Algebraically coloring a map

I noticed you can take an uncolored map like:

You can set up a set of equations to solve for, as follows... if region X is next to regions Y, Z,... one equation is X=Y*Z*...
For the above:

Note that C is not considered adjacent to D...

And there are a few solutions to the above, but we can add extra conditions such as A!=B, because A and B are adjacent and so on until we get down to one solution...

Now we can use a map of these complex numbers to color values by putting a color wheel on the complex plane:

And use that as a key to fill in the colors:

In this case it gave 5 colors not 4 as for the famous 4 color theorem, but I still think it's an interesting process...