Set Partition Complexes

Discrete & Computational Geometry, Sep 2008

The Hom complexes were introduced by Lovász to study topological obstructions to graph colorings. The vertices of Hom(G,K n ) are the n-colorings of the graph G, and a graph coloring is a partition of the vertex set into independent sets. Replacing the independence condition with any hereditary condition defines a set partition complex. We show how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes. It was conjectured by Babson and Kozlov, and proved by Čukić and Kozlov, that Hom(G,K n ) is (n−d−2)-connected, where d is the maximal degree of a vertex of G. We generalize this to set partition complexes.

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Set Partition Complexes

Discrete Comput Geom Set Partition Complexes Alexander Engström 0 0 A. Engström ( ) Department of Mathematics, Royal Institute of Technology , 100 44 Stockholm , Sweden The Hom complexes were introduced by Lovász to study topological obstructions to graph colorings. The vertices of Hom(G, Kn) are the n-colorings of the graph G, and a graph coloring is a partition of the vertex set into independent sets. Replacing the independence condition with any hereditary condition defines a set partition complex. We show how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes. It was conjectured by Babson and Kozlov, and proved by Cˇ ukic´ and Kozlov, that Hom(G, Kn) is (n − d − 2)-connected, where d is the maximal degree of a vertex of G. We generalize this to set partition complexes. Set partition complex; Hom complex; Topological combinatorics; Ramsey theory 1 Introduction The last years have seen a fast development of topological tools in combinatorics. The Lovász conjecture regarding Hom complexes was proved by Babson and Kozlov [ 2 ], and several related questions have been resolved [ 1, 4–8, 14, 16, 17 ]. The Hom complexes encode graph colorings, which we view as partitions into independent sets. Our first goal is to introduce the new concept of set partition complexes and then explain in an independent section how they are related to Hom complexes. Then we prove a connectivity theorem for set partition complexes, which, when specialized to Hom complexes, recovers the theorem of Cˇukic´ and Kozlov [ 4 ]. In the last section we give an example of connections to Ramsey theory and hints on future research. 2 Notation and Basics For any positive integer n, we let [n] denote the set {1, 2, . . . , n}. The simplex with vertex set V is denoted by ΔV , and the vertex set of a simplicial complex Σ is denoted by V (Σ ). If V ⊆ V (Σ ), then the induced subcomplex Σ [V ] is the set of simplices from Σ with vertices in V , and Σ \ V = Σ [V (Σ ) \ V ]. A topological space T is n-connected if for all 0 ≤ i ≤ n, any continuous map from the i-sphere to T can be extended to a continuous map from the (i + 1)-ball to T . Hence arcwise connected and 0-connected are the same. We define all nonempty spaces to be (−1)-connected and all spaces to be n-connected for n ≤ −2. There are several good surveys on combinatorial topology, see, for example, [ 3, 13 ]. The vertex set of a graph G is denoted by V (G) and the edge set by E(G). An independent set is a subset of vertices of a graph with no edges between them. If A and B are sets, then KA is the complete graph with vertex set A, and KA,B is the complete bipartite graph with vertex set A B and all edges between A and B. Usually Kn = K[n] and Km,n = K[m],[m+n]\[m] are used. 3 The Set Partition Complex In this section we discuss the construction of set partition complexes. The set partition complex is a subcomplex of a direct product of simplices. For any finite simplicial complex Σ and finite set S, let Π (Σ, S) denote the direct product of simplices ΔS indexed by the vertices of Σ . A cell of Π (Σ, S) is then a direct product v∈V (Σ) σv of simplices, and its dimension is v∈V (Σ) dim σv . Definition 3.1 The set partition complex Part(Σ, S) is a subcomplex of Π (Σ, S) where a cell σ = v∈V (Σ) σv is in the subcomplex if {v ∈ V (Σ ) | s ∈ σv} ∈ Σ for all s ∈ S. First note that this really defines a subcomplex. If σ ∈ Π (Σ, S), σ ∈ Part(Σ, S), and σ ⊆ σ where σ = v∈V (Σ) σv ∈ Π (Σ, S) and σ = v∈V (Σ) σv , then {v ∈ V (Σ ) | s ∈ σv} ⊆ {v ∈ V (Σ ) | s ∈ σv} ∈ Σ for all s ∈ S and hence σ ∈ Part(Σ, S). The topology of Part(Σ, S) is inherited from the topology of Π (Σ, S), and it is a regular CW-complex (see [ 15 ] for a definition). We note that the set of functions η : V (Σ ) → 2S \ {∅} such that η−1(s) ∈ Σ for all s ∈ S indexes the cells of Part(Σ, S). This observation will be frequently used in the rest of the paper. Example 3.2 A geometrical realization of the set partition complex Part(Δ[ 4 ] \ {[ 4 ]}, [ 2 ]) is drawn in Fig. 1. For example, the vertices (1, 1, 2, 2), (1, 1, 1, 2), (2, 1, 1, 2), (2, 1, 2, 2) are on the boundary of the maximal cell indexed by the function η defined by η(1) = {1, 2}, η(2) = {1}, η(3) = {1, 2}, and η(4) = {2}. All the maximal cells are indexed by functions η of the form η(a) = {1}, η(b) = {2}, and η(c) = η(d) = {1, 2}, where {a, b, c, d} = {1, 2, 3, 4}, and hence are 2-dimensional. In this section we discuss the connection between Hom complexes and set partition complexes. Though not required in later sections, it provides a motivation for our construction. The study of set partition complexes originated from the theory of Hom complexes. For any graphs G and H , the polyhedral complex Hom(G, H ) has as its vertex set the graph homomorphisms from G to H . The most general definition of Hom complexes can be found in the survey [ 14 ], but we will mainly discuss the case where H is a complete graph. A graph homomorphism from a graph G into Kn corresponds to a proper coloring of the vertices of G with n colors. Among the vertices in G with the same color there are no edges, they form an independent set. A coloring of G is thus a partition of the vertex set into independent sets. Definition 4.1 For any graph G, the independence complex of G, Ind(G), is a simplicial complex with the same vertex set as G and with simplices the independent sets of G. The set partition complex Part(Ind(G), [n]) describes the n-colorings of G and will coincide with a certain Hom complex. Definition 4.2 The complex Hom(G, H ) is the subcomplex of v∈V (G) ΔV (H ) where a cell σ = v∈V (G) σv is in the subcomplex if, for every {u, v} ∈ E(G), there is a complete bipartite subgraph Kσu,σv in H . One can index the cells of Hom(G, H ) in almost the same way as for set partition complexes, and in fact this is a standard idea in the theory of Hom complexes. They are indexed by the functions η : V (G) → 2V (H ) \ {∅} such that Kη(u),η(v) ⊆ H for all {u, v} ∈ E(G). Proposition 4.3 As subcomplexes of [n]) and Hom(G, Kn) are the same. v∈V (G) ΔV (H ), the complexes Part(Ind(G), Proof Specializing the indexing of Hom(G, H ) with functions η to Hom(G, Kn), we get the set {η : V (G) → 2[n] \ {∅} | {u, v} ∈ E(G) ⇒ Kη(u),η(v) ⊆ Kn} = {η : V (G) → 2[n] \ {∅} | {u, v} ∈ E(G) ⇒ η(u) ∩ η(v) = ∅} = {η : V (G) → 2[n] \ {∅} | {u, v} ∈ E(G) ⇒ {u, v} ∈ η−1(w) for w ∈ [n]} = {η : V (G) → 2[n] \ {∅} | η−1(w) is independent in G for w ∈ [n]} = {η : V (G) → 2[n] \ {∅} | η−1(w) ∈ Ind(G) for w ∈ [n]}, which is exactly the functions that index Part(Ind(G), [n]). If d is the maximal degree of a vertex of G, Babson and Kozlov [ 1 ] conjectured that Hom(G, Kn) is (n − d − 2)-connected, and this was proved by Cˇ ukic´ and Kozlov [ 4 ]. Let χ˙ (G) be the largest number of colors a greedy algorithm might use to color G, so that χ (G) ≤ χ˙ (G) ≤ d + 1. A more formal definition of χ˙ (G) as gr(Ind(G)) will be given later. Engström [ 9 ] improved the result by showing that Hom(G, Kn) is (n − χ˙ (G) − 1)-connected, and Theorem 5.2 in the next section is a generalization of the result to set partition complexes. Hence one realizes that the structure of the independent sets and not the maximal degree implies the connectivity bounds for Hom complexes, and this inspired the study of set partition complexes. Other ways to generalize Hom complexes have been pursued, for example, by Živaljevic´ [ 19 ]. 