Hom complexes of graphs whose codomains are square-free

The Hom complex $\mathrm{Hom}(G, H)$ 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 $\mathrm{Hom}(G, H)$ 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 $\mathrm{Hom}(G,H)$ and realizable walks studied in the homomorphism reconfiguration problem.

💡 Research Summary

The paper investigates the homotopy type of Hom complexes Hom(G,H) when the codomain graph H is square‑free (i.e., contains no induced 4‑cycle) and both G and H are connected. The main result (Theorem 1.2) states that every connected component of Hom(G,H) is homotopy equivalent to exactly one of four spaces: a point, a circle, the graph H itself, or a connected double cover of H. Moreover, if G is non‑bipartite, each component is either contractible or homotopy equivalent to a circle.

The authors’ approach departs from earlier work that handled only the case H = Cₙ (cycle graphs). They introduce the 2‑fundamental group π₂¹(G) and the notion of a universal 2‑cover, originally developed in