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Topic: Could we apply the "Four Color Theorem" (Read 416 times) |
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Benny
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Could we apply the "Four Color Theorem"
« on: Sep 7th, 2009, 10:55am » |
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Question :
What is the minimum number of colors needed to color the plane so that no two points of the same color are exact one unit apart?
http://www.mathpuzzle.com/chrompln.html
It is at most 7 ... could it be 4, 5, or 6?
Could it be 4? Could we apply the "Four Color Theorem" here?
http://www.mathpages.com/home/kmath266/kmath266.htm
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Obob
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Re: Could we apply the "Four Color Theorem&qu
« Reply #1 on: Sep 9th, 2009, 4:57am » |
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Despite the terminology, the plane is likely not a planar graph (a graph in the plane without crossing edges). At the very least, if it is planar then some wierd embedding in the plane would be needed. If the four color theorem applied, we would have known the answer to be 4 already.
Since no unit-distance graph is known to have chromatic number higher than 4, it is certainly plausible that the number of colors needed could be 4. There is no reason that a coloring of the plane would have to be nearly as nice as the hexagonal coloring that demonstrates 7 colors suffice.
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| « Last Edit: Sep 9th, 2009, 4:59am by Obob » |
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