A team of six researchers has produced a new computer-assisted proof of the four-color theorem, the result that every contiguous map can be colored with four colors so that neighboring regions differ. The proof was posted online in March 2026 and is due to be presented at the annual Foundations of Computer Science conference in New York City next week.
The work builds on a problem that has challenged mathematicians since the 19th century. An early proof by Alfred Bray Kempe, announced in 1879, was shown to have a flaw 11 years later. The theorem was eventually proved in 1976 by Kenneth Appel and Wolfgang Haken using a computer to check a large set of graph configurations. A later computer-assisted proof, published in 1997, reduced the number of configurations that needed checking.
The latest team includes graph theorists Carsten Thomassen, Mikkel Thorup, Ken-ichi Kawarabayashi and Bojan Mohar, along with researchers Yuta Inoue and Atsuyuki Miyashita. Their approach identifies a set of 8,202 configurations in planar graphs and allows many to be reduced in parallel. The researchers say this makes the coloring process substantially more efficient than the earlier method, which required a number of steps that grew quadratically with the graph’s number of vertices.
Rather than concentrating only on areas with clusters of vertices that have few connections, the team examined flatter parts of a graph, where vertices are connected in triangular arrangements. The work also yielded new tools for studying structural properties of planar graphs, with potential applications to coloring problems on other surfaces, including a torus.
The new proof does not resolve a long-standing desire among some mathematicians for a shorter, computer-free explanation of why four colors suffice. Thomassen said he would like to see a proof that does not rely on a computer and intends to keep considering the question.
Sources: Smithsonian Magazine.