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

For simple graphs G and H, the Hom complex Hom(G,H) is a polyhedral complex whose vertices are the graph homomorphisms G→ H. It is known that 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 Hom(G,H) is homotopy equivalent to a disjoint union of points and circles whenever G is connected and H is a cycle.