FourColor Theorem

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Four-Color Theorem

• December 1, 2005

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History

• First theorized in 1840 by August Ferdinand
  Mobius
• Guthrie, in 1853, started to analyze what Mobius
  first conjectured
• Guthrie wrote to his brother Frederick at the
  University College in London.

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More History

• Frederick took the problem to his professor,
  Augustus DeMorgan.
• DeMorgan then wrote to Sir William Rowan
  Hamilton.
• The problem came back to Guthrie, who added more
  to the theory, but it was not officially written
  down until 1878.

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Prove It

• Guthrie discovered that any map in a plane can be
  colored using at most four colors.
• He discovered that the map had to contain parts
  with common boundaries, and that the map could be
  colored in such a way that no region bounded by
  another region shared the same color.

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Fallacious Proofs

• Kempe-1879
• Tait-1880
• Petersen-1891
• Heawood

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And the Winner is

• Kenneth Appel and Wolfgang Haken, in 1977,
  created a computer program that provided a proof
  of the four-color theorem.

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Only One Problem

• The computer program that they created took about
  1200 hours to run and is over a few thousands of
  lines long!!!

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Proper Definition

• FOUR COLOUR THEOREM.  For any subdivision of the
  plane into non-overlapping regions, it is always
  possible to mark each of the regions with one of
  the numbers  0, 1, 2, 3,  in such a way that no
  two adjacent regions receive the same number.

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Simple Diagrams
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And Now Some Practice?

• The Most Colorful Math of All

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More Simple Diagrams
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DeMorgans Theories

• DeMorgan claimed that if a fifth vertex should be
  added to the diagram, only three of the four
  vertices would be able to connect to this new
  vertex.

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Word of Caution

• The four color theorem applies only to planar (or
  spherical) "maps", not to regions drawn on other
  surfaces.
• Be careful not to confuse the four-color theorem
  with graph coloring problems involving the
  coloring of vertices as opposed to regions.
• http//bhs.broo.k12.wv.us/discrete/4Color.htm

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Now, Some Difficulty
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Algebraically

• Algebra in the Four-Color Theorem

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Now Some Coloring?
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Some Mathematical Humor

• Martin Gardner, in 1975, claimed that the picture
  on the prior slide could not be shaded using only
  four colors.
• However, he made this claim on April Fools Day.
  
• In fact the picture is four-colorable and was
  proven so by Wagon in 1998.

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New Proof

• Neil Robertson, Daniel P. Sanders, Paul Seymour
  and Robin Thomas have created a manmade proof of
  the four color theorem.
• They originally tried to prove Appel and Hakens
  proof by hand but found it too complicated and
  tedious.

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An Interactive Website

• Four-Color Theorem Puzzle

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Sources

• http//mathworld.wolfram.com/Four-ColorTheorem.htm
  l
• http//www.cs.uidaho.edu/casey931/mega-math/gloss
  /math/4ct.html
• http//www.mathpages.com/home/kmath266/kmath266.ht
  m
• http//www.math.gatech.edu/thomas/FC/fourcolor.ht
  ml
• http//www.geocities.com/dharwadker/
• http//bhs.broo.k12.wv.us/discrete/4Color.htm
• http//www.puzzle.jp/four_color_problem-e.html