  We prove that $\{\overline{K_3}, H\}$-free graphs are not counterexamples to
Hadwiger's Conjecture, where $H$ is any one of 33 graphs on seven, eight, or
nine vertices, or $H=K_8$. This improves on past results of
Plummer-Stiebitz-Toft, Kriesell, and Bosse. The proofs are mostly
computer-assisted.
