Homotopy types of Hom complexes of graph homomorphisms whose codomains are cycles

TL;DR

This paper extends known results about the homotopy types of Hom complexes from cycles to cases where the domain graph is connected and the codomain is a cycle, providing explicit constructions and criteria.

Contribution

It generalizes the homotopy classification of Hom complexes to broader graph pairs and introduces explicit universal covers and a criterion for their homotopy types.

Findings

Hom complexes are homotopy equivalent to points or circles when $G$ is connected and $H$ is a cycle.

Explicit universal covers of connected components are constructed and shown to be contractible.

A simple criterion determines the homotopy type of each component.

Abstract

For simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex whose vertices are the graph homomorphisms $G\to H$ and whose edges connect the pairs of homomorphisms which differ in a single vertex of $G$. Hom complexes play an important role in an algebro-topological approach to the graph coloring problem. It is known that $\mathrm{Hom}(G,H)$ is homotopy equivalent to a disjoint union of points and circles when both $G$ and $H$ are cycles. We generalize this known result by showing that the same holds whenever $G$ is connected and $H$ is a cycle. To this end, we explicitly construct the universal cover of each connected component of $\mathrm{Hom}(G,H)$ and prove that it is contractible. Additionally, we provide a simple criterion to determine whether the connected component containing a given homomorphism is homotopy equivalent to a point or circle.