Fold and Mycielskian on homomorphism complexes

Homomorphism complexes were introduced by Lov\'asz to study topological obstructions to graph colorings. We show that folding in the second parameter of the homomorphism complex yields a homotopy equivalence. We study how the Mycielski construction changes the homotopy type of the homomorphism complex. We construct graphs showing that the topological bound obtained by odd cycles can be arbitrarily worse than the bound provided by Hom(K_2,G)