Homotopy types of the Hom complexes of graphs

For a fixed finite graph T, we show that there is no homotopy invariant of Hom(T,G) which gives an upper bound for the chromatic number of G. More precisely, for a non-bipartite graph G, we construct a graph H such that Hom(T,G) and Hom(T,H) are homotopy equivalent but χ(H) is much larger than χ(G). The equivariant homotopy type of Hom(T,G) is also considered.