The Five Color Theorem (without Kempe chains)

Submission for the #SoME2 competition. Most animations were done in manim (https://www.manim.community/), and the 3d images were rendered using svg3d (https://github.com/prideout/svg3d).

Proofs:

The degree of a vertex or a face is the number of edge incidences (edges that meet a vertex or face twice are counted twice). Each edge has two vertex incidences and two face incidences, so both the sum of vertex degrees and the sum of face degrees equals 2E.

Eliminating "bad borders" means that each vertex has degree at least 3 (unless there are just two faces). Then, the sum of degrees is at least 3V, and hence 3V ≤ 2E. Plugging that into Euler's formula to eliminate V yields the inequality E ≤ 3F-6 (when F ≥ 3).

You can't have a neighborly map of six countries because you would need 6 choose 2 = 15 borders, but 3(6)-6 = 12. The average degree of the faces is 2E/F ≤ 6 - 12/F, which is less than 6.

Errata:

Whoops, looks like Franklin's paper was actually from 1934, sorry!