Obstructions for three-coloring graphs with one forbidden induced subgraph

The complexity of coloring graphs without long induced paths is a notorious problem in algorithmic graph theory, an especially intruiging case being that of 3-colorability. So far, not much was known about certification in this context.

We prove that there are only finitely many 4-critical P₆-free graphs, and give the complete list that consists of 24 graphs. In particular, we obtain a certifying algorithm for 3-coloring P₆-free graphs, which solves an open problem posed by Golovach et al. Here, P₆ denotes the induced path on six vertices.

Our result leads to the following dichotomy theorem: if H is a connected graph, then there are finitely many 4-critical H-free graphs if and only if H is a subgraph of P₆. This answers a question of Seymour. The proof of our main result involves two distinct automatic proofs, and an extensive structural analysis by hand.