The Four Color Theorem (4CT)

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The Four Color Theorem(4CT)

• Emily Mis
• Discrete Math Final Presentation

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Origin of the 4CT

• First introduced by Francis Guthrie in the early
  1850s
• Communicated to Augustus De Morgan in 1852
• First printed reference of the theorem in 1878 in
  the Proceedings of the London Mathematical
  Society
• The original problem was stated to be
• the greatest number of colors to be used in
  coloring a map so as to avoid identity of color
  in lineally contiguous districts is four.
• -Frederick Guthrie to the
  Royal Society of Edinburgh (1880)

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Graph Theory for the Four Color Conjecture
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Graph Theory for the Four Color Conjecture
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Graph Theory for the Four Color Conjecture
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Any Planar Map Is Four-Colorable

• Planar graph - a graph drawn in a plane without
  any of its edges crossing or intersecting
• Each vertex (A,B,C,D,E) represents a region in a
  graph
• Each edge represents regions that share a boundary

In any plane graph each vertex can be assigned
exactly one of four colors so that adjacent
vertices have different colors. -Four-Color
Conjecture
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Proving the Conjecture

• A. B. Kempe in 1879
• Found to be flawed by Heawood in 1890
• Introduced a new technique now called Kempes
  chains

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Kempes Chains
A four-sided region R is surrounded by Regions 1
- 4 that have already been colored by the four
available colors
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Kempes Chains
A four-sided region R is surrounded by Regions 1
- 4 that have already been colored by the four
available colors
First consider all regions colored b and
d. Either there is a chain of regions colored b
and d connecting Region 2 and Region 4, or no
such chain exists. If no chain exists, you can
change the color of either Region 4 or Region 2
in order to free up the other color for the
center region R.
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Proving the Conjecture

• A. B. Kempe in 1879
• Found to be flawed by Heawood in 1890
• Introduced a new technique now called Kempes
  chains

• P. G. Tait in 1880
• Found to be flawed by Peterson in 1891
• Found an equivalent formulation of the 4CT in
  terms of three-edge coloring

No two edges coming from the same vertex share
the same color
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Proving the Conjecture

• A. B. Kempe in 1879
• Found to be flawed by Heawood in 1890
• Introduced a new technique now called Kempes
  chains
• 1900s brought proofs on limited sets of regions
• Increased to a 90-region proof in 1976 by Mayer

• P. G. Tait in 1880
• Found to be flawed by Peterson in 1891
• Found an equivalent formulation of the 4CT in
  terms of three-edge coloring

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other concerned parties

• Sir William Hamilton
• Arthur Cayley
• Lewis Carroll

"A is to draw a fictitious map divided into
counties. B is to color it (or rather mark the
counties with names of colours) using as few
colours as possible. Two adjacent counties must
have different colours. A's object is to force B
to use as many colours as possible. How many can
he force B to use?"
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other concerned parties

• Sir William Hamilton
• Arthur Cayley
• Lewis Carroll

"A is to draw a fictitious map divided into
counties. B is to color it (or rather mark the
counties with names of colours) using as few
colours as possible. Two adjacent counties must
have different colours. A's object is to force B
to use as many colours as possible. How many can
he force B to use?"
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The New Proof of the 4CT

• Completed by Appel and Haken in 1976
• Based on Kempes chains
• Required 1200 hours of computation
• Used mostly to perform reductions and discharges
  on planar configurations using Kempes original
  idea of chains
• Introduced a collection of 1476 reducible
  configurations
• These configurations are an unavoidable set that
  must be tested to show that they are reducible
• No member of this set can appear in a minimal
  counterexample

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Discharging - Why do it?

• Discharging -- moving a charge along a circuit
  of connected vertices in order to cancel positive
  and negative values as much as possible
• Sites where a positive value remains are often
  part of a reducible configuration
• G is the smallest maximal plane graph which
  cannot be four-colored
• each vertex in G gets a charge (6-deg v)
• from Euler, we know that the sum of all G is 12
• A charge is then moved around the circuit to
  change the charges of individual vertices
• Discharging is used to show that a certain set S
  is an unavoidable set

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Were not in Kansas anymore

• A new version of the computer-based proof was
  produced by Robertson, Sanders, Seymour and
  Thomas in 1996
• Used a quadratic algorithm for four-color planar
  graphs
• Decreased the size of the unavoidable set to 633
• The fears
• Is this a movement towards computer-based proof
  for traditional mathematical proofs?
• Does this proof qualify as a proof based on the
  original definition of a proof?

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Works Cited

• Thomas, Robin (1998) An Update on the Four-Color
  Theorem, Notices of the AMS, 45 7848-859
• Brun, Yuriy The Four Color Theorem, MIT
  Undergraduate Journal of Mathematics pp 21-28
• Calude, Andreea (2001)The Journey of the Four
  Colour Theorem Through Time, The New Zealand
  Mathematics Magazine, 38327-35
• Cayley (1879) On the colouring of maps,
  Proceedings of the Royal Geographical Society and
  Monthly Record of Geography New Monthly Series,
  14259-261
• Robertson et al (1996) A New Proof of the
  Four-Colour Theorem, Electronic Research
  Announcements of the AMS, 2117-25
• Mitchem, John (1981) On the History and Solution
  of the Four-Color Map Problem, The Two-Year
  College Mathematics Journal, 122108-116
• Bernhart (1991) Review of Every Planar Map is
  Four Colorable by Appel and Haken, American
  Mathematical Society, Providence RI, 1989
• May, Kenneth (1965) The Origin of the Four-Color
  Conjecture, Isis, 563346-348
• Saaty, Thomas (1967) Remarks on the Four Color
  Problem the Kempe Catastrophe, Mathematics
  Magazine, 40131-36