In this paper we study the homotopy type of $\Hom(C_m,C_n)$, where $C_k$ is the cyclic graph with $k$ vertices. We enumerate connected components of $\Hom(C_m,C_n)$ and show that each such component is either homeomorphic to a point or homotopy equivalent to $S^1$.
Moreover, we prove that $\Hom(C_m,L_n)$ is either empty or is homotopy equivalent to the union of two points, where $L_n$ is an $n$-string, i.e., a tree with $n$ vertices and no branching points.
In this article, we define a family of regular bipartite graphs and show that the homotopy type of the independence complexes of this family is the wedge sum of spheres of certain dimensions.
In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set P_r of r-element multichains from P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there exactly 2^r such functions which yield subdivisions of the order complex of P of which 2^{r-1} are pairwise different. Within this class are for example the order complexes of the interval and the zig-zag poset of P and the rth edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are …
Consider a family $\mathcal F$ of $C_{2r+1}$-free graphs, where $r\geq 2$. Suppose that each graph in $\mathcal F$ has minimum degree linear in its number of vertices. Thomassen showed that such a family has bounded chromatic number, or, equivalently, that all graphs in $\mathcal F$ are homomorphic to a complete graph of bounded size. Considering instead homomorphic images which are themselves $C_{2r+1}$-free, we construct a family of dense $C_{2r+1}$-free graphs with no $C_{2r+1}$-free homomorphic image of bounded size. This provides the first nontrivial lower bound on the homomorphism thresh…
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their efficiency on some simplicial complexes, the devoid complexes of graphs, which are simplicial complexes parametrized by graphs. Additionally, we compute the homotopy type of dominance complexes of chordal graphs.
Schrijver identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$. Björner and de Longueville proved that the neighborhood complex of the stable Kneser graph $SG_{n,k}$ is homotopy equivalent to a $k-$sphere. In this article, we prove that the homotopy type of the neighborhood complex of the Kneser graph $KG_{2,k}$ is a wedge of $(k+4)(k+1)+1$ spheres of dimension $k$. We construct a maximal subgraph $S_{2,k}$ of $KG_{2,k}$, whose neighborhood complex is homotopy equivalent to the neighborhood complex of $SG_{2,k}$. Further, we prove tha…
We prove that the topological connectivity of a graph homomorphism complex Hom($G,K_m$) is at least $m-D(G)-2$, where $\displaystyle D(G)=\max_{H\subseteq G}δ(H)$. This is a strong generalization of a theorem of Cukić and Kozlov, in which $D(G)$ is replaced by the maximum degree $Δ(G)$. It also generalizes the graph theoretic bound for chromatic number, $\displaystyleχ(G)\leq D(G)+1$, as $\displaystyleχ(G)=\min\{ m:\text{Hom}(G,K_m)\neq\varnothing\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c/n$ for a…
Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the natural map between the group of homeomorphisms of $X_f$ and the group of homotopy classes of self-homotopy equivalences of $X_f$ is precisely $f$. In addition, realization problems involving homology, homotopy groups and groups of automorphisms are considered.
We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special emphasis on finite products of chains. For the special case of a Boolean algebra, we observe that the corresponding homomorphism complex is isomorphic to the subcomplex of cubical cells in a permutahedron. Thus, this work can be interpreted as a generalization of the study of these complexes. We provide a detailed investigation when our poset is a product of…
We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes of categorical product of complete graphs are wedge sum of circles, upto homotopy. Further, we show that if we perturb a graph $G$ in a certain way, then the independence complex of this new graph is homotopy equivalent to the suspension of the independence complex of $G$.
We show that if a graph $G$ involves a certain square grid graph as a full subgraph, then a certain operation on it yields a simplicial suspension of the independence complex of $G$. This generalizes a result of Csorba. As a corollary, we determine the simple homotopy types of the independence complexes of some grid graphs.
