In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes.
To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of a graph. Then, for an arbitrary lattice ${\mathcal L}$ we describe a formal deformation of the barycentric subdivision of the atom crosscut complex $Γ({\mathcal L})$ to its order complex $Δ(\bar{\mathcal L})$. We proceed by proving that the complex of sets bounded from below ${\mathcal J}({\mathcal L})$ can also be collapsed to $Δ(\bar{\mathcal L})$.
Finally, as a pinnacle of our project, we apply all these results to certain graph complexes. Namely, by describing an explicit formal deformation, we prove that, for any graph $G$, the neighborhood complex ${\mathcal N}(G)$ and the polyhedral complex $\text{\tt Hom}(K_2,G)$ have the same simple homotopy type in the sense of Whitehead.
The groupoid of projectivities, introduced by M. Joswig, serves as a basis for a construction of parallel transport of graph and more general $Hom$-complexes. In this framework we develop a general conceptual approach to the Lovasz Hom-conjecture, recently resolved by E. Babson and D. Kozlov, and extend their result from graphs to simplicial complexes. The paper also provides new evidence that the language and methods of groupoids, after being successfully tested in other major mathematical fields, offer new insights and perspectives for combinatorial applications.
In this article, we give conditions on a graph under which the Lovász' original bound of the graph can be improved by increasing the topological connectivity of its neighbourhood complex. We also work out conditions under which computing the topological connectivity of hom complex of a pair of graphs can be simplified. In particular, hom complex as a covariant functor acting on a double mapping cylinder of graphs is a homotopy pushout of hom complex functor applied to its subgraphs. We give applications of this result where the computation of hom complexes is simplified. Finally, we explain wh…
To any two graphs G and H one can associate a cell complex Hom(G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovasz Conjecture which states that if Hom(C_{2r+1},G) is k-connected, then χ(G)\geq k+4, where r,k\in Z, r\geq 1, k\geq -1, and C_{2r+1} denotes the cycle with 2r+1 vertices. The proof requires analysis of the complexes Hom(C_{2r+1},K_n). For even n, the obstructions to graph colorings are provided by the presence of torsion in H^*(Hom(C_{2r+1},K_n);Z). For odd n, the obstructions are expressed as vanishing of certain powers of Stiefel-Whitn…
A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like ``holonomy'', ``parallel transport'', ``bundles'', ``combinatorial curvature'' etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial ``Theorema Egregium'' for cubical complexes non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool for extending Babson-Kozlov-Lovasz graph coloring results to more gen…
For a pair $(H_1,H_2)$ of graphs, Lovász introduced a polytopal complex called the Hom complex $\text{Hom}(H_1,H_2)$, in order to estimate topological lower bounds for chromatic numbers of graphs. The definition is generalized to hypergraphs. Denoted by $K_r^r$ the complete $r$-graph on $r$ vertices. Given an $r$-graph $H$, we compare $\text{Hom}(K_r^r,H)$ with the box complex $\mathsf{B}_{\text{edge}}(H)$, invented by Alon, Frankl and Lovász. We verify that $\text{Hom}(K_r^r,H)$ and $\mathsf{B}_{\text{edge}}(H)$, both are equipped with right actions of the symmetric group on $r$ letters $S_r$…
We show that the vanishing of certain cohomology groups of polyhedral complexes imply upper bounds on Ramsey numbers. Lovasz bounded the chromatic numbers of graphs using Hom complexes. Babson and Kozlov proved Lovasz conjecture and developed a Hom complex theory. We generalize the Hom complexes to Ramsey complexes. The main theorem states that if certain cohomology groups of the Ramsey complex Ram(dDelta_{p^k}, Sigma) are trivial, then the vertices of the simplicial complex Sigma cannot be n-colored such that every color correspond to a face of Sigma. In a corollary, we give an explicit descr…
By Lovasz' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z_2-space Hom(K_2,G) plus two. We show that the cohomological index of Hom(K_2,G) is also greater than the cohomological index of the Z_2-space Hom(C_{2r+1}, G) for r>0. This gives a new and simple proof of the strong form of the graph colouring theorem by Babson and Kozlov, which had been conjectured by Lovasz, and at the same time shows that it never gives a stronger bound than can be obtained by Hom(K_2, G). The proof extends ideas introduced by Zivaljevic in a previous ele…
For any two graphs $G$ and $H$ Lovász has defined a cell complex $Hom(G,H)$ having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with $G$ a cycle of odd length. More specifically, we show that: if $Hom(C_{2r+1},G)$ is $k$-connected, then $χ(G)\geq k+4$. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney classes.
The Hom-complexes were introduced by Lovasz to study topological obstructions to graph colorings. It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that Hom(G,K_n) is (n-d-2)-connected, where d is the maximal degree of a vertex of G. We give a short proof of the conjecture.
In [MZ04] Matousek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovasz's original bound [L78] can be restated as $\chr G \geq \ind (\B(G)) +2$. Sarkaria's bound [S90] can be formulated as $\chr G \geq \ind (\B_0(G)) +1$. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. Some of the results was announced in [MZ04].
For two general polytopal complexes the set of face-wise affine maps between them is shown to be a polytopal complex in an algorithmic way. The resulting algorithm for the affine hom-complex is analyzed in detail. There is also a natural tensor product of polytopal complexes, which is the left adjoint functor for Hom. This extends the corresponding facts from single polytopes, systematic study of which was initiated in [6,12]. Explicit examples of computations of the resulting structures are included. In the special case of simplicial complexes, the affine hom-complex is a functorial subcomple…
We show that the n-th power of the first Stiefel-Whitney class of the Z_2-operation on the graph complex Hom(C_{2r+1},K_{n+2})$ is zero, confirming a conjecture by Babson and Kozlov. This proves the strong form of their graph colouring theorem, which they had only proven for odd n. Our proof is also considerably simpler than their proof of the weak form of the theorem, which is also known as the Lovász conjecture.
