Every planar graph can have its vertices colored with four colors in such a way that no edge connects two vertices of the same. On the history and solution of the fourcolor map problem jstor. If the components are divided into sets a1 and b1, a2 and b2, et cetera, then let a iaiand b ibi. The four color map theorem and why it was one of the most controversial mathematical proofs. Two regions that have a common border must not get the same color.
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. Graph theory spring 2004 dartmouth college on writing proofs 1 introduction what constitutes a wellwritten proof. Students should also be aware of kuratowskys theorem, and the four color theorem. Every connected graph with at least two vertices has an edge. In particular, we present kempes proof of the fourcolor theorem. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism. For more detailed information visit the math 355 wiki page.
For a more detailed and technical history, the standard reference book is. Prove that if uis a vertex of odd degree in a graph, then there exists a path from uto another vertex vof the graph where valso has odd degree. Part i covers basic graph theory, eulers polyhedral formula, and the first published false proof of the fourcolour theorem. Neuware in mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. The mathematical reasoning used to solve the theorem lead to many practical applications in mathematics, graph theory, and computer science. This is usually done by constructing the dualgraphof the map, and then appealing to the compactness theorem of propositional. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete graph.
Introductory graph theory by gary chartrand, handbook of graphs and networks. Next, in a planar graph, we see that there must be a vertex with degree at most 5. Graphs, colourings and the fourcolour theorem oxford science. Else, 2e total degree 3v which contradicts with the fact e 3v 6. G, this means that every face is an open subset of r2 that. Diestel is excellent and has a free version available online. If t is a minimal counterexample to the four color theorem, then no good configuration appears in t. If both summands on the righthand side are even then the inequality is strict. The four color theorem is a theorem of mathematics. For an nvertex simple graph gwith n 1, the following are equivalent and. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color. For graph theory, wikipedia gives a good overview, and you can skip the really. They are called adjacent next to each other if they share a segment of the border, not just a point. Kempes proof was accepted for a decade until heawood showed an error using a map.
In graph theory, graph coloring is a special case of graph labeling. If v is any vertex of g which is not in g1, then g1 is a component of the subgraph g. According to the theorem, in a connected graph in which every vertex has at most. If there is time, it is good to study the proof of kuratowskis theorem. The four color theorem coloring a planar graph youtube. Obviously the above graph is not 3colorable, but it is 4 colorable. 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. 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. In mathematics, the four color theorem, or the four color map theorem, states that, given any. For every internally 6connected triangulation t, some good configuration appears in t.
The five color theorem is implied by the stronger four color theorem, but. Note that this map is now a standard map each vertex meets exactly three edges. The four color theorem, or the four color map theorem, states that given any separation of the plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. The four color theorem asserts that every planar graph and therefore every map on the plane or sphere no matter how large or complex, is 4 colorable. The fourcolor theorem states that any map in a plane can be colored using. The very best popular, easy to read book on the four colour theorem is. Why doesnt this figure disprove the four color theorem.
We can now state the 4color theorem in the language of graph theory. Our proof proceeds by induction on, and, for each, we will use induction on n. Both these proofs are computerassisted and quite intimidating. Let v be a vertex in g that has the maximum degree. Graphs, colourings and the fourcolour theorem and millions of other books. Tait, in 1880, showed that the four color theorem is equivalent to the statement that a certain type of graph called a snark in modern terminology must be nonplanar. While theorem 1 presented a major challenge for several generations of mathematicians, the corresponding statement for ve colors is fairly easy to see.
Theorem 1 fourcolor theorem every planar graph is 4colorable. The four colour theorem nrich millennium mathematics project. Four color theorem 4ct states that every planar graph is four colorable. The situation was partially remedied 20 years later, when robertson, sanders, seymour, and thomas published a new proof of. History, topological foundations, and idea of proof softcover reprint of the original 1st ed. The 6color theorem nowitiseasytoprovethe6 colortheorem. History, topological foundations, and idea of proof. Four, five, and six color theorems nature of mathematics.
In graph theory, brooks theorem states a relationship between the maximum degree of a graph and its chromatic number. One aspect of the fourcolor theorem, which was seldom covered and relevant to the field of visual communication, is the actual effectiveness of the distinct 4 colors scheme chosen to define its mapping. If gis a connected planar graph on nitely many vertices, then. From the above two theorems it follows that no minimal counterexample exists, and so the 4ct is true. Before i ever knew what the four color theorem was, i noticed that i could divide up a map into no more than four colors. A simpler statement of the theorem uses graph theory.
It is not the easiest book around, but it runs deep and has a nice unifying theme of studying how. The proof was reached using a series of equivalent theorems. A graph is planar if it can be drawn in the plane without crossings. The reader should be able to understand each step made by the author without struggling. Much of the material in these notes is from the books graph theory by reinhard diestel and introductiontographtheory bydouglaswest. Many graph theory books are available for readers who may want to. The four colour conjecture was first stated just over 150 years ago, and finally. 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. We want to color so that adjacent vertices receive di erent colors. February 1, 2008 abstract a simpler proof of the four color theorem is presented. Graphs, trees, paths and cycles, connectedness, chromatic number, planarity conditions, genus of a graph, the five color theorem. A simpler proof of the four color theorem is presented. This video was cowritten by my super smart hubby simon mackenzie. Despite the seeming simplicity of this proposition, it was only proven in 1976, and then only with the aid of computers.
There are two proofs given by appel,haken 1976 and robertson,sanders,seymour,thomas 1997. Planar map is fourcolorable, a book claiming a complete and detailed proof with a. I use this all the time when creating texture maps for 3d models and other uses. In 1943, hugo hadwiger formulated the hadwiger conjecture, a farreaching generalization of the fourcolor problem that still remains unsolved. Show that if every component of a graph is bipartite, then the graph is bipartite. The translation from graph theory to cartography is readily made by noting. A simple but rather vague answer is that a wellwritten proof is both clear and concise.
A path from a vertex v to a vertex w is a sequence of edges e1. Recall that a graph is a collection of points, calledvertices, and a. What are some good books for selfstudying graph theory. Avertexcoloring of agraphisanassignmentofcolorstotheverticesofthegraph. The notes form the base text for the course mat62756 graph theory. For each vertex that meets more than three edges, draw a small circle around that vertex and erase the portions of the edges that lie in the circle. Let g be the smallest planar graph in terms of number of vertices that cannot be colored with five colors.
Theorem b says we can color it with at most 6 colors. Formal proofthe four color theorem institute for computing. 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. Four color theorem abebooks abebooks shop for books. It was first proven by appel and haken in 1976, but their proof was met with skepticism because it heavily relied on the use of computers. Then there is some vertex vin our graph with degree at. In graph theoretic terminology, the fourcolor theorem states that the vertices of every. Graphs, colourings and the fourcolour theorem oxford. Canterbury, published his proof of the fourcolor theorem in the journal of.