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 ca (...truncated)