Answers to some problems about graph coloring test graphs

We prove that a graph whose chromatic number is 2 is a homotopy test graph. We also prove that there is a graph K with two involutions γ1 and γ2 such that (K,γ1) is a Stiefel-Whitney test graph, but (K,γ2) is not. These are answers to some of the problems suggested by Kozlov.