The homotopy type of complexes of graph homomorphisms between cycles

TL;DR

This paper investigates the homotopy types of complexes of graph homomorphisms between cycles and strings, revealing their topological structures and classifying their connected components.

Contribution

It provides a complete classification of the homotopy types of $ ext{Hom}(C_m,C_n)$ and $ ext{Hom}(C_m,L_n)$, including enumeration and topological characterization.

Findings

Connected components of $ ext{Hom}(C_m,C_n)$ are points or $S^1$

$ ext{Hom}(C_m,L_n)$ is either empty or two points

Homotopy types depend on cycle and string parameters

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.