Quantum Gravity 200.9: Chromatic Number Via Total Gaussian Curvature

[OsherD]

>From Osher Doctorow

Wolfram's and Wikipedia's "Heawood Conjecture" yield the now proven
relationship between the Chromatic Number c(g) and the Euler
characteristic X which holds except for sphere, plane, and Klein
bottle:

1) c(g) = floor[ 7/2 + sqrt(49 - 24X)/2]

where g is genus (in particular number of holes if all else constant),
floor(x) = greatest integer < = g.

Since the Gauss-Bonnet Theorem gives the Total Gaussian Curvature K_t
by:

2) K_t = 2pi X(M), M compact boundryless 2-dimensional Riemannian
manifold, for example surface in 3-space

(see for example "Gauss-Bonnet formula" Wolfram), and since the Euler
characteristic X (see Wikipedia's "Euler characteristic) is:

3) X = 2 - 2g

we have a direct equation relating the Chromatic Number and the
Gaussian Curvature under the above conditions.

For example, for a Torus, g = 1 (one hole), X = 0 from (3), and c(g) =
7 from (1), so K_t is 0 from (2).

Osher Doctorow