nLab four-colour theorem

Idea

The four-colour theorem, or four colour map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be coloured using at most four colours so that no two adjacent regions have the same colour. Regions are considered adjacent if they share a boundary segment.

Related concepts

• Kepler conjecture

References

See also

• Wikipedia, Four color theorem

On the logical equivalence between the four-colour theorem and a statement about the transition from the small N limit to the large N limit for Lie algebra weight systems on Jacobi diagrams via the 't Hooft double line construction:

• Dror Bar-Natan, Lie Algebras and the Four Color Theorem, Combinatorica 17-1(1997) 43–52 (arXiv:q-alg/9606016, doi:10.1007/BF01196130)

On a reformulation of the four-colour theorem as a statement about typing in lambda calculus:

• Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps (arXiv:1512.06751).

• Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs (arXiv:1804.10540).

Fawcett showed that the four-color theorem is equivalent to the existence of a left adjoint (that is, preserving epimorphisms) from planar graphs to Set:

• Barry Fawcett, A categorical characterization of the four-colour theorem, Canad. Math. Bull. Vol 29 (4), 1986.