In this paper we study implications of folds in both parameters of Lovász' Hom(-,-) complexes. There is an important connection between the topological properties of these complexes and lower bounds for chromatic numbers. We give a very short and conceptual proof of the fact that if G-v is a fold of G, then Bd(Hom(G,H)) collapses onto Bd Hom(G-v,H), whereas Hom(H,G) collapses onto Hom(H,G-v). We also give an easy inductive proof of the only nonelementary fact which we use for our arguments: if $ϕ$ is a closure operator on P, then $Δ(P)$ collapses onto $Δ(ϕ(P))$.
The main result of this paper is a proof of the following conjecture of Babson & Kozlov: Theorem. Let G be a graph of maximal valency d, then the complex Hom(G,K_n) is at least (n-d-2)-connected. Here Hom(-,-) denotes the polyhedral complex introduced by Lovász to study the topological lower bounds for chromatic numbers of graphs. We will also prove, as a corollary to the main theorem, that the complex Hom(C_{2r+1},K_n) is (n-4)-connected, for $n\geq 3$.
This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lovász in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This conjecture stated that the \emph{Kneser graph} $\KG_{m,n}$, the graph with all $k$-element subsets of $\{1,2,...,n\}$ as vertices and all pairs of disjoint sets as edges, has chromatic number $n-2k+2$. Several other proofs have since been published (by Bárány, Schrijver, Dolnikov, Sarkaria, Kriz, Greene, and others), all of them based on some version of the Borsuk--U…
We investigate a notion of $\times$-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph $\times$-homotopy is characterized by the topological properties of the $\Hom$ complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; $\Hom$ complexes were introduced by Lovász and further studied by Babson and Kozlov to give topological bounds on chromatic number. Along the way, we also establish some structural properties of $\Hom$ complexes involving products and exponentia…
We show that the nerve complex of n arcs in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time O(n log n). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine the dihedral group action on homology, and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we use the connection to cyclic polytopes to …
$Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$ and $H$. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that $Hom(K_m,K_n)$ is homotopy equivalent to a wedge of $(n-m)$-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph $G$, and integers $m\geq 2$ and $k\geq -1$, we have $\varpi_1^k(\thom(K_m,G))\neq 0$, then $χ(G)\geq k+m$; here $Z_…
We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $\Hom$ complexes in terms of spaces of equivariant poset maps and $Γ$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $…
For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well known result of Lovasz that if N[G] is k-connected, then the chromatic number of G is at least k + 3. We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(log d), compared to the expected dimension d of the complex itself.
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways these can be used to obtain lower bounds for the chromatic numbers of graphs, presented in a recent series of papers \cite{BK03a,BK03b,BK03c,CK1,CK2,K4,K5}. The structural theory is developed and put in the historical context, culminating in the proof of the Lovász Conjecture, which can be stated as follows: For a graph G, such that the complex Hom(C_{2r+1}…
The theory of complexes of directed trees was initiated by Kozlov to answer a question by Stanley, and later on, results from the theory were used by Babson and Kozlov in their proof of the Lovasz conjecture. We develop the theory and prove that complexes on directed acyclic graphs are shellable. A related concept is that of independence complexes: construct a simplicial complex on the vertex set of a graph, by including each independent set of vertices as a simplex. Two theorems used for breaking and gluing such complexes are proved and applied to generalize results by Kozlov. A fruitful rest…
In 2002, A. Björner and M. de Longueville showed the neighborhood complex of the $2$-stable Kneser graph ${KG(n, k)}_{2-\textit{stab}}$ has the same homotopy type as the $(n-2k)$-sphere. A short time ago, an analogous result about the homotopy type of the neighborhood complex of almost $s$-stable Kneser graph has been announced by J. Osztényi. Combining this result with the famous Lovász's topological lower bound on the chromatic number of graphs has been yielded a new way for determining the chromatic number of these graphs which was determined a bit earlier by P. Chen. In this paper we prese…
In this paper we study a pair of numerical parameters associated to a graph $G$. One the one hand, one can construct $\text{Hom}(K_2, G)$, a space of homomorphisms from a edge $K_2$ into $G$ and study its (topological) connectivity. This approach dates back to the neighborhood complexes introduced by Lovász in his proof of the Kneser conjecture. In another direction Brightwell and Winkler introduced a graph parameter called the warmth $ζ(G)$ of a graph $G$, based on asymptotic behavior of $d$-branching walks in $G$ and inspired by constructions in statistical physics. Both the warmth of $G$ an…
The neighborhood complexes of graphs were introduced by Lovász in his proof of the Kneser conjecture. He showed that a certain topological property of $N(G)$ gives a lower bound for the chromatic number of $G$. In this paper, we study a combinatorial description of the fundamental groups of the neighborhood complexes. For a positive integer $r$, we introduce the $r$-fundamental group $π_1^r(G,v)$ of a based graph $(G,v)$ and the $r$-neighborhood complex $N_r(G)$ of $G$. The $1$-neighborhood complex is the neighborhood complex. We show that the even part $π_1^{2r}(G,v)_{ev}$, which is a subgrou…
