Obstructions for three-coloring graphs with one forbidden induced subgraph

Abstract

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.