----- Original Message -----

From: "acroorca2002" <orc@orcoast.com>

Sent: Saturday, December 06, 2003 1:32 PM

Subject: [BoundaryPoint] Re: Four Color Maps

> well i know this has been a perennially interesting question eric

> but how about first showing us a theoretical

example of a topology

> requiring a 5th color
> & then at least

we will know what to seek on your behalf in reality

>
> otherwise i

think we may only continue to peck away at this huge

> offering
>

& finally shrug our shoulders

> even if we had the geographical

knowledge to apply to it

Here is my stab at it. Hopefully, one day the below picture will hang in the Louvre. Start with A, B and C. Color them each a seperate color. Now, surround by D. It must be a fourth color. Now make E, F and G which must be red, yellow and blue, though not necessarily in that order. Now suppose E claims a (purple) swath of land cutting two of the boundaries AC and CB. Come to think of it, it is actually unnecessary for F and G to exist. EFG could be one big red ring on the outside. Then to make that swath, it would have to be a new color, so EFG must now be that color. So the new map would show A, B, C and D, as is and change my EFG outside to just one E and make it purple.

> --- In

href="mailto:BoundaryPoint@yahoogroups.com">BoundaryPoint@yahoogroups.com, "Eric Choate" <choatune@y...>

>

wrote:

> > The presence of the Azerbaijani exclave (and many others)

means

> that
> > the present set of national boundaries on

earth does not satisfy

> the
> > prerequesites to apply the

Four Color Theorem. Nevertheless, a

> four
> > color map

of the Caucasus can still be made. It's just that it's

> >

possible to draw borders with exclaves that do make it impossible

> to
> > color countries with only four colors.
> >
> >

But there are national exclaves that I am unaware of. Most would

>

> > small enough that they would be omitted on a world map that is

of a

> > size that I could find. Does anyone know of an exclave

that if it

> > were included would force more than four colors?
> >
> > Similarly, I'm not that familiar with the
> > states/provinces/cantons/oblasts/etc., of every country. is there
> a
> > country that cannot produce a four-color map of its

political

> > subdivisions?
> >
> > What

about US states with counties? Or I'm sure with all the

> >

gerrymandering in state legislatures, a congressional district map

> >

somewhere needs a fifth color. Anyone know of one?

> >
> > eric
>
>
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