Four colour theorem pdf merge

Youngs 19101970 were able to prove in 1968 that all of heawoods estimates, for the chromatic number. The four color theorem is a theorem of mathematics. Although flawed, kempes original purported proof of the four color theorem provided some of the basic tools later used to prove it. The four color theorem states that any given separation of a plane into contiguous regions, producing a figure named a map, no more than four colors are required to color the regions of the map so.

The fourcolour map problem to prove that on any map only four colours are needed to separate countries is celebrated in mathematics. The purpose of this question is to collect generalizations, variations, and strengthenings of the four. Birkhoff, whose work allowed franklin to prove in 1922 that the fourcolor conjecture is true for maps with at most twentyfive regions. Haken the main lemma below trivially follows from the four colour theorem, the in terest in the lemma is. Two regions that have a common border must not get the same color. The four color theorem was finally proven in 1976 by kenneth appel and wolfgang haken, with some assistance from john a. A path from a vertex v to a vertex w is a sequence of edges e1. I decided to use this lesson because it is fun and demonstrates the difference between a proof and a conjecture, that there can be more then one way to solve a problem. The four colour theorem mactutor history of mathematics. The fourcolour theorem, that every loopless planar graph admits a vertex colouring with at. The four colour conjecture first seems to have been made by francis guthrie. And it can be adapted to a proof for the four colour theorem that was believed to be correct for ten years and then it was shown to be faulty. The fourcolour theorem any planar graph may be properly coloured using no more than four colours. The same method was used by other mathematicians to make progress on the fourcolor.

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. From the above two theorems it follows that no minimal counterexample exists, and so the 4ct is true. Csc165 week 7 department of computer science, university. Formal proofthe four color theorem semantic scholar. In mathematics, the four color theorem, or the four color map theorem, states that, given any. Also, as the theorem states, two areas need to share a common border, just a common interception is not enough. For every internally 6connected triangulation t, some good configuration appears in t. Theorem 3 four colour theorem every loopless planar graph admits a vertexcolouring with at most four different colours. The appelhaken proof began as a proof by contradiction. The four color theorem asserts that every planar graph can be properly colored by four colors.

History the four color theorem was proven in 1976 by kenneth appel and wolfgang haken. 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. It resisted the attempts of able mathematicians for over a century and when it was successfully proved in 1976 the computer proof was controversial. The 4color theorem is fairly famous in mathematics for a couple of reasons. They will learn the fourcolor theorem and how it relates to map coloring. While theorem 1 presented a major challenge for several generations of mathematicians, the corresponding statement for five colors is fairly easy to see. Some arithmetical restatements of the four color conjecture core.

Part ii ranges widely through related topics, including mapcolouring on surfaces with holes, the famous theorems of kuratowski, vizing, and brooks, the conjectures of hadwiger and hajos, and much more besides. It is an outstanding example of how old ideas combine with new discoveries and techniques in different fields of mathematics to provide new approaches to a problem. The statement of the theorem may be introduced as follows. The four colour theorem is a game of competitive colouring in. They are called adjacent next to each other if they share a segment of the border, not just a point. The proof was reached using a series of equivalent theorems. Appel and hakens approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallestsized counterexample to the four color theorem. Here we give another proof, still using a computer, but simpler than appel and hakens in several respects. An investigation for pupils about the classic four colour theorem. Thus, the formal proof of the four color theorem can be given in the following section. Students will gain practice in graph theory problems and writing algorithms. This was the first time that a computer was used to aid in the proof of a major theorem.

Pdf a victorian age proof of the four color theorem. The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than five colors in such a way that no two adjacent regions receive the same color. A formal proof has not been found for the four color theorem since 1852 when francis guthrie first. It was the first major theorem to be proved using a computer. Let g be the smallest planar graph in terms of number of vertices that cannot be colored with five colors. First the maximum number of edges of a planar graph is obatined as well as the. Let v be a vertex in g that has the maximum degree. Hales, formal proofs, notices of the ams, this issue 11 h heesch, untersuchungen zum vierfarbenproblem, 1969 12 a b kempe, on the geographical problem of the four colours, american journal of mathematics 23 1879, 193200 n robertson, d sanders, p seymour, and r thomas, the four colour theorem, j combinatorial theory, series b 70. A simpler proof of the four color theorem is presented. Investigation four colour theorem teaching resources. If t is a minimal counterexample to the four color theorem, then no good configuration appears in t. Thats why 2 colors would be enough for the following graph, the 2 red and the 2 blue areas dont count as each others neighbors. A computerchecked proof of the four colour theorem 1 the story.

Unfortunately combining definitions also makes much of the coq. Graph theory, fourcolor theorem, coloring problems. What is particularly striking is that gerhard ringel 1919 and j. Theorem of the day the fourcolour theorem any planar graph may be properly coloured using no more than four colours. Recall that the four colour theorem is equivalent to the statement that bridgeless cubic planar graphs are threeedgecolourable. We want to color so that adjacent vertices receive di erent colors. Challenge yourself to colour in the pictures so that none of the colours touch.

Generalizations of the fourcolor theorem mathoverflow. A fascinating paper by louis kauffman establishes the equivalence of the fourcolour theorem the assertion that any planar graph can have its vertices coloured with one of four colours, such that neighbouring vertices have different colours to the following seemingly unrelated combinatorial statement let n be a positive integer, and let a and b be two complete parenthesisations of the. Having fun with the 4color theorem scientific american. Eventually errors were found, and the problem remained open on into the twentieth century. This is usually done by constructing the dualgraphof the map, and then appealing to the compactness theorem of propositional. Pdf the four color theorem download full pdf book download. The four colour theorem nrich millennium mathematics project. Some background and examples, then a chance for them to have a go at. It is an outstanding example of how old ideas combine with new discoveries and techniques in different fields of mathematics to provide new approaches to a. Fourcolor theorem in terms of edge 3coloring, stated here as theorem 3. Pdf arthur cayley frs and the fourcolour map problem.