OBSTRUCTIONS FOR THREE-COLORING AND LIST THREE-COLORING H-FREE GRAPHS

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-3-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.