The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different. Four color theorem simple english wikipedia, the free. The four color theorem has been notorious for attracting a large number of false proofs and disproofs in its long history. The fourcolor theorem history, topological foundations. This paper focuses on assigning colors to the vertices1 of a plane graph with the goal of proving the fourcolor theorem without a computer. If gis a connected planar graph on nitely many vertices, then. An expanded version of the proof was later reprinted in 1. They will learn the fourcolor theorem and how it relates to map coloring. Nevertheless, parts of the proof still cannot be veri. It is an outstanding example of how old ideas can be combined with new discoveries. This report gives an account of a successful formalization of the proof of the four colour theorem, which was fully checked by the coq v7. If t is a minimal counterexample to the four color theorem, then no good configuration appears in t.
Four color theorem the fourcolor theorem states that. To understand the problem better, lets consider some examples. Jan 11, 2017 in 1976 appel and haken achieved a major break through by thoroughly establishing the four color theorem 4ct. The four color theorem is equivalent to the proposition that every planar triangulation with more than three vertices is the union of two connected bipartite graphs, each with no isthmus. Robin thomas has nodes of the maximal planar graph with four colors listed many. A false proof of the fourcolor theorem week 5 ucsb 2014 in todays talk, were going to study the fourcolor theorem. Let v be a vertex in g that has the maximum degree. Pdf a formal proof of the four color theorem peter. The format proof 3 was achieved by kenneth appel and wolfgang haken and was published in 1976. Feb 18, 20 very simple proof of this theorem, it has been around without a sustainable proof for more than 120 years. Graph theory and the fourcolor theorem week 4 ucsb 2015 through the rest of this class, were going to refer frequently to things called graphs. I, as a trained algebraic topologist, was asked to comment on this. In 1976 appel and haken achieved a major break through by thoroughly establishing the four color theorem 4ct.
A historical overview of the four color theorem mark walters may 17, 2004 certainly any mathematical theorem concerning the coloring of maps would be relevant and widely applicable to modernday cartography. The four color theorem states that any mapa division of the plane into any number of regionscan be colored using no more than four colors in such a way that no two adjacent regions share the same color. 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. Last doubts removed about the proof of the four color theorem. From the above two theorems it follows that no minimal counterexample exists, and so the 4ct is true. Formal proofthe fourcolor theorem georges gonthier the tale of a brainteaser francisguthrie certainlydidit, whenhe coinedhis innocent little coloring puzzle in 1852. A formal proof has not been found for the four color theorem since 1852 when francis guthrie first conjectured the four color theorem. A graph is planar if it can be drawn in the plane without crossings. His starting point is the secondgeneration proof by robertson et al. 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. Another part, however, was more rational skepticism. Take any map, which for our purposes is a way to partition. The four color theorem is one of many mathematical puzzles which share the characteristics of being easy to state, yet hard to prove. The four color theorem is particularly notable for being the first major theorem proved by a computer.
Specifically, if you have a ry chain and a rg chain, then there can be an edge between the y and the g which throws a wrench in the flipping and recoloring process, because the y and the g both need to get turned into r, but they are adjacent. Their proof is based on studying a large number of cases for which a computerassisted search for hours is required. The four colour conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. A false proof of the fourcolor theorem week 5 ucsb 2014 in todays talk, were going to study the four color theorem. Pdf the four color theorem a new proof by induction. Introduction since 1852 when francis guthrie first conjectured the four color theorem 1, a formal proof has not been found for the four color theorem.
The most spectacular example is gonthiers formal proof of the four color theorem. The formal proof proposed can also be regarded as an algorithm to color a planar graph using four colors. Pdf a simpler proof of the four color theorem is presented. The five color theorem is obviously weaker than the four color theorem, but it is much easier to prove. This proof is largely based on the mixed mathematicscomputer proof 26 of. I dont think seperating the map into four different colors can be considered an application. Coloring the four color theorem this activity is about coloring, but dont think its just kids stuff. The search continues for a computerfree proof of the four color theorem.
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. The four color theorem has been known since 18521,2, and has just proved with the help of computer. In 1976, kenneth appel and wolfgang haken 2 published their proof of the four. It can be shown that g g g must have a vertex v v v shared by at. Take any map, which for our purposes is a way to partition the plane r2 into a collection of connected regions r. The shortest known proof of the four color theorem today still has over 600 cases. This gives us one last rephrasing of the four color theorem. My brother in law and i were discussing the four color theorem. Gonthier, georges 2005, a computerchecked proof of the four colour theorem pdf. In this degree project i cover the history of the four color theorem, from the origin, to the first proof by appel and haken 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.
Introduction many have heard of the famous four color theorem. First, we can associate a graph to a map in the following way. Xiangs formal proof of the four color theorem 1 a formal proof of the four color theorem limin xiang department of information science, kyushu sangyo university 31 matsukadai 2chome, higashiku, fukuoka 88503, japan email. Graph theory and the fourcolor theorem week 4 ucsb 2015. A new proof of the four color theorem published by robertson, sanders, seymour, and thomas based on the same outline.
The four color theorem, sometimes known as the four color map theorem or guthries problem, is a problem in cartography and mathematics. Naturally, i was acquainted with the four color 1 a latin word meaning the whole of something, a collective entirety. The implications of accepting this method as a general proof rightly raised questions about what it means to prove a theorem. 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. In this paper, we introduce graph theory, and discuss the four color theorem. Avertexcoloring of agraphisanassignmentofcolorstotheverticesofthegraph. This proof of the five color theorem is based on a failed attempt at the four color proof by alfred kempe in 1879. Using a similar method to that for the formal proof of. The purpose of these remarks is to raise the question of whether the 4ct is really a theorem. I was reading up on the four color theorem and am wondering if there is any practical application of it. Pdf a computerchecked proof of the four colour theorem. In this note, we study a possible proof of the four colour theorem, which is the proof contained in potapov, 2016, since it is claimed that they prove the equivalent for three colours, and if you can colour a map with three colours, then you can colour it with four, like three starts being the new minimum. What makes the new result particularly significant from a reliability point of view is that the proof assistant gonthiers employed, called coq, is a widelyused general purpose utility, which can be verified experimentally, unlike the specialpurpose programs used in the earlier proofs of the four color theorem. As for the four color theorem, nothing could be further from the truth.
