Colouring classification on H-subgraph-free graphs

Determine a complete computational complexity classification of Graph Colouring on H-subgraph-free graphs: for each fixed graph H, decide whether Graph Colouring restricted to the class of graphs that forbid H as a subgraph is polynomial-time solvable or NP-complete, thereby resolving the status that remains open for these finitely-bounded monotone classes.

Background

The paper studies complexity on finitely-bounded monotone (subgraph-closed) classes via a C123 framework. While many problems admit full dichotomies in this setting, Graph Colouring is singled out as a prominent exception not currently subsumed by the framework.

The authors explicitly note that even the more modest goal of classifying Colouring for H-subgraph-free graphs (i.e., forbidding a single subgraph H) remains unresolved, highlighting a longstanding gap compared to problems like Independent Set or List Colouring that fit the framework and admit classifications.