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We introduce new methods for understanding the topology of $\Hom$ complexes(spaces of homomorphisms between two graphs), mostly in the context of groupactions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functorsfrom graphs to posets, and introduce a functor $(-)^1$ from posets to graphsobtained by taking atoms as vertices. Our main structural results establishuseful interpretations of the equivariant homotopy type of $\Hom$ complexes interms of spaces of equivariant poset maps and $\Gamma$-twisted products ofspaces. When $P = F(X)$ is the face poset of a simplicial complex $X$, thisprovides a useful way to control the topology of $\Hom$ complexes.  Our foremost application of these results is the construction of new familiesof `test graphs' with arbitrarily large chromatic number - graphs $T$ with theproperty that the connectivity of $\Hom(T,G)$ provides the best possible lowerbound on the chromatic number of $G$. In particular we focus on two infinitefamilies, which we view as higher dimensional analogues of odd cycles. Thefamily of `spherical graphs' have connections to the notion of homomorphismduality, whereas the family of `twisted toroidal graphs' lead us to establish aweakened version of a conjecture (due to Lov\'{a}sz) relating topological lowerbounds on chromatic number to maximum degree. Other structural results allow usto show that any finite simplicial complex $X$ with a free action by thesymmetric group $S_n$ can be approximated up to $S_n$-homotopy equivalence as$\Hom(K_n,G)$ for some graph $G$; this is a generalization of a result ofCsorba. We conclude the paper with some discussion regarding the underlyingcategorical notions involved in our study.

Topology of Hom complexes and test graphs for bounding chromatic number