The neighborhood complex $\N(G)$ of a graph $G$ were introduced by L. Lov{á}sz in his proof of Kneser conjecture. He proved that for any graph $G$, \begin{align} \label{abstract} χ(G) \geq conn(\N(G))+3. \end{align} In this article we show that for a class of exponential graphs the bound given in (\ref{abstract}) is sharp. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy. We were also able to find a class of exponential graphs, which are homotopy test graphs. Hedetniemi's conjecture states that the chromatic number of the categorical produ…
In this paper, we determine the homotopy type of the Morse complex of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if $K$ contains two leaves that share a common vertex, then the Morse complex is strongly collapsible and hence has the homotopy type of a point. We also show that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of Ayala et al. In addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$. This result is used to…
To estimate the lower bound for the chromatic number of a graph $G$, Lovász associated a simplicial complex $\mathcal{N}(G)$ called the neighborhood complex and relates the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent ei…
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 $Δ(ϕ(P))$.
For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the homotopy type of $r$-independence complexes of certain families of graphs including complete $s$-partite graphs, fully whiskered graphs, cycle graphs and perfect $m$-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.
In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. As a key ingredient of our work, we introduce the class of cointerval simplicial complexes and investigate their combinatorial and topological properties. As a concrete illustration of these structural results, we introduce and study nonnesting monomial ideals, an interesting family of combinatorially defined ideals.
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{\mathca…
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 $S_r$…
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].
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph ${\tt G}$ is the simplicial complex whose faces are the multipaths of ${\tt G}$. We compute the Euler characteristic, and associated generating function, of the multipath complex for some families of graphs, including transitive tournaments and complete bipartite graphs. Then, we compute the homotopy type of multipath complexes of linear graphs, polygons, small grids and transitive tournaments. We show that they are all contractible or wedges of spheres. We introduce a new technique for deco…
For $n \in \mathbb{N}$, let $h(n)$ denote the number of simplicial complexes on $n$ vertices up to homotopy equivalence. Here we prove that $h(n) \geq 2^{2^{0.02n}}$ when $n$ is large enough. Together with the trivial upper bound of $2^{2^n}$ on the number of labeled simplicial complexes on $n$ vertices this proves a conjecture of Kalai that $h(n)$ is doubly exponential in $n$.
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 introduce a model structure on the category of graphs, which is Quillen equivalent to the category of $\mathbb{Z}_2$-spaces. A weak equivalence is a graph homomorphism which induces a $\mathbb{Z}_2$-homotopy equivalence between their box complexes. The box complex is a $\mathbb{Z}_2$-space associated to a graph, considered in the context of the graph coloring problem. In the proof, we discuss the universality problem of the Hom complex.
The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a pair of graphs $T$ and $G$, considered in the graph coloring problem. It is known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds for the chromatic number of $G$. For a fixed finite graph $T$, we show that there is no homotopy invariant of ${\rm Hom}(T,G)$ which gives an upper bound for the chromatic number of $G$. More precisely, for a non-bipartite graph $G$, we construct a graph $H$ such that ${\rm Hom}(T,G)$ and ${\rm Hom}(T,H)$ are homotopy equivalent but $χ(H)$ is much larger than $χ(G)$. Th…
We introduce WI-posets as intermediate objects in the study of Z_2-homotopy types of graph complexes. It turns out that (almost) all graph complexes associated to a graph can be viewed as avatars of the same object, as long as their Z_2-homotopy types are concerned. Among the applications are a proof that each finite, free Z_2-complex is a graph complex and an evaluation of Z_2-homotopy types of complexes Ind(C_n) of independence sets in a cycle C_n. The main tools used in the paper are Quillen fiber theorem and Bredon criterion for Z_2-equivalence of Z_2-complexes.
We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed (reconfigured) into the other by changing one color at a time, maintaining an H-coloring throughout. This is the same as finding a path in the Hom(G,H) complex. For H=K_k this is the problem of finding paths between k-colorings, which was shown to be in P for k<=3 and PSPACE-complete otherwise by Cereceda et al. 2011. We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H…
We determine the homotopy types of the independence complexes of the $(n \times 6)$-square grid graphs. In fact, we show that these complexes are homotopy equivalent to wedges of spheres.
