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Color Problems

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Outline of the Five Color Theorem Proof

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(See Stein, *Mathematics: A Man Made Universe *or Kinsey and Moore,* Symmetry, Shape and Space.*)

**Theorem:** Any map on the sphere (or the plane) can
be colored with **five or fewer** colors.
**Definition:** A map is called *regular* if
every vertex has degree 3.

"**Lemma 4**": **If** we can
color every regular map on the sphere with five or fewer colors **then**
we can color any map on the sphere with five or fewer colors. Proof.

**"Lemma 5": ** A regular map on the
sphere has at least one region with 5 or fewer edges.

**Proof:** We can count the edges with the vertices, and since there are exactly 3 edges at every vertex we have

3V =2E or V = 2/3 E (*).

But since V+R=E+2 we have R-2= E-V,

and by the equality (*) we have

E - 2/3E = E -V = R -2,

so 1/3E = R - 2.

Now we suppose all regions
have 6 or more edges.

Then 6R ≤ 2E , R ≤ 1/3E .

But then

R ≤ 1/3E = R-2 or R ≤ R-2 which is absurd.

End of proof !

[See also Kinsey amd Moore for a treatment for coloring on any surface.]

"**Lemma 6**": Any regular map covering the sphere
(or the plane) can be colored with five or fewer colors.

*The Four Color Theorem is not as easy!*