In mathematics, the fourcolor theorem, or the fourcolor map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so. An investigation for pupils about the classic four colour theorem. The fourcolor theorem history, topological foundations. The four colour conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976.
Elementary mathematical analysis by herberg theodore abebooks. A simpler proof of the four color theorem is presented. I, as a trained algebraic topologist, was asked to comment on this. Investigation four colour theorem teaching resources. If t is a minimal counterexample to the four color theorem, then no good configuration appears in t. The four color theorem 4ct essentially says that the vertices of a planar graph may be colored with no more than four different colors. Planar graphs, seven color theorem seven color theorem if we are interested in other manifolds, replace 2 with 22g in eulers formula, giving. Famous theorems of mathematicsfour color theorem wikibooks. The four color theorem is a theorem of mathematics. Unscientific unamerican, and other april fools jokes in sa history. In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a border segment, not just a point. The fourcolor theorem states that any map in a plane can be colored using four colors in. History, topological foundations, and idea of proof softcover reprint of the original 1st ed.
The history, and some of the math behind the 4 color theorem. Fourcolor theorem scimath faq do we need more than four colors to color a twodimensional map. The fourcolor theorem stands at the intersection of science and art. The four colour theorem mactutor history of mathematics. Then, we will prove eulers formula and apply it to prove the five color theorem. For every internally 6connected triangulation t, some good configuration appears in t. Four color theorem simple english wikipedia, the free. The four colour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by appel and haken, using a computer. Eulers formula and the five color theorem min jae song abstract. This page gives a brief summary of a new proof of the four color theorem and a four coloring algorithm found by neil robertson, daniel p. A graph is a set of points called vertices which are connected in pairs by rays called edges.
Many famous mathematicians have worked on the problem, but the proof eluded formulation until the 1970s, when it. The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human. Neuware in mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. The four color theorem, sometimes known as the four color map theorem or guthries problem, is a problem in cartography and mathematics. Georges gonthier, a mathematician who works at microsoft research in cambridge, england, described how he had used a new computer technology called a mathematical assistant to verify a proof of the famous four color theorem, hopefully putting to rest any doubts about. Kenneth may, a twentieth century mathematics historian, explains that books on cartography and the history of mapmaking do not mention the fourcolor property. Hardly any general history book has much on the subject, but the last chapter in. Mastorakis abstractin this paper are followed the necessary steps for the realisation of the maps coloring, matter that stoud in the attention of many mathematicians for a long time. The four color theorem was one of the first major theorem that was proved by the computer. Last doubts removed about the proof of the four color theorem. To dispel any remaining doubts about the appelhaken proof, a simpler proof using the same. Four color theorem encyclopedia article citizendium. The four color map theorem states that on a plane, which is divided into nonoverlapping contiguous regions, the regions can be colored with four colors in such a way that all regions are colored and no two adjacent regions have the same color.
In this paper, we introduce graph theory, and discuss the four color theorem. Ppt fourcolor theorem powerpoint presentation free to. The four color theorem was proved in 1976 by kenneth appel and wolfgang haken after many false proofs and counterexamples unlike the five color theorem, a theorem that states that five colors are enough to color a map, which was proved in the 1800s. Coloring the four color theorem this activity is about coloring, but dont think its just kids stuff. After proving this equivalence, we have an algebraic statement that is true, because the four color theorem has been established, but which if given a purely algebraic proof would provide a computerindependent proof of the four color theorem. A computerchecked proof of the four colour theorem pdf. Pdf this is a historical survey of the four colour theorem and a. An algebraic reformulation of the four color theorem. This investigation will lead to one of the most famous theorems of. Let g be a the smallest planar graph by number of vertices that has no proper 6coloring. The four color theorem can be stated purely topologically, without any reference to graph theory. An equivalent combinatorial interpretation is this theorem was proved with the aid of a computer in 1976. Let g be the smallest planar graph in terms of number of vertices that cannot be colored with five colors.
The four colour theorem nrich millennium mathematics project. Kempes proof was accepted for a decade until heawood showed an error using a map with 18. Elementary mathematical analysis by herberg, theodore and a great selection of related books, art and collectibles available now at. The four color theorem stands at the intersection of science and art. Okay, i ripped that off from star trek, episode 15, but i like to.
Some history could you please give me the history of this topic. It provided a lot of interesting information and was a great read. This is usually done by constructing the dualgraphof the map, and then appealing to the compactness theorem of propositional. In a complete graph, all pairs are connected by an edge. Wilson defines the problem and explains some of the methods used by those trying to solve it. The proof was reached using a series of equivalent theorems. An update on the fourcolor theorem robin thomas 848 n otices of the ams v olume 45, number 7 e very planar map of connected countriescan be colored.
A historical overview of the fourcolor theorem sigmaa history. Each region must be contiguous that is it may not be partitioned as are. Naturally, i was acquainted with the four color 1 a latin word meaning the whole of something, a collective entirety. I believe because of the new technology, the proof of four colo theorem will be improved in later time. Transum, friday, november, 2015 the four colour theorem states that it will take no more than four different colours to colour a map or similar diagram so that no two regions sharing a border are coloured in the same colour. A graph is planar if it can be drawn in the plane without crossings. Then we prove several theorems, including eulers formula and the five color theorem. Four color, also known as four color comics and one shots, was an american comic book anthology series published by dell comics between 1939 and 1962. Some basic graph theory is featured to ensure that the reader can follow. The more complex the mind, the greater the need for play. The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different colours, was proved in 1976 by appel and haken, using a computer. I used this book as a resource for my history of mathematics paper on the fourcolor theorem.
