The intuitive statement of the four color theorem, i. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as physics, chemistry, psychology, sociology, and theoretical computer science. Graph theory and the fourcolor theorem week 7 ucsb 2015 in this class, we are going to prove the fourcolor theorem. We can prove the following slightly stronger theorem, which illustrates the same idea. Theorem 1 if g is a simple graph whose maximum vertexdegree is d, then xg. If both summands on the righthand side are even then the inequality is strict. In graphtheoretic terms, the theorem states that for loopless planar, the chromatic number of its dual graph is.
The five color theorem is implied by the stronger four color theorem, but. The heawoods five color theorem as well as in particular four color theorem are very much essential for the concept of map coloring which are included in this chapter elegantly. Let v be a vertex in g that has the maximum degree. A path from a vertex v to a vertex w is a sequence of edges e1. This chapter also includes the detailed discussion of coloring of planar graphs.
A tree t is a graph thats both connected and acyclic. With so many common borders, we require a more mathematicallyoriented description of the problem. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as. There is also a platformindependent professional edition, which can be annotated, printed, and shared over many devices. Every planar graph can have its vertices colored with four colors in such a way that no edge connects two vertices of the same. 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.
Basic concepts in graph theory a subgraph,, of a graph,, is a graph whose vertices are a subset of the vertex set of g, and whose edges are a subset of the edge set of g. Graphs and trees, basic theorems on graphs and coloring of. This is usually done by constructing the dualgraphof the map, and then appealing to the compactness theorem of propositional. It might take more than 4 colors to color a map on, say, the surface of a bagel where you need 7 for some maps. 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. An easy application of the four color theorem shows that if crg 1. Xiangs formal proof of the four color theorem 2 paper.
Werner, verified the 1996 proof robertson, sanders, seymour, thomas proof of the theorem in coq see mathworld on the 4 color theorem. The formal proof proposed can also be regarded as an. Brooks theorem recall that the greedy algorithm shows that. Generalizations of the fourcolor theorem mathoverflow. List of theorems mat 416, introduction to graph theory. Vertex coloring is an assignment of colors to the vertices of a graph. This is certainly an important contribution, but its not like its the first proof of the theorem. List of theorems mat 416, introduction to graph theory 1.
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. A graph is bipartite if and only if it has no odd cycles. A graph is kcolorableif there is a proper kcoloring. Any graph produced in this way will have an important property. To avoid notational ambiguities, we shall always assume tacitly that v \e. The elements v2vare called vertices of the graph, while the. Mar 05, 20 we can now state the 4 color theorem in the language of graph theory. 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. More than any other field of mathematics, graph theory poses some of the deepest and most. Francis guthrie 1852 the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Brooks theorem 2 let g be a connected simple graph whose maximum vertexdegree is d. According to the theorem, in a connected graph in which every vertex has at most. This proof was first announced by the canadian mathematical society in 2000 and subsequently published by orient longman and universities press of india in 2008. 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.
Kempes proof for the four color theorem follows below. Then we prove several theorems, including eulers formula and the five color. Graph theory has experienced a tremendous growth during the 20th century. Students will gain practice in graph theory problems and writing algorithms. Part ii ranges widely through related topics, including mapcolouring. This proof was first announced by the canadian mathematical society in 2000 and. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism. Let g be the smallest planar graph in terms of number of vertices that cannot. Then vertex 5 adjacent to 6 and 1 must be colour c. A subgraph is a spanning subgraph if it has the same vertex set as g. In this paper, we introduce graph theory, and discuss the four color theorem. In graph theory, graph coloring is a special case of graph labeling. Neuware in mathematics, the four color theorem, or the four color map theorem, states that given any separation of a. The very best popular, easy to read book on the four colour theorem is.
Graph theory, four color theorem, coloring problems. Dirac introduced the concept of color criticality in order to simplify graph coloring theory. Thus, the vertices or regions having same colors form independent sets. A kcoloring of g is an assignment of k colors to the vertices of g in such a way that adjacent vertices are assigned different. The notes form the base text for the course mat62756 graph theory. In section 2, some notations are introduced, and the formal proof of the four color theorem is given in section 3.