An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $Λ$ has the partial order defined by $λ\le λ+ μ$. The Frobenius complex is the order complex of an open interval of $Λ$ with respect to this partial order. The reduced homology of the Frobenius complex is related to the torsion group of the monoid algebra $K[Λ]$. In this paper, we pay attention to homotopy types of Frobenius complexes, and we express the homotopy types of the Frobenius complexes of $Λ$ in terms of those of $Λ_1$ and $Λ_2$ when $Λ$ is an affine monoid obtained by gluin…
The notion of $\times$-homotopy from \cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space $\Hom_*(G,H)$ with the homotopy groups of $\Hom_*(G,H^I)$. Here $\Hom_*(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $\Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that $π_i \big(\Hom_*(G,H) \big) \cong [G,Ω^i H]_{\times}$, where $ΩH$ is the graph of based closed paths in $H$ and $[G,K]_…
Box complex is a $\mathbb{Z}_2$-space associated to a graph, and it is known that a certain $\mathbb{Z}_2$-homotopy invariant of it, called the $\mathbb{Z}_2$-index, gives an effective lower bound for the chromatic number. On the other hand, we show that any $\mathbb{Z}_2$-homotopy invariant of the box complex is not equivalent to the chromatic number. Namely, we construct a graph homomorphism $f:X \rightarrow Y$ such that it gives rise to a $\mathbb{Z}_2$-homotopy equivalence between their box complexes, but $X$ and $Y$ have different chromatic numbers. To see this, we show that some deformat…
We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex $K$ belongs to this class, then the polyhedral join $\mathcal{Z}^*_{K}(\underline{X}, \emptyset)$ is homotopy equivalent to a wedge sum of CW complexes of the form $Σ^r X_{i_1} * X_{i_2} * \cdots * X_{i_k}$, where $\underline{X}$ is a family $\{X_i\}_{i \in V(K)}$ of CW complexes and $Σ$ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product $G[H]$ of a graph $H$ over …
We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial $G$-actions …
If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $Ωf$ and $Ωf\circΩg$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\unde…
A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian (Adv. in Math., 218 (2008), 87-104) in the framework …
We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0,1]-valued functions on unit cubes of increasing dimension, each corresponding to a dimension of the abstract simplicial complex. We show that convergence in homomorphism density implies convergence in a cut-metric, and vice versa, as well as showing that simplicial complexes sampled from the limit objects closely resemble its structure. Applying this framework, we als…
It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, its homotopy type, if not contractible, would be a link invariant and it would imply that the extreme Khovanov homology of any link diagram does not contain torsion. We prove the conjecture in many special cases and f…
We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regular CW-complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes.
A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can be misleading. An intrinsic homotopy theory, not based on such realisation but agreeing with it, is introduced. The applications developed here are aimed at image analysis in metric spaces and have connections with digital topology and mathematical morphology. A metric space X has a structure of simplicial complex at each (positive) resolution e; the resulti…
We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence of normalized counting measures (counting connected components according to homotopy type), and we show that this sequence of random probability measures converges in probability to a deterministic probability measure. We then investigate the dependence of the limiting homotopy measure on the coloring probability $p$, and our results show a qualitative chang…
The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for posets is given. Finally, in an appendix we outline connections with ring-theory and survey some uses of sequential Cohen-Macaulayness in commutative algebra.
We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters. In more detail, let $Q_n$ be the vertex set of the hypercube graph with $2^n$ vertices, equipped with the shortest path metric. Equivalently, $Q_n$ is the set of all binary strings of length $n$, equipped with the Hamming distance. The Vietoris-Rips complex of $Q_n$ at scale parameter zero is $2^n$ points, and the Vietoris-Rips complex of $Q_n$ at scale parameter one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris-Rips complex of $Q_…
Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the h…
