81. Graph Theory White Pages: Percy John Heawood On the fourcolour map theorem, Quart. J. Pure Appl. On extended congruencesconnected with the four-colour map theorem, Proc. London Math. http://www.cs.columbia.edu/~sanders/graphtheory/people/Heawood.PJ.html

Percy John Heawood http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heawood.html Students: Some Old Graph Theory Publications: Map-colour theorem, Quart. J. Pure Appl. Math. On the four-colour map theorem, Quart. J. Pure Appl. Math. Map colour theorem, Quart. J. Pure Appl. Math. On extended congruences connected with the four-colour map theorem, Proc. London Math. Soc. Failures in congruences connected with the four-colour map theorem, Proc. London Math. Soc. Graph Theory Publications: Map-colour theorem. Proc. London Math. Soc. (2) Note on a correction in a paper on map-congruences. J. London Math. Soc. Note on a correction in a paper on map-congruences. J. London Math. Soc. Graph Theory Books: The Graph Theory White Pages www.graphtheory.com JLG additions and corrections through Registration

82. Abstract For Talk: Robertson Colloquium Talk Neil Robertson. The fourcolour theorem. Abstract for the ColloquiumApril 22, 1999 at 430. That planar maps can be properly colored http://www.math.binghamton.edu/dept/ComboSem/abstract.19990422coll.html

Neil Robertson The four-colour theorem Abstract for the Colloquium April 22, 1999 at 4:30

83. Full List Of Dr Woodall's Papers The fourcolour theorem, Bull. Inst. Math. Appl. 14 (1978), 245-249.The four-colour theorem, Mathematical Spectrum 11 (1979), 69 http://www.maths.nottingham.ac.uk/personal/drw/papers.html

Full list of papers by D. R. Woodall latest at end) Papers related to teaching Inductio ad absurdum?, The Mathematical Gazette (1975), 64-70, and (in Czech) Pokroky Matematiky Fyziky a Astronomie Teaching through discussion groups, The Mathematical Gazette Finite sums, matrices and induction, The Mathematical Gazette School, student and popular articles, and working papers The Marlborough College Eclipse Expedition, J. Brit. Astron. Assoc. (1961), 369-372, and (with minor variations) Report of the Marlborough College Natural History Society Spectrum Analysis at Marlborough College, J. Brit. Astron. Assoc. Parallel Curves, Eureka The paradox of the surprise examination, Eureka Impossible objects, Eureka A criticism of the Football League Eigenvector, Eureka On badly behaved fish fingers, Eureka (1971), 37-39, and (with minor variations) The Mathematical Intelligencer The four-colour conjecture is proved, Manifold (1977), 14-22; reprinted as "The 4-colour theorem" in Seven Years of Manifold 1968-1980 (I. Stewart and J. Jaworski, eds, Shiva Publishing Ltd, 1981), 69-75. The four-colour theorem

84. Ideas, Concepts, And Definitions four Color theorem. (See also The Mathematics Behind the Maps, The MostColorful Math of All, and The Story of the Young Map Colorer.). http://www.c3.lanl.gov/mega-math/gloss/math/4ct.html

Four Color Theorem (See also: The Mathematics Behind the Maps The Most Colorful Math of All , and The Story of the Young Map Colorer The Four Color Problem was famous and unsolved for many years. Has it been solved? What do you think? The story of the problem Since the time that mapmakers began making maps that show distinct regions (such as countries or states), it has been known among those in that trade, that if you plan well enough, you will never need more than four colors to color the maps that you make. The basic rule for coloring a map is that no two regions that share a boundary can be the same color. (The map would look ambiguous from a distance.) It is okay for two regions that only meet at a single point to be colored the same color, however. If you look at a some maps or an atlas, you can verify that this is how all familiar maps are colored. Mapmakers are not mathematicians, so the assertion that only four colors would be necessary for all maps gained acceptance in the map-making community over the years because no one ever stumbled upon a map that required the use of five colors. When mathematicians picked up the thread of the conversation, they began by asking questions like: Are you sure that four colors are enough? How do you know that no one can draw a map that requires five colors? What is it about the way that regions are arranged and touch each other in a map that would make such a thing true? When the question came to the European mathematics community at the end of the 19th century, it was perceived as interesting but solvable. Prominent and experienced mathematicians who tackled the problem were surprised by their inability to solve it. Take for example, this account from

85. The Four Color Theorem The four Color theorem. How many different colors are sufficient tocolor the countries on a map in such a way that no two adjacent http://www.mathpages.com/home/kmath266/kmath266.htm

