Friday, November 14, 2008

Hypothesis about the 4 Color Theorem: Correction

There a problem with a previous statement I made. I stated that the outer circle forms another "layer", implying that the new layer borders only the layer above and below it. This would mean every element has a single left and a single right neighbor and multiple linearly arranged neighbors above and below. This is only true for some maps. But with more complex maps, however you divide the map into "layers", you will end up with: elements that extend themselves into several layers (not just the ones above and below);  or elements that border more than one element in the same layer (what I call a "wrap around" ).

It's terribly annoying, because it's like I can sense the reason for why only 4 colors are required but I can't quite see it! I guess this is what pretty much defines a theorem versus a proof... My hope was that arranging everything into isolated and linear numeric sequences (some form of "layering") would provide me with a clear vision of the why. But these so called "wrap arounds" keep throwing me off. 

If I could just stop bouncing through Kaliningrad!

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