Four color theorem
Higher surfaces

One can also consider the coloring problem on surfaces other than the plane. The problem on the sphere or cylinder is equivalent to that on the plane. For closed (orientable or non-orientable) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

where the outermost brackets denote the floor function.

Alternatively, for an orientable surface the formula can be given in terms of the genus of a surface, g:

This formula, the Heawood conjecture, was proposed by P. J. Heawood in 1890 and, after contributions by several people, proved by Gerhard Ringel and J. W. T. Youngs in 1968. The only exception to the formula is the Klein bottle, which has Euler characteristic 0 (hence the formula gives p = 7) but requires only 6 colors, as shown by Philip Franklin in 1934.

For example, the torus has Euler characteristic χ = 0 (and genus g = 1) and thus p = 7, so no more than 7 colors are required to color any map on a torus. This upper bound of 7 is sharp: certain toroidal polyhedra such as the Szilassi polyhedron require seven colors.

A Möbius strip requires six colors (Tietze 1910) as do 1-planar graphs (graphs drawn with at most one simple crossing per edge) (Borodin 1984). If both the vertices and the faces of a planar graph are colored, in such a way that no two adjacent vertices, faces, or vertex-face pair have the same color, then again at most six colors are needed (Borodin 1984).

For graphs whose vertices are represented as pairs of points on two distinct surfaces, with edges drawn as non-crossing curves on one of the two surfaces, the chromatic number can be at least 9 and is at most 12, but more precise bounds are not known; this is Gerhard Ringel 's Earth–Moon problem.
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