# Homotopy Type of Complexes of Graph Homomorphisms between Cycles

@article{Cukic2006HomotopyTO, title={Homotopy Type of Complexes of Graph Homomorphisms between Cycles}, author={Sonja Lj. Cukic and Dmitry N. Kozlov}, journal={Discrete \& Computational Geometry}, year={2006}, volume={36}, pages={313-329} }

In this paper we study the homotopy type of Hom(Cm,Cn), where Ck is the cyclic graph with k vertices. We enumerate connected components of Hom(Cm,Cn) and show that each such component is either homeomorphic to a point or homotopy equivalent to S1. Moreover, we prove that Hom(Cm,Ln) is either empty or is homotopy equivalent to the union of two points, where Ln is an n-string, i.e., a tree with n vertices and no branching points.

## 17 Citations

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- MathematicsContributions Discret. Math.
- 2008

Graphs are constructed showing that the topological bound obtained by odd cycles can be arbitrarily worse than the bound provided by Hom(K_2,G).

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We compute the cohomology groups of the spaces of colorings of cycles, i.e., of the prodsimplicial complexes Hom (Cm, Kn). We perform the computation first with Z2, and then with integer…

### Homology of Hom Complexes

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- 2012

The hom complex Hom (G,K) is the order complex of the poset composed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible…

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- MathematicsComb.
- 2013

SGn,k is the stable Kneser graph (Schrijver graph) of stable n-subsets of a set of cardinality 2n+k and it is derived that there is a graph G with χ(G)=χ(SG n,k) such that the complex Hom(SGm,k,G) is non-empty and connected.

### Cohomology of colorings of cycles

- Mathematics
- 2005

We compute the cohomology groups of the spaces of colorings of cycles, i.e., of the prodsimplicial complexes ${\tt Hom}(C_m,K_n)$. We perform the computation first with ${\Bbb Z}_2$, and then with…

### Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes

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### Set Partition Complexes

- MathematicsDiscret. Comput. Geom.
- 2008

It is shown how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes.

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The neighborhood complex of a graph was introduced by Lovász to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and…

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