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

Given finite simple graphs $G$ and $H$, the Hom complex $\mathrm{H}\mathrm{o}\mathrm{m}(G,H)$ is a polyhedral complex having the graph homomorphisms $G\to H$ as the vertices. We determine the homotopy type of each connected component of $\mathrm{H}\mathrm{o}\mathrm{m}(G,H)$ 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 $\mathrm{H}\mathrm{o}\mathrm{m}(G,H)$ is homotopy equivalent to a wedge sum of circles. We further show that, given any graph homomorphism $f\text{\hspace{0.17em}⁣}:G\to H$ to a square-free $H$, one can determine the homotopy type of the connected component of $\mathrm{H}\mathrm{o}\mathrm{m}(G,H)$ containing $f$ algorithmically.