The fourcolour theorem the chromatic number of a planar graph is at most four. Clearly every kchromatic graph contains akcritical subgraph. 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. These four regions were linked by seven bridges as shown in the diagram. It took more than 100 years between conjecture and proof for this theorem. I made this resource as a hook into the relevance of graph theory d1. In mathematics, the four color theorem, or the four color map theorem, states that, given any. Eg, then the edge x, y may be represented by an arc joining x and y. Four color map problem an introduction to graph theory.
The notes form the base text for the course mat62756 graph theory. It is not the easiest book around, but it runs deep and has a nice unifying theme of studying how. Even those with only a mild interest in coloring problems or graphs or topology will have fun reading this book. So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without. If a graph ghas no subgraphs that are cycle graphs, we call gacyclic. Iv, we will show how to construct the solutions to this graph problem. The four colour theorem the chromatic number of a planar graph is at most four.
We assume that there exists a minimal graph that is not four colorable, thus every smaller graph can be four colored, for coloring graphs we will use the colors. Applications of graph coloring graph coloring is one of the most important concepts in graph theory. 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. Diestel is excellent and has a free version available online. 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 have the same color, or for short, every planar graph is fourcolorable thomas 1998, p. A former president of the british society for the history of mathematics, he has written and edited many books on the history of. The graph mapping allows us to leverage the tools of spectral graph theory, which gives an immediate way to decompose graphs into disconnected components. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism. We could put the various lectures on a chart and mark with an \x any pair that has students in common. Connected a graph is connected if there is a path from any vertex. A circuit starting and ending at vertex a is shown below. Wilson introduction to graph theory longman group ltd. 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.
Inclusionexclusion, generating functions, systems of distinct representatives, graph theory, euler circuits and. But k3,3 is toroidal, that is it can be embedded on the torus. This problem is an outgrowth of the wellknown fourcolour map problem, which asks whether the countries on every map can be. 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. 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. Theorem 1 fourcolor theorem every planar graph is 4 colorable. Coloring the four color theorem this activity is about coloring, but dont think its just kids stuff. Graph theory 121 circuit a circuit is a path that begins and ends at the same vertex.
He points out that many advances in graph theory were made. It was the first major theorem to be proved using a computer. Graphs, colourings and the fourcolour theorem oxford. Graph theory, part 2 7 coloring suppose that you are responsible for scheduling times for lectures in a university. 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. You want to make sure that any two lectures with a common student occur at di erent times.
In graphtheoretic terms, the theorem states that for loopless planar, the chromatic number of its dual graph is. The intuitive statement of the four color theorem, i. The elements v2vare called vertices of the graph, while the e2eare the graphs edges. About onethird of the course content will come from various chapters in that book. Each person is a vertex, and a handshake with another person is an edge to that person.
Graph theory combinatorics and optimization university. May 17, 2015 i made this resource as a hook into the relevance of graph theory d1. Formal proofthe four color theorem american mathematical. Applications of graph coloring graph coloring is one of the most important concepts in. A tree t is a graph thats both connected and acyclic. 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. Four color theorem simple english wikipedia, the free. It is used in many realtime applications of computer science such as. The concept of a graph was known already to euler in the early eighteenth century, but it was the notorious four colour problem, posed by f.
In an acyclic graph, the endpoints of a maximum path have only one neighbour on the path and therefore have degree 1. Four colour map problem an introduction to graph theory. Their magnum opus, every planar map is fourcolorable, a book claiming a complete and detailed proof with a. But hang on a second what if our graph has more than one node and more than one edge. Xiangs formal proof of the four color theorem 2 paper. He is currently a visiting professor at the london school of economics. Free graph theory books download ebooks online textbooks. Website with complete book as well as separate pdf files with each individual chapter. A historical overview of the fourcolor theorem sigmaa history. 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.
Guthrie in the midnineteenth century, that spurred the development of this simple concept into a flourishing theory. The concept of a graph was known already to euler in the early eighteenth century, but it was the notorious fourcolour problem, posed by f. In section 2, some notations are introduced, and the formal proof of the four color theorem is given in section 3. Graph coloring is nothing but a simple way of labelling graph components such as vertices, edges, and regions under some constraints. Moreover, when just one graph is under discussion, we usually denote this graph by g. We use the symbols vg and eg to denote the numbers of vertices and edges in graph g. A gentle introduction to graph theory basecs medium. Introduction to graph theory allen dickson october 2006 1 the k. What are some good books for selfstudying graph theory.
