The Four Color Theorem
(4CT)
Emily Mis
Discrete Math Final Presentation
Origin of the 4CT
• First introduced by Francis Guthrie in the early 1850’s
– 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)
Graph Theory for the Four Color Conjecture
Graph Theory for the Four Color Conjecture
Graph Theory for the Four Color Conjecture
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
“Proving” the Conjecture
• A. B. Kempe in 1879
– Found to be flawed by Heawood in 1890
– Introduced a new technique now called Kempe’s
chains
Kempe’s Chains
A four-sided region R is
surrounded by Regions 1 - 4
that have already been colored
by the four available colors
Kempe’s 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.
“Proving” the Conjecture
• A. B. Kempe in 1879
– Found to be flawed by Heawood in 1890
– Introduced a new technique now called Kempe’s
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
“Proving” the Conjecture
• A. B. Kempe in 1879
– Found to be flawed by Heawood in 1890
– Introduced a new technique now called Kempe’s
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
• 1900’s brought proofs on limited sets of
regions
– Increased to a 90-region proof in 1976 by Mayer
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?"
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?"
The New Proof of the 4CT
• Completed by Appel and Haken in 1976
– Based on Kempe’s chains
– Required 1200 hours of computation
• Used mostly to perform reductions and
discharges on planar configurations using
Kempe’s 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
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
We’re 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”?
Works Cited
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Thomas, Robin (1998) An Update on the Four-Color Theorem, Notices
of the AMS, 45: 7:848-859
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, 38:3:27-35
Cayley (1879) On the colouring of maps, Proceedings of the Royal
Geographical Society and Monthly Record of Geography New Monthly
Series, 1:4:259-261
Robertson et al (1996) A New Proof of the Four-Colour Theorem,
Electronic Research Announcements of the AMS, 2:1:17-25
Mitchem, John (1981) On the History and Solution of the Four-Color
Map Problem, The Two-Year College Mathematics Journal, 12:2:108116
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,
56:3:346-348
Saaty, Thomas (1967) Remarks on the Four Color Problem: the Kempe
Catastrophe, Mathematics Magazine, 40:1:31-36