The homotopy type of complexes of graph homomorphisms between cycles

Details The homotopy type of complexes of graph homomorphisms between cycles The homotopy type of complexes of graph homomorphisms between cycles

2004-08-02

arxiv.org/abs/math/0408015v3

Mathematics

Combinatorics

15 pages, 8 figures; Final version, to appear in Journal of Discrete and Computational Geometry

Scientific paper

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.

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