It had been noticed that it only required four colors to fill in the different contiguous shapes on a map of regions or countries or provinces in a flat surface known as a plane such that no two adjacent regions with a common boundary had the same color. We know that degv four color theorem robin thomas 848 n otices of the ams v olume 45, number 7 e very planar map of connected countriescan be colored using four colors in such a way that countries with a common. History, topological foundations, and idea of proof 9781461272540 by fritsch, rudolf and a great selection of similar new, used and collectible books available now at great prices. The four colour conjecture was first stated just over 150 years ago, and finally. This book discusses a famous problem that helped to define the field now known as topology. They will learn the fourcolor theorem and how it relates to map. This investigation will lead to one of the most famous theorems of mathematics and some very interesting results. A historical overview of the fourcolor theorem mark walters may 17, 2004. What happens if we try to generalize the four color theorem to other. All structured data from the file and property namespaces is available under the creative commons cc0 license. Last doubts removed about the proof of the four color theorem at a scientific meeting in france last december, dr. The four color map theorem is easy to understand and hard to prove. The four color theorem states that any plane separated into regions, such as a political map of the counties of a state, can be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Before continuing with the history of the four colour conjecture we will complete details of francis guthrie.
Contents introduction preliminaries for map coloring. Graph theory, fourcolor theorem, coloring problems. In fact the picture is fourcolorable and was proven so by wagon in 1998. The first statement of the four colour theorem appeared in 1852 but surprisingly it wasnt until 1976 that it was proved with the aid of a computer. We can easily produce a 6 coloring with one color for each vertex. History, topological foundations, and idea of proof. Applications of the four color problem mariusconstantin o. The four color theorem people school of mathematics.
Pdf the journey of the four colour theorem through time. The mathematical reasoning used to solve the theorem lead to many practical applications in mathematics, graph theory, and computer science. Intuitively, the four color theorem can be stated as given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two regions which are adjacent have the same color. Elementary mathematical analysis by herberg theodore. Two regions that have a common border must not get the same color. The four color theorem is a theorem in mathematics that states that given any map you need at most 4 different colors to color each patch of the map so that it is guaranteed that no patches next to each other have the same color. I purchased this book as a resource for my history of mathematics paper on the fourcolor theorem. Students will gain practice in graph theory problems and writing algorithms. In fact a substantial part of graph theory developed in trying to prove the four color theorem. Science a thoroughly accessible history of attempts to prove the four color theorem.
The journey of the four colour theorem through time. Science a thoroughly accessible history of attempts to prove the fourcolor theorem. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. As for the fourcolor theorem, nothing could be further from the truth.
Four, five, and six color theorems in 1852, francis guthrie pictured above, a british mathematician and botanist was looking at maps of the counties in england and discovered that he could always color these maps such that no adjacent country is the same color with at most four colors. Many famous mathematicians have worked on the problem, but the proof eluded formulation until the 1970s, when it was. It says that in any plane surface with regions in it people think of them as maps, the regions can be colored with no more than four colors. The fourcolor theorem states that any map in a plane can be colored using fourcolors in. Their magnum opus, every planar map is fourcolorable, a book claiming a complete and detailed proof with a microfiche. It is an outstanding example of how old ideas can be combined with new discoveries. Four color theorem and five color theorem stack exchange. What is the minimum number of colors required to print a map so. Georges gonthier, a mathematician who works at microsoft research in cambridge, england, described how he had used a new computer technology called a mathematical assistant to verify a proof of the famous four color theorem, hopefully putting to rest any doubts. So, it is by no means necessary that a proof of the four color theorem should even mention graphs. The six color theorem 62 the six color theorem theorem. The mathematical reasoning used to solve the theorem lead to many practical applications. I was very interested in the material and enjoyed the writing. From the above two theorems it follows that no minimal counterexample exists, and so the 4ct is true.
Four, five, and six color theorems nature of mathematics. Kenneth may, a twentieth century mathematics historian, explains that \ books on cartography and the history of mapmaking do not mention the fourcolor property, though. A free powerpoint ppt presentation displayed as a flash slide show on id. Mastorakis abstractin this paper are followed the necessary steps for the realisation of the. Hardly any general history book has much on the subject, but the last chapter in katz called computers and applications has a section on graph theory, and the four colour theorem is mentioned twice. One aspect of the fourcolor theorem, which was seldom covered and relevant to the field. A historical overview of the fourcolor theorem mark walters may 17, 2004 certainly any mathematical theorem concerning the coloring of maps would be relevant and widely applicable to modernday cartography. Revised edition by joseph miller thomas and a great selection of related books, art and collectibles available now at. We want to color so that adjacent vertices receive di erent colors. This elegant little book discusses a famous problem that helped to define the field now known as graph theory. Let v be a vertex in g that has the maximum degree. But the independent verification had convinced people that the theorem was finally proved. The title is a reference to the four basic colors used when printing comic books cyan, magenta, yellow and black at the time.
They are called adjacent next to each other if they share a segment of the border, not just a point. Files are available under licenses specified on their description page. Unscientific unamerican, and other april fools jokes in. In this way, the controversy over the modern methods used in the proof of the fourcolor theorem had also spread to disciplines outside of mathematics. Nov, 2015 the four colour theorem states that it will take no more than four different colours to colour a map or similar diagram so that no two regions sharing a border are coloured in the same colour. Wilsons lucid history weaves together lively anecdotes, biographical sketches, and a nontechnical account of the mathematics. The fourcolor theorem history, topological foundations, and. Here we give another proof, still using a computer, but simpler than appel and hakens in several respects. Since the four color theorem has been proved by a computer they reduced all the planar graphs to just a bunch of different cases, about a million i think, most of the books show the proof of the five color theorem which has a noncomputer proof.