# The homotopy type of complexes of graph homomorphisms between cycles

Research paper by Sonja Lj. Cukic, Dmitry N. Kozlov

Indexed on: 12 Sep '05Published on: 12 Sep '05Published in: Mathematics - Combinatorics

#### Abstract

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.