
Re: 4 colors problem
Posted:
Mar 5, 2014 5:56 PM


"The 4color theorem doesn't claim that every partial 4coloring of a planar graph can be completed to a full 4coloring."
The unit pattern (in the previous post I called it the unit function, but meant "unit pattern") guarantees a full 4coloring, so it's reliable.
"The claim of the theorem is that of _all_ possible valid colorings, at least one of them uses at most 4 colors."
The unit pattern guarantees to use no more than 4 colors.
"Thus, to show that a given planar graph is a counterexample to the 4color theorem, it's not sufficient to produce a partial 4coloring which can't be completed to a full coloring."
The unit pattern uses a partial 4coloring which completes to a full coloring for anysized map and for any number of countries.
"You would have to show that _all_ possible valid colorings use more than 4 colors, not just the one that you think makes sense."
What does this mean?
With a=blue, b=green, c=red, d=yellow, Color the following map accordingly,
c d a b a b c d c d a b a b c d c d a b
which fulfills the promise of the 4 color theorem. Now, let's say the second and third countries on the third row unites, so they have to share the same color. Now we have,
c d a b a b c d c d d b a b c d c d a b
How would you color this to fulfill the 4 color theorem using any coloring order?

