it is sometimes convenient to talk about face numbers in the language of graphs. Definition 2.6. The i-th face number of a graph is the i-th face number of its clique complex. Likewise, the clique vector of a graph is the face vector of its clique complex. The face numbers defined here are shifted by one from what is often used for simplicial complexes. This is done because we are primarily concerned with flag complexes, or equivalently, clique complexes of graphs, where it is more natural to index i as the number of vertices in a clique of the graph, following Eckhoff [3]. The graph concept corresponding to the dimension of a simplicial complex is the clique number. Definition 2.7. The clique number of a graph is the number of vertices in its largest clique. Note that the clique number of a graph is one larger than the dimension of its clique complex. 3. The Kruskal-Katona and Frankl-Furedi-Kalai¨ theorems For the general case of simplicial complexes, the question of which face vectors are possible is answered by the Kruskal-Katona theorem [6, 5]. Stating the theorem requires the following lemma. Lemma 3.1. Given any positive integers m and k, there is a unique s and unique nk > nk−1 > ··· > nk−s ≥ k − s > 0 such that n n n m = k + k−1 + ··· + k−s : k k − 1 k − s The representation described in the lemma is called the k-canonical representa- tion of m. Theorem 3.2 (Kruskal-Katona). For a simplicial complex C, let n n n m = c (C) = k + k−1 + ··· + k−s k k k − 1 k − s 4 ANDY FROHMADER be the k-canonical representation of m. Then n n n c (C) ≤ k + k−1 + ··· + k−s : k+1 k + 1 k k − s + 1 Furthermore, given a vector (1; c1; c2; : : : ; ct) which satisfies this bound for all 1 ≤ k < t, there is some complex that has this vector as its face vector. To construct the complexes which demonstrate that the bound of the Kruskal- Katona theorem is attained, we need the reverse-lexicographic (\rev-lex") order. To define the rev-lex order of i-faces of a simplicial complex on n vertices, we start by labelling the vertices 1; 2;::: . Let N be the natural numbers, let A and B be distinct subsets of N with jAj = jBj = i, and let ArB be the symmetric difference of A and B.