The numerous attempts to solve the fourcolour problem have influenced the development of certain branches of graph theory. Prove that a complete graph with nvertices contains nn 12 edges. 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. In section 2, some notations are introduced, and the formal proof of the four color theorem is given in section 3. A handchecked case flow chart is shown in section 4 for the proof, which can be regarded as an algorithm to color a planar graph using four colors so. Each person is a vertex, and a handshake with another person is an edge to that person. The proof is computerassisted in the sense that for two lemmas in the article we did not give proofs, and instead asserted that we have verified those statements using a computer. Graphs, colourings and the fourcolour theorem oxford. The four color theorem is a theorem of mathematics. Is there a proper coloring that uses less than four colors. Graph theory, fourcolor theorem, coloring problems. That is to say, he showed that any map on the sphere whatever could be colored with four colors.
The four color theorem asserts that every planar graph can be properly colored by four colors. 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. As part of my cs curriculum next year, there will be some graph theory involved and this book covers much much more and its a perfect introduction to the subject. Then we prove several theorems, including eulers formula and the five color theorem. Get your students to attempt to colour in the maps using the least number of colours they. A kcoloring of a graph is a proper coloring involving a total of k colors. Through a considerable amount of graph theory, the four color theorem was reduced to a nite, but large number 8900 of special cases. It is used in many realtime applications of computer science such as. This is usually done by constructing the dualgraphof the map, and then appealing to the compactness theorem of propositional logic. Graph theory is also concerned with the problem of coloring maps such that no two adjacent regions of a map share the same color. 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.
They are called adjacent next to each other if they share a segment of the border, not just a point. The core of kempes theorem relied on a proven theorem, a nowcommon graph theory. In early july of 1879, alfred bray kempe announced in nature that he had a proof of the four colour conjecture. The notes form the base text for the course mat62756 graph theory. The four color problem is discussed using terms in graph theory, the study graphs. In 1858, in the same month as he presented his famous. In this paper we present a story as a comic based on graph theory. In a graph, no two adjacent vertices, adjacent edges, or adjacent regions are colored with minimum number of colors. Coloring the four color theorem this activity is about coloring, but dont think its just kids stuff.
Four color theorem simple english wikipedia, the free. A tree t is a graph thats both connected and acyclic. History the four color theorem was proven in 1976 by kenneth appel and wolfgang haken. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism. Four color problem has contributed to important research in graph theory, such as chromatic numbers of graphs. Suppose you are given four cubes with each of the six faces painted with one of the colors red, white, green, or yellow. 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. Pdf arthur cayley frs and the fourcolour map problem. 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. We could put the various lectures on a chart and mark with an \x any pair that has students in common. 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.
The intuitive statement of the four color theorem, i. Four color map problem an introduction to graph theory. This problem is an outgrowth of the wellknown four colour 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. In this paper, we introduce graph theory, and discuss the four color theorem. The fourcolour map problem also has links with the theory of polyhedra, and cayle y had a lifelong interest in this subject. Another problem of topological graph theory is the mapcolouring problem.
In 1976 an affirmative answer to the fourcolour problem, with the use of a computer, was announced cf. The fourcolor theorem abbreviated 4ct now can be stated as follows. It could alternatively just be used as maths enrichment at any level. The notorious fourcolor problem university of kansas. A computerchecked proof of the four colour theorem georges gonthier. Similarly, an edge coloring assigns a color to each. The regions aeb and befc are adjacent, as there is a common edge be between those two regions.
You want to make sure that any two lectures with a common student occur at di erent times to avoid a con ict. Introduction to graph theory dover books on mathematics. A graph is a set of vertices, where a pair of vertices are connected with an edge if some relation holds between the two. Here we give additional details for one of those lemmas, and we include the original computer.
An example of a plane graph with a 4coloring is given in the left half of figure 1. In the complete graph, each vertex is adjacent to remaining n1 vertices. It was the first major theorem to be proved using a computer. The book is really good for aspiring mathematicians and computer science students alike. G, this means that every face is an open subset of r2 that. Combinatorics combinatorics applications of graph theory. In graphtheoretic terms, the theorem states that for loopless planar, the chromatic number of its dual graph is. Xiangs formal proof of the four color theorem 2 paper. A graph g is said to be planar if it can be represented on a plane in such a fashion that the vertices are all distinct points, the edges are simple curves, and no two edges meet one another except at their terminals. Part i covers basic graph theory, eulers polyhedral formula, and the first published false proof of the fourcolour theorem.
The purpose of this question is to collect generalizations, variations, and strengthenings of the four color theorem with a description of their status. This is the unique graph of four mutually adjoining. Kempe 18451922 published a solution of the fourcolor problem. Introduction to graph theory and applications 1 introduction to graph theory and applications. The elements v2vare called vertices of the graph, while the e2eare the graphs edges. 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. 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. If a graph ghas no subgraphs that are cycle graphs, we call gacyclic. A graph that has a kcoloring is said to be kcolorable. 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. Graph coloring is nothing but a simple way of labelling graph components such as vertices, edges, and regions under some constraints.
Appel and haken published an article in scienti c american in 1977 which showed that the answer to the problem is yes. The four colour theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the regions of the. It took more than 100 years between conjecture and proof for this theorem. Generalizations of the fourcolor theorem mathoverflow. Four colour map problem an introduction to graph theory. Let the vertices of a graph such that every two adjacent vertices have different color is a very common problem in the graph theory. In graph theoretic terms, the theorem states that for loopless planar, the chromatic number of its dual graph is. So what we seek is a kcoloring of our graph with k as small as possible. Two regions that have a common border must not get the same color. Prove that in this group, there are four people who can be seated at a round table so that so that each person knows both his neighbours. They will learn the fourcolor theorem and how it relates to map coloring.
We discuss the value of using graph theory in constructing narratives. The proof of the four color theorem is the first computerassisted proof in mathematics. Graph theory, part 2 7 coloring suppose that you are responsible for scheduling times for lectures in a university. In graph theory, graph coloring is a special case of graph labeling. I made this resource as a hook into the relevance of graph theory d1. Also, hamilton made contributions to graph theory such as the idea of a hamiltonian circuit, i. Applications of graph coloring graph coloring is one of the most important concepts in graph theory. The fourcolour theorem the chromatic number of a planar graph is at most four. This number is called the chromatic number and the graph is called a properly colored graph. The four color theorem is an important result in the area of graph coloring. For example, k4, the complete graph on four vertices, is planar, as figure 4a shows. Assume that a complete graph with kvertices has kk 12. A general theory of reducibility was established by george david birkho. This investigation will lead to one of the most famous theorems of mathematics and some very interesting results.1057 950 515 1099 674 1074 1154 705 894 341 271 807 907 1198 938 1039 427 1440 768 1062 126 1064 1517 465 1167 149 1132 926 1436 301 1350 67 739