## Thursday, October 1, 2015

### vague idea of simple proof of four color theorem

I thought if you have a map like so, regions enclosed by edges:
There are two basic cases, the one below, left is where there is at least one region that touches two or more edges. and below right no country touches the edge at more than one border...
So I thought we can make a border around these maps and color one country as follows...

The above you use the country touching more than one border to divide the map into 2, parts one with a red  and orange border and the other with a blue and orange border...

And the above you color one country touching the border and divide the border like so into one smaller map with an orange and red border...

So you see with any map with these kinds of 2 color borders it's possible to color in one country and the remaining unfilled area of the map is again a map with a 2 color border... So I think the proof would go that you assume that your original map is 4 colorable, and gradually color one country at a time with the above criteria until you are left with a map with only one country with a two color border, which would of course be 4 colorable... I know there are complications as to when you need the fourth color but this is just a rough idea...