Graphs, colourings and the fourcolour theorem oxford. In the complete graph, each vertex is adjacent to remaining n1 vertices. It might take more than 4 colors to color a map on, say, the surface of a bagel where you need 7 for. In an undirected graph, an edge is an unordered pair of vertices. In mathematics, the four color theorem, or the four color map theorem, states that, given any.
Graph coloring vertex coloring let g be a graph with no loops. A simpler statement of the theorem uses graph theory. Let us construct an undirected graph for paris, in which the. 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. 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. Every planar graph can have its vertices colored with four colors in such a way that no edge connects. In graph theory, brooks theorem states a relationship between the maximum degree of a graph and its chromatic number. Appel and haken published an article in scienti c american in 1977 which showed that the answer to the problem is yes. Planar graphs are the tangency graphs of 2dimensional disk packings. The regions of any simple planar map can be coloured. If a graph ghas no subgraphs that are cycle graphs, we call gacyclic.
Nonplanar graphs can require more than four colors, for example this graph this is called the complete graph on ve vertices, denoted k5. Today we are going to investigate the issue of coloring maps and how many colors. For a more detailed and technical history, the standard reference book is. Through a considerable amount of graph theory, the four color theorem was reduced to a nite, but large number 8900 of special cases. Reinhard diestel graph theory 5th electronic edition 2016 c reinhard diestel this is the 5th ebook edition of the above springer book, from their series graduate texts in mathematics, vol.
Proof idea mathematical induction on the number of vertices of g. G of a graph g is the minimum k such that g is kcolorable. Their magnum opus, every planar map is fourcolorable, a book claiming a complete. Let g be the smallest planar graph in terms of number of vertices that cannot be colored with five colors. Pages in category theorems in graph theory the following 52 pages are in this category, out of 52 total. Part i covers basic graph theory, eulers polyhedral formula, and the first published false proof of the fourcolour theorem. More than any other field of mathematics, graph theory poses some of the deepest and most fundamental questions in pure mathematics while at the same time offering some of the must useful results directly applicable to real world problems. Avertexcoloring of agraphisanassignmentofcolorstotheverticesofthegraph. Nonplanar graphs can require more than four colors. Then we prove several theorems, including eulers formula and the five color theorem. While graph coloring, the constraints that are set on the graph are colors, order of coloring, the way of assigning color, etc. They will learn the four color theorem and how it relates to map coloring. The 6color theorem nowitiseasytoprovethe6 colortheorem. Four color theorem abebooks abebooks shop for books.
A counting theorem for topological graph theory 534. 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. A circuit starting and ending at vertex a is shown below. The more wellknown 4coloring theorem is much harder to prove. Department of mathematics and statistics, smith college, northampton, ma 01063. Connected a graph is connected if there is a path from any vertex to any other vertex.
Mastorakis abstractin this paper are followed the necessary steps for the realisation of the maps coloring, matter that stoud in the attention of many mathematicians for a long time. These notes include major definitions and theorems of the graph theory lecture held by prof. Graphs and trees, basic theorems on graphs and coloring of graphs. Graph is bipartite iff no odd cycle by sarada herke.
The four color theorem is a theorem of mathematics. The four color theorem 4ct essentially says that the vertices of a planar graph may be colored with no more than four different. We can now state the 4color theorem in the language of graph theory. Vertex 1 cannot have colour a, so label it with colour b. A ball packing is a collection of balls with disjoint interiors. Applications of the four color problem mariusconstantin o. 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. So the following is a generalization of four color theorem. Graph, g, is said to be induced or full if for any pair of. A coloring is proper if adjacent vertices have different colors. 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. We present a new proof of the famous four colour theorem using algebraic and topological methods.
A graph is rpartite if its vertex set can be partitioned into rclasses so no edge lies within a class. The tangency graph of a ball packing takes the balls as vertices and connect two vertices if and only if they are tangent. A coloring is given to a vertex or a particular region. A computerchecked proof of the four colour theorem 1 the story. Gonthier, georges 2008, formal proofthe fourcolor theorem pdf, notices. Rationalization we have two principal methods to convert graph concepts from. Francis guthrie 1852 the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map. In graph theoretic terminology, the four color 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. The four color theorem 28 march 2012 4 color theorem 28 march 2012. Werner, verified the 1996 proof robertson, sanders, seymour, thomas proof of the theorem in coq see mathworld on the 4.