They will learn the fourcolor theorem and how it relates to map coloring. The fact that the proof also involved an initial manual case analysis 4 that was large 10,000 cases, difficult to verify, and. The shortest known proof of the four color theorem today still has over 600 cases. Four color theorem simple english wikipedia, the free. Our prepress department will prepare a print ready pdf from your supplied file and send you via email within 1 business day. A formal proof has not been found for the four color theorem since 1852 when francis guthrie first conjectured the four color theorem. Four color theorem ohio state department of mathematics. A simpler proof of the four color theorem is presented. Jul 11, 2016 with an amusing history spanning over 150 years, the four color problem is one of the most famous problems in mathematics and computer science. First the maximum number of edges of a planar graph is obatined as well as the. Pdf a simpler proof of the four color theorem is presented. 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. Feb 18, 20 very simple proof of this theorem, it has been around without a sustainable proof for more than 120 years. 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.
Pdf the four color theorem franciszek jagla academia. Part of the appealof the four color problem is that its statement theorem 1. The format proof 3 was achieved by kenneth appel and wolfgang haken and was published in 1976. This theorem was proved with the aid of a computer in 1976. An algebraic reformulation of the four color theorem.
Using a similar method to that for the formal proof of the five color theorem, a formal proof is proposed in this paper of the four color theorem, namely, every planar graph is four colorable. A bad idea, we think, directed people to a rough road. With an amusing history spanning over 150 years, the four color problem is one of the most famous problems in mathematics and computer science. Definition of four color theorem in the dictionary. Avertexcoloring of agraphisanassignmentofcolorstotheverticesofthegraph. Jan 11, 2017 in 1976 appel and haken achieved a major break through by thoroughly establishing the four color theorem 4ct. This paper focuses on assigning colors to the vertices1 of a plane graph with the goal of proving the fourcolor theorem without a computer. Applications of the four color problem mariusconstantin o. 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. Here we give another proof, still using a computer, but simpler than appel and hakens in several respects.
In 1976 appel and haken achieved a major break through by thoroughly establishing the four color theorem 4ct. A computerchecked proof of the four colour theorem 1 the story. Theorem 3 four colour theorem every loopless planar graph admits a vertexcolouring with at most four different colours. I was wondering if proof by induction or contradiction is better, but i decided for proof by induction, as this is easier to translate in actual code then.
Thinking about graph coloring problems as colorable vertices and edges at a high level allows us to apply graph co. Well, besides the obvious application to cartography, graph coloring algorithms and theory can be applied to a number of situations. Using a similar method to that for the formal proof of. The author thanks tibor jord an for calling our attention to the work 1 by b ohme et al references.
The vernacular and tactic scripts run on version v8. Take any map, which for our purposes is a way to partition the plane r2 into a collection of connected regions r. Guthrie, who first conjectured the theorem in 1852. Gonthier, georges 2005, a computerchecked proof of the four colour theorem pdf. 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. Pdf a formal proof of the four color theorem peter. We want to color so that adjacent vertices receive different colors. Dec, 2015 this video should give you a basic understanding of why the four colour theorem holds good. Last doubts removed about the proof of the four color theorem at a scientific meeting in france last december, dr. Pdf the four color theorem a new proof by induction. It is obvious that three colors are inadequate, and it is not at all difficult to prove that five colors are sufficient to color a map. Then, we will prove eulers formula and apply it to prove the five color theorem. The four color theorem can be stated purely topologically, without any reference to graph theory. To dispel any remaining doubts about the appelhaken proof, a simpler proof using the same.
Pdf proofing is the fastest and least expensive way to proof your print file before going to the press. This problem is sometimes also called guthries problem after f. Four, five, and six color theorems nature of mathematics. 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. The regions of any simple planar map can be coloured with only four colours, in such a way that any two adjacent regions.
The regions of any simpleplanar map can be colored with only four colors, in such a way thatanytwoadjacentregionshavedi. February 1, 2008 abstract a simpler proof of the four color theorem is presented. Then we prove several theorems, including eulers formula and the five color theorem. Put your pen to paper, start from a point p and draw a continuous line and return to p again.
A short note on a possible proof of the fourcolour theorem. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. What is the importance of the four color theorem for math. Robin thomas has nodes of the maximal planar graph with four colors listed many. Eulers formula and the five color theorem min jae song abstract. A graph is planar if it can be drawn in the plane without crossings. The proof theorem 1the four color theorem every planar graph is fourcolorable. Kempe proved the fivecolor theorem theorem 2 below and discovered what became known as kempe chains, and tait found an equivalent formulation of the fourcolor theorem in terms of edge. Aug 08, 2010 pdf proofing is the fastest and least expensive way to proof your print file before going to the press. So, it is by no means necessary that a proof of the four color theorem should even mention graphs. The four color theorem for any subdivision of the plane into nonoverlapping regions, it is always possible to mark each of the regions with four different colors in such a way that no two adjacent regions receive the same color. The four color theorem states that the regions of a map a plane separated into contiguous regions can be marked with four colors in such a way that regions sharing a border are different colors. In 1976, kenneth appel and wolfgang haken 2 published their proof of the four.
