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 $\gamma_1$ and $\gamma_2$ such that $(K,\gamma_1)$ is a Stiefel-Whitney test graph, but $(K,\gamma_2)$ is not. These are answers to some of the problems suggested by Kozlov.