It has also applications to such other branches of mathematics as group theory. The elements v2vare called vertices of the graph, while the e2eare the graphs edges. One of the most challenging theoretical problems, the fourcolour problem see below belongs to the domain of graph theory. Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. The four colour conjecture was first stated just over 150 years ago, and.
Part i covers basic graph theory, eulers polyhedral formula, and the first published false proof of the fourcolour theorem. This number is called the chromatic number and the graph is called a properly colored graph. In any plane graph each vertex can be assigned exactly one of four colors so that. Graph coloring is nothing but a simple way of labelling graph components such as vertices, edges, and regions under some constraints.
The problem is, given m colors, find a way of coloring the vertices of a graph such that no two adjacent vertices are colored using same color. It includes all the elementary graph theory that should be included in an introduction to the subject, before concentrating on specific topics relevant to the fourcolour problem. I made this resource as a hook into the relevance of graph theory d1. In graphtheoretic terminology, the fourcolor theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, every planar graph is fourcolorable thomas 1998, p.
It was the first major theorem to be proved using a computer. 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. Graphs, colourings and the fourcolour theorem oxford. K6nig 1 published the first book on graph theory with notions later used to formulate conjectures equivalent to the fourcolor problem. This problem inspired the great swiss mathematician leonard euler to create graph theory, which led to the development of topology. Get your students to attempt to colour in the maps using the least number of colours they can, without any adjacent sections being the same colour. We say g is kchoosable if all lists lv have the cardinality k and g is llist colourable for all possible assignments of such lists. The theory of transportation networks can be regarded as a chapter of the theory of directed graphs. This problem is an outgrowth of the wellknown fourcolour map problem, which asks whether the countries on every map can be coloured by using just four colours in such a way that countries sharing an edge have different colours. What are the reallife applications of four color theorem. Part of thecomputer sciences commons, and themathematics commons this dissertation is brought to you for free and open access by the iowa state university capstones, theses and dissertations at iowa state university.
They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism. This problem was first posed in the nineteenth century, and it was quickly conjectured that in all cases four colors suffice. Here we give another proof, still using a computer, but simpler than appel and hakens in several respects. The mathematical depth of it isnt that great, but this just made it all the more interesting as i preferred reading about the history of it. This is another important book which led to the research into problem solving and. A ball packing is a collection of balls with disjoint interiors. In graph theoretic terminology, the fourcolor theorem states that the vertices of every planar.
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. Additionally, the graphs under consideration are planar. Until recently various books and papers stated that the problem of fourcoloring. The fourcolour theorem, that every loopless planar graph admits a vertexcolouring with at most four different. The climax of the book is a new proof of the famous four colour theorem. The four color problem dates back to 1852 when francis guthrie, while trying to color the map of counties of england noticed that four colors sufficed. It includes all the elementary graph theory that should be included in an introduction to the subject, before concentrating on specific topics relevant to the four colour problem.
Two regions that have a common border must not get the same color. Coloring problems in graph theory kevin moss iowa state university follow this and additional works at. On the history and solution of the fourcolor map problem jstor. In 1858, in the same month as he presented his famous. 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. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color. Diestel is excellent and has a free version available online. We hope this book will continue to evoke interest in the four color problem, in its computer aided solution, and perhaps in finding an alternative way to prove it. They are called adjacent next to each other if they share a segment of the border, not just a point.
Coloring the four color theorem this activity is about coloring, but dont think its just kids stuff. Long an attractive topic for amateur mathematicians, the 4cc even featured in martin gardners infamous hoax column in the april 1975 edition of scientific american. There are two classical conjectures from erdos, rubin and taylor 1979 about. In graph theoretic terms, the theorem states that for loopless planar, the chromatic number of its dual graph is. Graph coloring problem is to assign colors to certain elements of a graph subject to certain constraints vertex coloring is the most common graph coloring problem. A tree t is a graph thats both connected and acyclic. The four color problem is examined in graph theory, where the vertex set is the regions of a map and an edge connects two vertices exactly when they share a border. 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. If a graph ghas no subgraphs that are cycle graphs, we call gacyclic. A former president of the british society for the history of mathematics, he has written and edited many books on the history of mathematics, including lewis carroll in numberland, and also on graph theory, including introduction to graph theory and four colours suffice. Francis guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked. It is not the easiest book around, but it runs deep and has a nice unifying theme of studying how. The tangency graph of a ball packing takes the balls as vertices and connect two vertices if and only if they are tangent. What are some good books for selfstudying graph theory.
