Recall that a graph is a collection of points, calledvertices, and a. Much of the material in these notes is from the books graph theory by reinhard diestel and introductiontographtheory bydouglaswest. So its this book of problems you will constantly run into in. We discuss such problems in chapter 6, where we try to colour the vertices of a. Today we are going to investigate the issue of coloring maps and how many colors are required. 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. 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. This problem inspired the great swiss mathematician leonard euler to create graph theory, which led to the development of topology. Text53 graph theory presents a natural, readerfriendly way to learn some of the essential ideas of graph theory starting from first principles. In mathematics, the four color theorem, or the four color map theorem, states that, given any. I was wondering if proof by induction or contradiction is better, but i decided for proof by induction, as this is easier to translate in actual code then. If youre interested in exploring some of the ideas of graph theory.
The 6color theorem nowitiseasytoprovethe6 colortheorem. Tymoczko in the article four color problem and its significance philosophic journal of philosophy, 1979. Appel and haken published an article in scienti c american in 1977 which showed that the answer to the problem is yes. A catalog record for this book is available from the library of congress. Ngo introduction to graph coloring the authoritative reference on graph coloring is probably jensen and toft, 1995. Much of the material in these notes is from the books graph theory by reinhard diestel and. This gives us one last rephrasing of the fourcolor theorem. Jun 30, 2016 cs6702 graph theory and applications notes pdf book anna university semester seven computer science and engineering slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. The proof was reached using a series of equivalent theorems. It is used in many realtime applications of computer science such as.
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. When a planar graph is drawn in this way, it divides the plane into regions called faces. 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. As a bonus, wefind that any graph which becomes planar upon the. A simpler statement of the theorem uses graph theory. Let g be a the smallest planar graph by number of vertices that has no proper 6coloring. We may assume g 3, since the result is easy otherwise. The intuitive statement of the four color theorem, i. When a connected graph can be drawn without any edges crossing, it is called planar. Alfred kempe proves the fourcolor theorem 4ct four colors suffice to color. The concept of graphs in graph theory stands up on some basic terms such as point, line, vertex, edge. For an nvertex simple graph gwith n 1, the following. The subject of graph theory had its beginnings in recreational math problems see number game, but it has grown into a significant area of mathematical research, with applications in chemistry, operations research, social sciences, and computer science. Let g be the smallest planar graph in terms of number of vertices that cannot be colored with five colors.
Pdf a generalization of the 5color theorem researchgate. 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. Recall that a graph is a collection of points, calledvertices, and a collection ofedges, which are connections between two vertices. Induction hypothesis now assume that any simple planar graph on v. We prove the result by induction on the number of vertices. For example the vertex 1, 3, 4 in q6 is coded by 1, 0, 1, 1, 0, 0, 0.
Graph theory, branch of mathematics concerned with networks of points connected by lines. The format is similar to the companion text, combinatorics. In the complete graph, each vertex is adjacent to remaining n1 vertices. Of course, both this result and the fivecolor theorem of the next section are subsumed by the fourcolor theorem. Part i covers basic graph theory, eulers polyhedral formula, and the first published false proof of the fourcolour theorem. Four color theorem simple english wikipedia, the free. Burnsides theorem will allow us to count the orbits, that is, the different colorings, in a variety of problems. We prove that six colors will suffice for every planar graph. According to the theorem, in a connected graph in which every vertex has at most. This problem was first posed in the nineteenth century, and it was quickly conjectured that in all cases four colors suffice. 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. Put your pen to paper, start from a point p and draw a continuous line and return to p again. What are some good books for selfstudying graph theory.
The elements v2vare called vertices of the graph, while the e2eare the graphs edges. Kempes proof for the four color theorem follows below. Associated to each edge are two distinguished vertices called ends. In fact, this proof is extremely elaborate and only recently discovered and is known as. Draw, if possible, two different planar graphs with the same number of vertices, edges, and faces. 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. Brooks theorem recall that the greedy algorithm shows that. Their magnum opus, every planar map is fourcolorable, a book claiming a complete and. The source of the maptograph illustrations i used in this post is this pdf. Let g be a the smallest planar graph by number of vertices that has no proper 6 coloring. The six color theorem 62 the six color theorem theorem.
