Four color theorem

So in the end, the proof was false.

So this wasn't a counterexample to the four-color theorem but just a way in which Kempe's wasn't complete enough to actually prove it.

Exactly.

So it made clear that any solution to the four-color conjecture at that point would be far more complicated than the solution that Kempe had put forward.

Graph theorists are interested in graphs that map onto all different sorts of shapes, and this general structure is really common.

So now that you've identified these configurations within those areas and you have a technique for identifying new ones, the authors are really excited about other kinds of questions that they can tackle using these techniques that just weren't possible because in prior proofs, the configurations were so specific to this one question of four-coloring a planar graph, and we're focusing on kind of like weird parts of those graphs that don't have that sort of generalizable aspect to them.

them.We mentioned at the beginning and throughout planar graphs, which means on a flat surface or on a sphere, but if you try to apply this to a torus, which is a donut, a topologically different surface, the four-color theorem doesn't apply, and I think your story said that you need seven to color a donut-

Yeah, exactly