Presentation on theme: "The Four Color Theorem (4CT)"— Presentation transcript:
1The Four Color Theorem (4CT) Emily MisDiscrete Math Final Presentation
2Origin of the 4CTFirst introduced by Francis Guthrie in the early 1850’sCommunicated to Augustus De Morgan in 1852First printed reference of the theorem in 1878 in the Proceedings of the London Mathematical SocietyThe 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 LondonBrother Frederick (later to become a physicist) was a current student of De Morgan’sNoticed in a map of England
6Any Planar Map Is Four-Colorable Planar graph - a graph drawn in a plane without any of its edges crossing or intersectingEach vertex (A,B,C,D,E) represents a region in a graphEach 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 1879Found to be flawed by Heawood in 1890Introduced a new technique now called Kempe’s chains4-sided region RThe colors a,b,c, and d have been used to color the surrounding areasLook for a b-d chain (region 4 and region 2) if they are not in a chain of alternating colors, then the chain colorsCan be reversed so that region 4 and region 2 are both colored b -- then only 3 colors are used in the areas aroundR and R can be colored in d.
8Kempe’s ChainsA four-sided region R is surrounded by Regions that have already been colored by the four available colors
9Kempe’s ChainsA four-sided region R is surrounded by Regions that have already been colored by the four available colorsFirst 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 1879Found to be flawed by Heawood in 1890Introduced a new technique now called Kempe’s chainsP. G. Tait in 1880Found to be flawed by Peterson in 1891Found an equivalent formulation of the 4CT in terms of three-edge coloringNo two edges comingfrom the same vertexshare the same color
11“Proving” the Conjecture A. B. Kempe in 1879Found to be flawed by Heawood in 1890Introduced a new technique now called Kempe’s chains1900’s brought proofs on limited sets of regionsIncreased to a 90-region proof in 1976 by MayerP. G. Tait in 1880Found to be flawed by Peterson in 1891Found an equivalent formulation of the 4CT in terms of three-edge coloring
12other concerned parties Sir William HamiltonArthur CayleyLewis Carroll"A is to draw a fictitious map divided into counties.B is to color it (or rather mark the counties with names ofcolours) using as few colours as possible.Two adjacent counties must have different colours.A's object is to force B to use as many coloursas possible. How many can he force B to use?"
13other concerned parties Sir William HamiltonArthur CayleyLewis Carroll"A is to draw a fictitious map divided into counties.B is to color it (or rather mark the counties with names ofcolours) using as few colours as possible.Two adjacent counties must have different colours.A's object is to force B to use as many coloursas possible. How many can he force B to use?"
14The New Proof of the 4CT Completed by Appel and Haken in 1976 Based on Kempe’s chainsRequired 1200 hours of computationUsed mostly to perform reductions and discharges on planar configurations using Kempe’s original idea of chainsIntroduced a collection of 1476 reducible configurationsThese configurations are an unavoidable set that must be tested to show that they are reducibleNo member of this set can appear in a minimal counterexample
15Discharging - Why do it?Discharging -- moving a charge along a circuit of connected vertices in order to cancel positive and negative values as much as possibleSites where a positive value remains are often part of a reducible configurationG is the smallest maximal plane graph which cannot be four-coloredeach vertex in G gets a charge (6-deg v)from Euler, we know that the sum of all G is 12A charge is then moved around the circuit to change the charges of individual verticesDischarging is used to show that a certain set S is an unavoidable set
16We’re not in Kansas anymore A new version of the computer-based proof was produced by Robertson, Sanders, Seymour and Thomas in 1996Used a quadratic algorithm for four-color planar graphsDecreased the size of the unavoidable set to 633The 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”?
17Works CitedThomas, 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 21-28Calude, Andreea (2001)The Journey of the Four Colour Theorem Through Time, The New Zealand Mathematics Magazine, 38:3:27-35Cayley (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, 2:1:17-25Mitchem, 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, 1989May, 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