We want to color so that adjacent vertices receive di erent colors. An algebraic reformulation of the four color theorem. This proof was controversial because most of the cases were checked by a computer program, not by hand. Assume the ftc is true and form g0 1 and g 0 2 from g as described above. To dispel any remaining doubts about the appelhaken proof, a simpler proof using the same. This coloring uses at most three colors for the ring, leaving us a free color for the kernel face, so the original map is also fourcolorable. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete graph. Clearly, each of g0 1 and g 0 2 is connected and has no odd cycles. This was followed by false proofs given independently by kempe 1879 and tait 1880. The 6 color theorem nowitiseasytoprovethe6 colortheorem. A new noncomputer direct algorithmic proof for the famous four color theorem based on new concept spiralchain coloring of maximal planar graphs has been proposed by the author in 2004 6. For every internally 6connected triangulation t, some good configuration appears in t. Formal proofthe four color theorem american mathematical.
Graph theory, fourcolor theorem, coloring problems. Francis guthrie 1852 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. While i certainly hope that there does not exist a purely human, machinefree, proof of the four color theorem if there is, it would mean that this theorem is, a posteriori, trivial. Pdf a simple proof of the fourcolor theorem researchgate. A very nice one is due to dror barnatan, in terms of lie algebras. Pdf the four color theorem download full pdf book download. Let g be the smallest planar graph in terms of number of vertices that cannot be colored with five colors. At first, the new york times refused to report on the appelhaken proof. The four color problem is central to the intersection of algebra, topology, and statistical mechanics.
A bad idea, we think, directed people to a rough road. The four color theorem was proved by appel, haken, and koch 1, 2 using substantial computing resources. The proof was reached using a series of equivalent theorems. Then we prove several theorems, including eulers formula and the five color theorem. Four color theorem wikimili, the best wikipedia reader. In this way, the controversy over the modern methods used in the proof of the four color theorem had also spread to disciplines outside of mathematics. A computerchecked proof of the four colour theorem georges gonthier microsoft research cambridge this report gives an account of a successful formalization of the proof of the four colour theorem, which was fully checked by the coq v7. Kempe published what he and them athematics communittyh ought was a proof of the four color theorem. Oct 22, 2019 the implications of accepting this method as a general proof rightly raised questions about what it means to prove a theorem. Although the traditional proof uses a computer and gonthier uses a computer, the two computer processes di.
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. Their conquest of the four color theorem came almost a century after the world had accepted the first proof of the theorem. Several other people have independently programmed it. In particular, were going to consider a proof of the fourcolor theorem, given by kempe in 1879. Although flawed, kempes original purported proof of the four color theorem provided some of the basic tools later used to prove it. In fact, its earliest proof occurred by accident, as the result of a flawed attempt to prove the four color theorem. As for the fourcolor theorem, nothing could be further from the truth. It was the first major theorem to be proved using a computer. We know that degv proof that four colors suffice their proof has been accepted by most mathemati cians, and the old four color problem has given way in mathemat ics to the new four color theorem 4ct. 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. Platonic solids 7 acknowledgments 7 references 7 1. The four color theorem graphs the solution of the four color problem more about coloring graphs coloring maps history the history of the four color theorem i 1976.
We show that the mathematical proof of the four color theorem yields a perfect interpretation of the standard model of particle physics. Pdf four proofs for the four color theorem ibrahim cahit. The hope of the present authors is that a direct proof of theorem 1 will be shorter than the known proofs of the four color theorem, thereby providing a shorter proof. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. Pdf four color theorem proof peter didomenica academia. 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. The four color theorem, or the four color map theorem, states that given any. V by eulers theorem, we conclude that the average degree of a vertex is. Students will gain practice in graph theory problems and writing algorithms. This proof turned out to be fallacious, and kempe is remembered mostly for this fallacious proof, which is. February 1, 2008 abstract a simpler proof of the four color theorem is presented.
This proof is largely based on the mixed mathematicscomputer proof 26 of robertson et al, but contains original contributions as well. 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. The shortest known proof of the four color theorem. A short note on a possible proof of the fourcolour theorem. Four, five, and six color theorems nature of mathematics. From this until 1880, there was limited progress in proving the four color theorem, but in 1880 alfred bray kempe published his proof of the four color theorem. Their proof is based on studying a large number of cases for which a computer. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in heawoods original short paper that contained. We want to color so that adjacent vertices receive different colors. Computer proof assistants have been under development for decades see box early milestones, but only recently has it become a practical matter to prove major theorems formally. An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by haken and appel. Their proof relies on checking a large number of cases by computer, sparking ongoing debate over what a proof really is.
The steps of the proof enable us to construct the triemann surface and particle frame which forms the gauge. Erasing an appropriate pair of opposite edges disposes of the square con. Toward a language theoretic proof of the four color theorem. Indeed, thanks to the four color theorem, people are still debating. This investigation will lead to one of the most famous theorems of. The 6color theorem nowitiseasytoprovethe6 colortheorem. Pdf the four color theorem franciszek jagla academia. In mathematics, the four color theorem, or the four color map theorem, states that, given any.
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