The exhaustive classification of planar 4-critical $P_7$-free graphs

A graph is planar if it can be embedded in the plane without crossings. A graph is 4-critical if its chromatic number is 4 and deleting any edge lowers its chromatic number. A graph is $P_7$-free if it has no induced subgraph isomorphic to the path $P_7$. Exhaustive classification conjecture. The 52 graphs from Table~ are the only planar 4-critical $P_7$-free graphs.

This conjecture concerns the unresolved general planar $P_7$-free case. The authors determined all planar 4-critical $P_7$-free graphs with at most 30 vertices, and the largest graph found in that range has 13 vertices; whether any further graphs exist remains open.