A path from a vertex v to a vertex w is a sequence of edges e1. Their magnum opus, every planar map is fourcolorable, a book claiming a. In fact, this proof is extremely elaborate and only recently discovered and is known as. By the way, a natural follow up would be a four color algorithm. Four colour map problem an introduction to graph theory. Pdf arthur cayley frs and the fourcolour map problem.
He points out that many advances in graph theory were made during the process of proving the fourcolor theorem. Thinking about graph coloring problems as colorable vertices and edges at a high level allows us to apply graph co. The fourcolour map problem also has links with the theory of polyhedra, and cayle y had a lifelong interest in this subject. The fourcolour theorem the chromatic number of a planar graph is at most four. Clearly a graph can be constructed from any map the regions being represented by. Clearly the deletion of connections cannot cause an ncolorable graph to require any additional colors, so in order to prove the four color theorem it would be sufficient to consider only complete graphs. It could alternatively just be used as maths enrichment at any level. Similarly, an edge coloring assigns a color to each. 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. He asked his brother frederick if it was true that any map can be colored using four colors in such a way that adjacent regions i.
Graph 1 has 5 edges, graph 2 has 3 edges, graph 3 has 0 edges and graph 4 has 4 edges. This investigation will lead to one of the most famous theorems of. Graph coloring set 1 introduction and applications. Four colors suffice is strewn with good anecdotes, and the author. The proof involved reducing the planar graphs to about 2000 examples where if the theorem was false, it was shown one of these would be a counterexample. The four colour theorem mactutor history of mathematics. Find all the books, read about the author, and more. A simpler statement of the theorem uses graph theory. A computerchecked proof of the four colour theorem 1 the story.
Four color map problem an introduction to graph theory. Planar graphs are the tangency graphs of 2dimensional disk packings. It develops a thorough understanding of the structure of graphs, the techniques used to analyze problems in graph theory and the uses of graph theoretical algorithms in mathematics, engineering and computer science. He is currently a visiting professor at the london school of economics. Introductory graph theory by gary chartrand, handbook of graphs and networks. In this paper, we introduce graph theory, and discuss the four color theorem. History the four color theorem was proven in 1976 by kenneth appel and wolfgang haken.
If v is any vertex of g which is not in g1, then g1 is a component of the subgraph g. Here are three of the best, ordered from most accessible to most mathematical. Thus, the formal proof of the four color theorem can be given in the following section. Well, besides the obvious application to cartography, graph coloring algorithms and theory can be applied to a number of situations. Generalizations of the fourcolor theorem mathoverflow. The fourcolor theorem states that any map in a plane can be colored using. The proof theorem 1the four color theorem every planar graph is fourcolorable. The book is designed to be selfcontained, and develops all the graphtheoretical tools. Then we prove several theorems, including eulers formula and the five color theorem. 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. In graph theory, graph coloring is a special case of graph labeling.
In a graph, no two adjacent vertices, adjacent edges, or adjacent regions are colored with minimum number of colors. Suppose you are given four cubes with each of the six faces painted with one of the colors red, white, green, or yellow. The number of colors needed to properly color any map is now the number of colors needed to color any planar graph. Another problem of topological graph theory is the mapcolouring problem. Buy graphs, colourings and the fourcolour theorem oxford science. Four color theorem simple english wikipedia, the free. It took more than 100 years between conjecture and proof for this theorem. It is the number of edges connected coming in or leaving out, for the graphs in given images we cannot differentiate which edge is coming in and which one is going out to a vertex. In mathematics, the four color theorem, or the four color map theorem, states that, given any.
In graphtheoretic terminology, the fourcolor theorem states that the vertices of every planar. The four colour conjecture first seems to have been made by francis guthrie. Introducing graph theory with a coloring theme, chromatic graph theory explores connections between major topics in graph theory and graph colorings as well as emerging topics. Part i covers basic graph theory, eulers polyhedral formula, and the first published false proof of the four colour theorem. Failed attempts to solve the 4cc inspired developments in modern graph theory over the next century but its solution eluded the efforts of everyone who attempted it. Im studying the four colour theorem as part of my degree, and found it a really good base to hair my understanding in the fundamental steps taken in order to solve the problem. Graphs, colourings and the fourcolour theorem oxford science. Every graph can be constructed by first constructing a complete graph and then deleting some connections edges. A graph g gv, e is called llist colourable if there is a vertex colouring of g in which the colour assigned to a vertex v is chosen from a list lv associated with this vertex.
1421 599 532 437 210 1204 1252 825 1147 931 549 1401 245 1339 994 1071 537 973 501 760 1365 904 103 526 263 632 1240 128 1300 504 385 740 136 1313 596 634 1052 725 772 211 342 108 1415 24 1395 936 1113