Re: Map Coloring

Ok I've created I guess it's kind of an algorithm to show how this
would work could someone help me refine this into a proof.

If the map has a Hamilton cycle find it. Draw it with a dashed line.
The cycle divides the plane into an inside and outside.
Two colors are needed inside and two are needed for the outside.
Start with one color on some region on the inside.
Every time you cross a solid line switch to the other color.
Repeat this process until the entire inside is colored.
Now pick a region in the outside. Color it the third color.
Every time you move to a new region and cross over a solid line switch
Crossing dashed lines will switch inside and outside regions.
The fourth color is the entire outside region.

thanks for the help

Could someone help me try to prove it by induction on that the inside
region needs two colors and the outside region needs two colors as
well. again thanks for the help.