5 Connectivity of Set Partition Complexes In this section we prove a connectivity theorem for set partition complexes. One of the parameters in the connectivity bound is related to the performance of a greedy set partition algorithm which we discuss next. Given a finite simplicial complex Σ , a greedy covering S1, S2, . . . , Sk of Σ is a sequence of disjoint sets Si ∈ Σ which partition V (Σ ) so that Si is a maximal face of Σ [Si ∪ Si+1 ∪ . . . ∪ Sk] for all 1 ≤ i ≤ k. What is the connection to greedy algorithms? Suppose that v1, v2, . . . , vm is an ordering of the vertex set of Σ . The output of the greedy algorithm is a partition S1, S2, . . . , Sk of V (Σ ) where all Si ∈ Σ . The algorithm starts with all Si = ∅. First, it puts v1 into S1. Then it treats v2, v3, . . . , vm like this: when it gets vi , it puts into the Sj with Sj ∪ {vi } ∈ Σ but Sl ∪ {vi } ∈ Σ for 1 ≤ l < j . Sometimes this kind of greedy algorithm is called a first-fit algorithm. Definition 5.1 We let gr(Σ ) denote the largest number of sets in a greedy covering of Σ . The rest of this section will be spent proving the following theorem. Theorem 5.2 The complex Part(Σ, [n]) is (n − gr(Σ ) − 1)-connected. As mentioned in the previous section, this generalizes Theorem 2.3 in [ 9 ], a strengthening of the main Theorem of [ 4 ], which was conjectured by Babson and Kozlov [ 1 ]. Before turning to the proof, we first need some properties of greedy coverings. Lemma 5.3 If V ⊆ V (Σ ), then gr(Σ [V ]) ≤ gr(Σ ). Proof It suffices to prove this when V and V (Σ ) differ by a vertex v. Let S1, S2, . . . , Sgr(Σ[V ]) be a greedy covering of Σ [V ]. If Si ∪ {v} ∈ Σ for 1 ≤ i ≤ gr(Σ [V ]), then S1, S2, . . . , Sgr(Σ[V ]), {v} is a greedy covering of Σ , and hence gr(Σ [V ]) + 1 ≤ gr(Σ ). Otherwise there is an Sj for which Sj ∪ {v} ∈ Σ and Si ∪ {v} ∈ Σ for 1 ≤ i < j . Then S1, S2, . . . , Sj ∪ {v}, . . . , Sgr(Σ[V ]) is a greedy covering of Σ , and again gr(Σ [V ]) ≤ gr(Σ ). Lemma 5.4 If σ is a maximal face of Σ , then gr(Σ ) > gr(Σ \ σ ). Proof Let S1, S2, . . . , Sgr(Σ\σ ) be a greedy covering of Σ \ σ . Then σ, S1, S2, . . . , Sgr(Σ\σ ) is a greedy covering of Σ . We prove the connectivity bound of Theorem 5.2 with a nerve lemma, and the next step is to gain control of certain subcomplexes of set partition complexes. Recall that we index a cell σ = v∈V (Σ) σv ∈ Part(Σ, [n]) with the function η : V (Σ ) → 2[n] \ {∅} defined by η(v) = σv . We will sometimes use η to denote the corresponding cell σ . If η(v) ⊇ η (v) for all v ∈ V (Σ ), then η ≥ η , and if also η = η , then η > η . Our proof of the next lemma relies on the notion of elementary collapses. In a regular CW-complex, the removal of a maximal cell τ , together with a cell τ which is on the boundary of τ and not on the boundary of any other cell, is an elementary collapse step and in particular a deformation retraction. There is a discrete version of Morse theory for regular CW-complexes based on this fact [ 11 ]. Lemma 5.5 If σ ⊂ σ ∈ Σ , then the complex Ψ := {η ∈ Part(Σ, [n]) | n ∈ η(u) ⇒ u ∈ σ } collapses onto a complex isomorphic to Ψ := {η ∈ Part(Σ \ (σ \ σ ), [n]) | n ∈ η(u) ⇒ u ∈ σ }. In particular, Ψ and Ψ are homotopy equivalent. Proof It suffices to prove this when σ \ σ = {v}. We prove the lemma by finding pairs of cells whose removal are elementary collapse steps. The elementary collapse steps are deformation retractions, and the homotopy type is preserved. Let η1, η2, . . . , ηk be an ordering of {η ∈ Ψ | n ∈ η(v)} such that if ηi > ηj , then i < j . Since v ∈ σ and n ∈ η(u) ⇒ u ∈ σ for all η ∈ Ψ , one can extend η so that n ∈ η(v) without falling out of Ψ . For 1 ≤ i ≤ k, define ηi∗ : V (Σ ) → 2[n] \ {∅} as ηi∗(w) = ηi (w) for w = v and ηi∗(v) = ηi (v) ∪ {n}. By construction there is no η ∈ Ψ \ {η1∗, η1, η2∗, η2, . . . , ηi∗−1, ηi−1} for which η > ηi∗ and for which η > ηi only when η = ηi∗. The successive pairwise removal of ηi∗, ηi in the order i = 1, 2, . . . , k collapses Ψ onto Ψ := {η ∈ Ψ | η(v) = {n}}. There is an isomorphism i : Ψ → Ψ by η(u) = (i(η))(u) for u ∈ V (Σ ) \ {v}. The only use of Lemma 5.5 in this paper is incorporated in Lemma 5.6. Lemma 5.6 If σ ∈ Σ , then the complex Ψ := {η ∈ Part(Σ, [n]) | n ∈ η(u) ⇒ u ∈ σ } is homotopy equivalent to Part(Σ \ σ, [n − 1]). Proof Set σ = ∅ and apply Lemma 5.5 to conclude that Ψ is homotopy equivalent to {η ∈ Part(Σ \ σ, [n]) | n ∈ η(u) ⇒ u ∈ ∅} = Part(Σ \ σ, [n − 1]). If Ψ1 and Ψ2 are two k-connected complexes and Ψ1 ∩ Ψ2 is (k − 1)-connected, then Ψ1 ∪ Ψ2 is k-connected. More generally, if Ψ1, Ψ2, . . . , Ψm are k-connected and every t -fold intersection of them is (k − t + 1)-connected, then Ψ1 ∪ Ψ2 ∪ · · · ∪ Ψm is k-connected by the connectivity nerve lemma [ 3 ] or by a standard spectral sequence argument. Now we are ready to prove Theorem 5.2, namely that Part(Σ, [n]) is (n − gr(Σ ) − 1)-connected. Proof The idea for our proof is to cover Part(Σ, [n]) with subcomplexes of Part(, )-type and then use the nerve lemma. We do induction on gr(Σ ) and on n − gr(Σ ). If gr(Σ ) = 1, then Σ is a simplex, so Part(Σ, [n]) is a product of simplices, hence contractible, and in particular (n − gr(Σ ) − 1)-connected. If n − gr(Σ ) = 0, then Part(Σ, [n]) is nonempty and (n − gr(Σ ) − 1)-connected, since n − gr(Σ ) − 1 = −1. Now suppose that gr(Σ ) > 1 and n > gr(Σ ), and let us define a cover of Part(Σ, [n]). Let S be the set of maximal faces of Σ , and let Ψσ = {η ∈ Part(Σ, [n]) | n ∈ η(u) ⇒ u ∈ σ } for all σ ∈ S. If η ∈ Part(Σ, [n]), then η−1(n) is a subset of an element of S, and thus Part(Σ, [n]) = ∪σ ∈S Ψσ . To apply the nerve lemma, we need to calculate the connectivity of intersections of subcomplexes in the cover. The complex Ψσ is homotopy equivalent to Part(Σ \ σ, [n − 1]) by Lemma 5.6 and ((n − 1) − (gr(Σ ) − 1) − 1)-connected by induction and Lemma 5.4. For any nonempty subset S of S, the intersection of Ψσ for σ ∈ S is η ∈ Part(Σ, [n]) n ∈ η(u) ⇒ u ∈ σ , σ ∈S which by Lemma 5.6 is homotopy equivalent to Part Σ \ σ, [n − 1] , σ ∈S and the latter is ((n − 1) − gr(Σ ) − 1)-connected by induction and Lemma 5.3. All the required connectivities are checked, and we apply the nerve lemma to conclude that Part(Σ, [n]) is (n − gr(Σ ) − 1)-connected. 6 A Ramsey Theory Example In this section we discuss a particular example, which illustrates a connection between set partition complexes and Ramsey theory, and use this to suggest future directions for research. The basic theorem of Ramsey theory states that, for any finite graph G and any number of colors r , there exists an n such that, for any edge coloring of Kn with r colors, there is a monochromatic copy of G (see, for example, [ 12 ] or Chap. 6 of [ 18 ]). While Hom complexes are used to study vertex colorings of graphs, set partition complexes can be used for describing edge colorings in the following way. Let Σ be the simplicial complex whose vertex set is the edge set of K4 and whose faces are the edges that do not form a triangle. Then the vertices of Part(Σ, [ 2 ]) are the two-colorings of the edges of K4 without any monochromatic triangles. From the drawing of Part(Σ, [ 2 ]) in Fig. 2 we see that it consists of six squares glued to form a cycle. In this paper we introduced set partition complexes, a generalization of Hom(G, Kn) complexes. A natural question is if there is a common generalization of set partition complexes and Hom(G, H ) complexes. An application of such a construction would be to answer coloring questions other than for graph vertices, for example, from Ramsey theory. In [ 10 ], for any two simplicial complexes Σ1 and Σ2, we will define a polyhedral complex whose vertices are indexed by the maps ϕ from V (Σ1) to V (Σ2) such that ϕ−1(σ ) ∈ Σ1 for all σ ∈ Σ2. The faces are the polyhedral cliques as for the Hom complexes and the set partition complexes. If Σ2 is of dimension zero, we get a set partition complex, and if Σ1 and Σ2 are the independence complexes of G and H , we get Hom(G, H ). 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Alexander Engström. Set Partition Complexes, Discrete & Computational Geometry, 2008, 357, DOI: 10.1007/s00454-008-9106-6