1. The Four Color Theorem (4CT)
Emily Mis
Discrete Math Final Presentation

2. 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)
Francis was a law student at University College London
Brother Frederick (later to become a physicist) was a current student of De Morgan’s
Noticed in a map of England

3. Graph Theory for the Four Color Conjecture

4. Graph Theory for the Four Color Conjecture

5. Graph Theory for the Four Color Conjecture

6. 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

7. “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
4-sided region R
The colors a,b,c, and d have been used to color the surrounding areas
Look for a b-d chain (region 4 and region 2) if they are not in a chain of alternating colors, then the chain colors
Can be reversed so that region 4 and region 2 are both colored b -- then only 3 colors are used in the areas around
R and R can be colored in d.

8. Kempe’s Chains
A four-sided region R is surrounded by Regions that have already been colored by the four available colors

9. Kempe’s Chains
A four-sided region R is surrounded by Regions 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.

10. “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

11. “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
1900’s 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

12. 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?"

13. 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?"

14. 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

15. 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

16. 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”?

17. Works Cited
Thomas, Robin (1998) An Update on the Four-Color Theorem, Notices of the AMS, 45: 7:
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:
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:
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:
Saaty, Thomas (1967) Remarks on the Four Color Problem: the Kempe Catastrophe, Mathematics Magazine, 40:1:31-36