Although flawed, kempes original purported proof of the four color theorem provided some of the basic tools later used to prove it. Four color theorem the fourcolor theorem states that any map in a plane can be colored using fourcolors in such a way that regions sharing a common boundary other than a single point do not share the same color. It was the first major theorem to be proved using a computer. The 6color theorem nowitiseasytoprovethe6 colortheorem. In mathematics, the four color theorem, or the four color map theorem, states that, given any.
Although heawood found the major flaw in kempes proof method in 1890, he was unable to go on to prove the four colour theorem, but he made a significant breakthrough and proved conclusively that all maps could be coloured with five colours. Thus, the formal proof of the four color theorem can be given in the following section. Of course, this is somewhat of a cheat, and we should be more explicit about what we mean by coloured map. This proof was controversial because most of the cases were checked by a computer program, not by hand. As for the fourcolor theorem, nothing could be further from the truth. What are the reallife applications of four color theorem. Pdf a simple proof of the fourcolor theorem researchgate. This video should give you a basic understanding of why the four colour theorem holds good. 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. In fact a substantial part of graph theory developed in trying to prove the four color theorem. Very simple proof of this theorem, it has been around without a sustainable proof for more than 120 years.
We want to color so that adjacent vertices receive di erent colors. The author thanks tibor jord an for calling our attention to. L1 we may assume that p is greater than or equal to 7. Their proof is based on studying a large number of cases for which a computer. Apr 26, 2006 a formal proof of the famous four color theorem that has been fully checked by the coq proof assistant. 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. The formal proof proposed can also be regarded as an algorithm to color a planar graph using four colors. 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. I think the importance of the four color theorem and its proof has to do with the notion of elegance in mathematics and basically how elegance relates to what mathematics is. 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.
In light of these, the goal of our present quick proof is that this perhaps not so wellknown proof is now available in a short and more or less selfcontained form. 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. This is usually done by constructing the dualgraphof the map, and then appealing to the compactness theorem of propositional. The formal proof proposed can also be regarded as an. Some basic graph theory is featured to ensure that the reader can follow and understand the proofs and procedures in the project. History, topological foundations, and idea of proof on free shipping on qualified orders. The four color theorem is one of many mathematical puzzles which share the characteristics of being easy to state, yet hard to prove.
Nevertheless, parts of the proof still cannot be veri. Download coq proof of the four color theorem from official. Last doubts removed about the proof of the four color theorem. The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand. We know that degv proof is about the 5 color theorem.
In this paper, we introduce graph theory, and discuss the four color theorem. Using a similar method to that for the formal proof of the five color theorem, a formal proof is proposed in this paper of the four color theorem, namely, every planar graph is fourcolorable. Do not redraw any part of the line but intersection is allowed. Part of the appeal of the four color problem is that. Pdf four proofs for the four color theorem ibrahim cahit. Another failed proof was published by tait in 1880. Lemma 2 every planar graph g contains a vertex v such that degv 5. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete graph. A machinechecked proof of the odd order theorem georges gonthier, andrea asperti, jeremy avigad, yves bertot, cyril cohen, francois garillot, st ephane le roux, assia mahboubi, russell oconnor, sidi. Let g be the smallest planar graph in terms of number of vertices that cannot be colored with five colors. Kempe proved the five color theorem theorem 2 below and discovered what became known as kempe chains, and tait found an equivalent formulation of the four color theorem in terms of edge.
The four color theorem has been known since 18521,2, and has just proved with the help of computer. Proof for the four color theorem 4ct suehwan jeong and junho yeo keywords. The search continues for a computerfree proof of the four color theorem. Let v be a vertex in g that has the maximum degree. In particular, were going to consider a proof of the fourcolor theorem, given by kempe in 1879. Formal proofthe four color theorem american mathematical. A formal proof of the famous four color theorem that has been fully checked by the coq proof assistant. Theorem 1 for any planar graph g, the chromatic number. Appel and haken published an article in scienti c american in 1977 which showed that the answer to the problem is yes. Students will gain practice in graph theory problems and writing algorithms.
The proof was reached using a series of equivalent theorems. 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. The regions of any simple planar map can be colored with only. In this note, we study a possible proof of the fourcolour 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. Information and translations of four color theorem in the most comprehensive dictionary definitions resource on the web. Contents introduction preliminaries for map coloring. Any planar map can be coloured with only four colours. Computerassisted proofs of the four color theorem 2, 18 and and the importance of computer formal proof methods are discussed in the next subsection. In this degree project i cover the history of the four color theorem, from the origin, to the. The four color theorem 4ct is the theorem stating that no more than four colors are required to color each part of a plane divided into finite parts so that no two adjacent parts have the same color. Graph theory, fourcolor theorem, coloring problems. This paper introduces the basic graph theory required to understand the four color theorem.