Homology tests for graph colorings
  We describe a simple homological test for obstructions to graph colorings. The main idea is to combine the framework of Hom-complexes with the following general fact: an arbitrary Z_2-space has nontrivial homology with Z_2-coefficients in the dimension equal to its Stiefel-Whitney height.   Actually, as a result we have a whole family of homology tests, one for each test graph. In general, these tests will give different answers, depending heavily on the choice of the test graph. We illustrate this phenomenon with some examples. 
Dmitry N. Kozlov