Homotopy types of Hom complexes of graph homomorphisms whose codomains are square-free

TL;DR

This paper characterizes the homotopy types of Hom complexes for graph homomorphisms into square-free graphs, showing they are wedge sums of circles and providing an algorithm to determine these types.

Contribution

It establishes the homotopy type of each connected component of Hom complexes into square-free graphs and offers an algorithmic method to determine these types.

Findings

Connected components are homotopy equivalent to wedge sums of circles.

Homotopy types can be determined algorithmically.

Results apply specifically to square-free graphs without 4-cycle subgraphs.

Abstract

Given finite simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex having the graph homomorphisms $G\to H$ as the vertices. We determine the homotopy type of each connected component of $\mathrm{Hom}(G,H)$ when $H$ is square-free, meaning that it does not contain the $4$-cycle graph $C_4$ as a subgraph. Specifically, for a connected $G$ and a square-free $H$, we show that each connected component of $\mathrm{Hom}(G,H)$ is homotopy equivalent to a wedge sum of circles. We further show that, given any graph homomorphism $f\colon G\to H$ to a square-free $H$, one can determine the homotopy type of the connected component of $\mathrm{Hom}(G,H)$ containing $f$ algorithmically.