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Re: 4 colors problem
Posted:
Mar 5, 2014 5:56 PM
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"The 4-color theorem doesn't claim that every partial 4-coloring
of a planar graph can be completed to a full 4-coloring."
The unit pattern (in the previous post I called it the unit function, but meant "unit pattern") guarantees a full 4-coloring, 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 4-color theorem, it's not sufficient to produce a partial
4-coloring which can't be completed to a full coloring."
The unit pattern uses a partial 4-coloring which completes to a full coloring
for any-sized 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?
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