The Four Color Theorem How many different colors are sufficient to color the countries on a map in such a way that no two adjacent countries have the same color? The figure below shows a typical arrangement of colored regions. Notice that we define adjacent regions as those that share a common boundary of non-zero length. Regions which meet at a single point are not considered to be "adjacent". The coloring of geographical maps is essentially a topological problem, in the sense that it depends only on the connectivities between the countries, not on their specific shapes, sizes, or positions. We can just as well represent each country by a single point (vertex), and the adjacency between two bordering countries can be represented by a line (edge) connecting those two points. It's understood that connecting lines cannot cross each other. A drawing of this kind is called a planar graph. A simple map (with just five "countries") and the corresponding graph are shown below. A graph is said to be n-colorable if it's possible to assign one of n colors to each vertex in such a way that no two connected vertices have the same color. Obviously the above graph is not 3-colorable, but it is 4-colorable. The Four Color Theorem asserts that every planar graph - and therefore every "map" on the plane or sphere - no matter how large or complex, is 4-colorable. Despite the seeming simplicity of this proposition, it was only proven in 1976, and then only with the aid of computers. Notice that the above graph is "complete" in the sense that no more connections can be added (without crossing lines). The edges of a complete graph partition the graph plane into three-sided regions, i.e., every region (including the infinite exterior) is bounded by three edges of the graph. Every graph can be constructed by first constructing a complete graph and then deleting some connections (edges). Clearly the deletion of connections cannot cause an n-colorable graph to require any additional colors, so in order to prove the Four Color Theorem it would be sufficient to consider only complete graphs.

86. GT Combinatorics Seminar It is well known that the case $k=5$ is equivalent to the four ColourTheorem, as proved by Wagner in 1937. About 60 years later http://www.math.gatech.edu/~ciucu/cosem/f01/sched.html

GT Combinatorics Seminar Fall 2001 August 31 at 4:00 pm in Skiles 255 SPEAKER: Michael Lacey, Georgia Tech TITLE: Large sets of integers not containing arithmetic progressions of length 5 September 7 at 4:00 pm in Skiles 255 SPEAKER: Balaji Gopalakrishnan, Georgia Tech TITLE: Solving Linear Programs Using Least-Squares Methods We will talk about some of the least-squares algorithms we have developed for solving linear programming problems. This is a joint work by Ellis Johnson, Joel Sokol and Earl Barnes. September 14 at 4:00 pm in Skiles 255 SPEAKER: Russell Lyons, Georgia Tech TITLE: Matroids and Probability I I will survey some new connections between matroids and probability theory. Most of these new connections are inspired by studies of the uniform probability measure on the spanning trees of a given finite connected graph. This probability measure and variants have been intensively studied, with many interesting connections to other areas of mathematics. Because of the ubiquity of matroids, however, there are now connections emerging to homology, group representations, ergodic theory, and phase transitions. Many open questions remain. There are continuous analogues that are heavily studied in the theory of random matrices, but I will stay on the discrete side. No background knowledge will be assumed. September 21 at 4:00 pm in Skiles 255 SPEAKER: Russell Lyons, Georgia Tech

87. Affiches Pour La "World Mathematical Year 2000" : Deuxième Prix It has taken more than 100 years before a correct proof for the fourcolour theorem has been found. The proof by Appel and Haken http://www.cmi.univ-mrs.fr/~damour/affiches/aff_math_B.html

CENTRE DE ET INFORMATIQUE DE PROVENCE Affiches pour la "World Mathematical Year 2000" The Bridges of Konigsberg Konigsberg (today: Kaliningrad) lies close to the baltic sea. Formerly a city of Eastern Prussia, it is now part of Russia. The 18th century philosopher Immanuel Kant is certainly the most prominent citizen of Konigsberg. The mathematical problem known as the "bridges of Konigsberg" is said to have been a popular riddle in 18th century Konigsberg. The challenge is to find out whether or not there is a way to walk over all seven bridges exactly once. By itself, this problem is indeed easy to understand and it could be solved, in principle, by systematically trying all possibilities. However, mathematics is more interested in understanding how the structure of the underlying problem determines whether such a way exists or not. We are therefore interested in an easy way to answer the question for all imaginable town maps with an arbitrary arrangement of rivers, land parts and connecting bridges. The text of the poster gives this general answer. It was first formulated by the mathematician Leonhard Euler (1707-1783). His approach is regarded as giving birth to modern graph theory, a branch of mathematics which has become increasingly important as a means of modelling networks, dependencies in production processes, logistic processes etc. An animated, interactive treatment of the bridges problem with more details and a biography of Euler can be found at

88. Fourier Transform From FOLDOC Fourier transform. mathematics A technique for expressing a waveformas a weighted sum of sines and cosines. Computers generally http://wwwacs.gantep.edu.tr/foldoc/foldoc.cgi?Fourier transform

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