Then we prove several theorems, including eulers formula and the five color. 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 can, without any adjacent sections being the same colour. A free powerpoint ppt presentation displayed as a flash slide. A ball packing is a collection of balls with disjoint interiors. While trying to color a map of the counties of england, francis guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. The river divided the city into four separate landmasses, including the island of kneiphopf. K6nig 1 published the first book on graph theory with notions later used to formulate conjectures equivalent to the fourcolor problem. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic. The four color theorem is one of many mathematical puzzles which share. The purpose of this question is to collect generalizations, variations, and strengthenings of the four color theorem with a description of their status.
Guthrie in the midnineteenth century, that spurred the. The elements v2vare called vertices of the graph, while the e2eare the graphs. This investigation will lead to one of the most famous theorems of mathematics and some very interesting. The four color theorem is an important result in the area of graph coloring. It is entertaining, erudite and loaded with anecdotes. Inclusionexclusion, generating functions, systems of distinct representatives, graph theory, euler circuits and walks, hamilton cycles and paths, bipartite graph, optimal spanning trees, graph coloring, polyaredfield counting. Prove that a complete graph with nvertices contains nn 12 edges. Then x and y are said to be adjacent, and the edge x, y. Part i covers basic graph theory, eulers polyhedral formula, and the first published false proof of the four colour theorem. Alexanderson, maa online i loved robin wilsons book.
In this paper, we introduce graph theory, and discuss the four color theorem. The book is really good for aspiring mathematicians and computer science students alike. The four color theorem asserts that every planar graph can be properly colored by four colors. A free powerpoint ppt presentation displayed as a flash slide show on id.
Introduction to graph theory applications math section. A path from a vertex v to a vertex w is a sequence of edges e1. 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. A graph g consists of a nonempty set of elements vg and a subset eg of the set of unordered pairs of distinct elements of vg. 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. Graph theory is also concerned with the problem of coloring maps such that no two adjacent regions of a map share the same color. As part of my cs curriculum next year, there will be some graph theory involved and this book.
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. Prove that in this group, there are four people who can be seated at a round table so that so that each person knows. In a graph, no two adjacent vertices, adjacent edges, or adjacent. Graph theory combinatorics and optimization university of. The dots are called nodes or vertices and the lines are.
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. The dots are called nodes or vertices and the lines are called edges. He asked his brother frederick if it was true that. It could alternatively just be used as maths enrichment at any level. The proof of the four color theorem is the first computerassisted proof in mathematics. In factit will pretty much always have multiple edges if.
The four color theorem is a theorem of mathematics. Generalizations of the fourcolor theorem mathoverflow. Draw a graph consisting of four disconnected vertices r, g, y, and w. Cs6702 graph theory and applications notes pdf book. Every connected graph with at least two vertices has an edge. Assume that a complete graph with kvertices has kk 12. You want to make sure that any two lectures with a common student occur at di erent times to avoid a con ict. History the four color theorem was proven in 1976 by kenneth appel and wolfgang haken. The elements of vg, called vertices of g, may be represented by points. This is usually done by constructing the dualgraphof the map, and then appealing to the compactness theorem of propositional logic. For example, the textbook graph theory with applications, by bondy and murty, is freely available see below. Alexanderson, maa online i loved robin wilsons book on the four color problem, because it gives the history as well as the arguments. In the complete graph, each vertex is adjacent to remaining n1 vertices. Introduction to graph theory dover books on mathematics.
This investigation will lead to one of the most famous theorems of mathematics and some very interesting results. Another problem of topological graph theory is the mapcolouring problem. 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. The tangency graph of a ball packing takes the balls as vertices and connect two vertices if and only if they are tangent. The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. In graph theory, graph coloring is a special case of graph labeling. 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. Jun 30, 2016 cs6702 graph theory and applications 1 cs6702 graph theory and applications unit i introduction 1. A graph g is kcriticalif its chromatic number is k, and every proper subgraph of g has chromatic number less than k.