Four-Color Theorem Re-proved After Nearly a Century, Unlocking Faster Graph Coloring
The Four-Color Theorem Gets a Rare New Proof
In 1852, Francis Guthrie wondered if four colors suffice to color any map. After a flawed 1879 proof and a controversial 1976 computer-assisted proof, a team of six mathematicians has produced a new computer proof of the four-color theorem. Posted online in March 2026, their work not only re-proves the theorem but also reveals a more efficient way to color maps and graphs, offering fresh insights into planar graphs.
“We can’t really work on a small project,” Kawarabayashi recalled thinking.
- pvillano
As much as I hate it, I don't think we'll ever get a proof of the four color theorem that isn't enumerating cases.
When you have an integral or the sum of an infinite series that comes out to pi, you know there must be some satisfying explanation involving a circle.
Contrast with "Examples of patterns that eventually fail" on math stackexchange[^1]. When a pattern ends at 906150257, you don't really expect the proof for that to be something beautiful. The reason for the exact value of an upper bound is that it isn't smaller and it isn't bigger.
There's a relationship between e, i, pi, and -1 comes from a deeper relationship between complex numbers and rotation.
The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.
- andrewla
The reason that the four-color theorem remains a fascinating problem for so many people and the source of so many cranks is not that the problem is easy to express.
It's that Kempe's proof, though in the end incorrect, is very elegant and digestible. Seeing why it is incorrect is actually really tricky! And once you have seen why it is incorrect it feels like it is just one or two corrections away from being a correct proof.
If you have not read the outline of Kempe's proof (the wikipedia article does a pretty fair job of it) then you definitely should. I promise you that you will waste at least a little bit of time trying to figure out how to fix the proof in two lines and become a world famous mathematician instantly.
- pvillano
It better not have 100s of individually checked configurations
Edit: damn it.
I was just thinking last night about the four color theorem in the context of the recent Navier-Stokes drama, and Tao's Mastodon post on the uselessness of inscrutable computer-generated formalizations. I would love for an AI company find a proof of the four-color theorem without individually checked configurations, and optimize it for human comprehensibility.