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) 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

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 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.

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 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

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 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, 12:2:108-116 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, 56:3:346-348 Saaty, Thomas (1967) Remarks on the Four Color Problem: the Kempe Catastrophe, Mathematics Magazine, 40:1:31-36