Summary. A planar graph, which is a graph that admits a planar embedding, has too little structure for our purposes because it does not specify a particular embedding. A plane graph carries a fixed embedding, which gives it a topological structure where combinatorics alone should suffice. A hypermap gives just the right amount of structure. It is a purely combinatorial object, but carries information that the planar graph lacks by encoding the relations among nodes, edges, and faces. This chapter is about hypermaps.

In the original proof of the Kepler conjecture, the basic combinatorial structure was that of a planar map, as defined by Tutte [47]. Although planar maps appear throughout that proof, they are lightweight objects, in the sense that no significant structural results are needed about them.

Gonthier makes hypermaps the fundamental combinatorial structure in his formal proof of the four-color theorem [16]. His formal proof eliminates topological arguments such as the Jordan curve theorem in favor of purely combinatorial arguments. When I learned of Gonthier's work, I significantly reorganized the proof by replacing planar maps with hypermaps, making them heavyweight objects, in the sense that significant structural results about them are needed.

As a result of these changes, many parts of the proof that were originally done topologically can now be done combinatorially, a change that significantly reduces the effort required to formalize the proof. These changes also make it possible to treat rigorously what was earlier done by geometric intuition.