Classification of Colouring on H-subgraph-free Graphs

Classify the computational complexity of the graph Colouring problem on H-subgraph-free graphs: for each fixed forbidden subgraph H, determine whether Colouring on the class of H-subgraph-free graphs is polynomial-time solvable or NP-complete, thereby providing a complete complexity classification for this setting.

Background

The paper studies complexity dichotomies for graph problems on finitely-bounded monotone classes defined by forbidden subgraphs, via the C123-framework. While many problems fit the framework and admit full classifications on such classes, the authors note that Colouring does not, and its complexity is not fully understood even in the simpler case of classes forbidding a single subgraph H.

This statement highlights a long-standing gap: although classifications exist for problems like Independent Set, Dominating Set, and List Colouring, a unified classification for Colouring on H-subgraph-free classes is still unresolved, motivating further research beyond the current framework.