I suppose it's not exactly mainstream, but Kronheimer and Mrowka are among those trying to prove the four color theorem using instanton homology and gauge theor

I suppose it's not exactly mainstream, but Kronheimer and Mrowka are among those trying to prove the four color theorem using instanton homology and gauge theory. It's certainly not finite graph theory, though (their idea of a 3-regular graph is the singular locus of a certain kind of orbifold!)