Hom complexes of graphs whose codomains are square-free

The Hom complex $\mathrm{H}\mathrm{o}\mathrm{m}(G,H)$ of graphs is a simplicial complex associated to a pair of graphs $G$ and $H$, and its homotopy type is of interest in the graph coloring problem and the homomorphism reconfiguration problem. In this paper, we show that if $G$ is a connected graph and $H$ is a square-free connected graph, then every connected component of $\mathrm{H}\mathrm{o}\mathrm{m}(G,H)$ is homotopy equivalent to a point, a circle, $H$ or a connected double cover over $H$. We also obtain a certain relation between the fundamental group of $\mathrm{H}\mathrm{o}\mathrm{m}(G,H)$ and realizable walks studied in the homomorphism reconfiguration problem.