  The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a
pair of graphs $T$ and $G$, considered in the graph coloring problem. It is
known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds
for the chromatic number of $G$.
  For a fixed finite graph $T$, we show that there is no homotopy invariant of
${\rm Hom}(T,G)$ 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 ${\rm Hom}(T,G)$ and ${\rm Hom}(T,H)$ are homotopy equivalent but
$\chi(H)$ is much larger than $\chi(G)$. The equivariant homotopy type of ${\rm
Hom}(T,G)$ is also considered.
