  A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We
characterize all graphs $H$ for which there are only finitely many minimal
non-three-colorable $H$-free graphs. Such a characterization was previously
known only in the case when $H$ is connected. This solves a problem posed by
Golovach et al. As a second result, we characterize all graphs $H$ for which
there are only finitely many $H$-free minimal obstructions for list
3-colorability.