If gis a connected planar graph on nitely many vertices, then. Cs6702 graph theory and applications notes pdf book. In this video lecture we will learn about theorems on graph, so first theorem is, the sum of degree of all the vertices is equal to twice the number of edges. These notes include major definitions and theorems of the graph theory lecture held by prof.
Graph theory 3 a graph is a diagram of points and lines connected to the points. An unlabelled graph is an isomorphism class of graphs. List of theorems mat 416, introduction to graph theory. The problem of map coloring arose as a topic of mathematical. Marcus, in that it combines the features of a textbook with those of a problem workbook. But fortunately, this is the kind of question that could be handled, and actually answered, by graph theory, even though it might be more interesting to interview thousands of people, and find out whats going on. Four, five, and six color theorems in 1852, francis guthrie pictured above, a british mathematician and botanist was looking at maps of the counties in england and discovered that he could always color these maps such that no adjacent country is the same color with at most four colors. The book is designed to be selfcontained, and develops all the graphtheoretical. Four, five, and six color theorems nature of mathematics. A graph is a pair of sets, whose elements called vertices and edges respectively. When a planar graph is drawn in this way, it divides the plane into regions called faces draw, if possible, two different planar graphs with the. In graph theory, brooks theorem states a relationship between the maximum degree of a graph and its chromatic number. Graph theory algorithmic, algebraic, and metric problems.
For an nvertex simple graph gwith n 1, the following are equivalent and. Much of this mathematics has developed a life of its own, and forms a fascinating part of the subject now known as graph theory. Put your pen to paper, start from a point p and draw a. Any simple planar graph can be properly colored with six colors. The number of colors needed to properly color any map is now the number of colors needed to color any planar graph. If the components are divided into sets a1 and b1, a2 and b2, et cetera, then let a iaiand b ibi.
In mathematics, the four color theorem, or 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. Show that if every component of a graph is bipartite, then the graph is bipartite. There is a part of graph theory which actually deals with graphical drawing and presentation of graphs, brie. G, as the smallest number ksuch that gis kcolorable. Then we prove several theorems, including eulers formula and the five color theorem. Cs6702 graph theory and applications notes pdf book anna university semester seven computer science and engineering slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. This chapter also includes the detailed discussion of coloring of planar graphs. Conversely any planar graph can be formed from a map in this way.
In recent years, graph theory has established itself as an important mathematical tool in. Graph theory has experienced a tremendous growth during the 20th century. Graphs, colourings and the fourcolour theorem oxford. It is not the easiest book around, but it runs deep and has a nice unifying theme of studying how. But we want to include the proofs, as the techniques they illustrate will have value later. A coloring is proper if adjacent vertices have different colors. List of theorems mat 416, introduction to graph theory 1. To understand the principles of the four color theorem, we must know some basic graph theory. In this paper, we introduce graph theory, and discuss the four color theorem. The four color theorem 28 march 2012 4 color theorem 28 march 2012. Apr 21, 2016 in this video lecture we will learn about theorems on graph, so first theorem is, the sum of degree of all the vertices is equal to twice the number of edges. In graphtheoretic terms, the theorem states that for loopless planar, the chromatic number of its dual graph is. Let v be a vertex in g that has the maximum degree.
Diestel is excellent and has a free version available online. 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. For many, this interplay is what makes graph theory so interesting. Avertexcoloring of agraphisanassignmentofcolorstotheverticesofthegraph. In fact, this proof is extremely elaborate and only recently discovered and is known as the 4colour map theorem. It has at least one line joining a set of two vertices with no vertex connecting itself. If a graph ghas no subgraphs that are cycle graphs, we call gacyclic. G, this means that every face is an open subset of r2 that. If both summands on the righthand side are even then the inequality is strict. Graph theory presents a natural, readerfriendly way to learn some of the essential ideas of graph theory starting from first principles. Applications of graph coloring graph coloring is one of the most important concepts in graph theory. Buy graphs, colourings and the fourcolour theorem oxford science publications on free shipping on qualified orders. First the maximum number of edges of a planar graph is obatined as well as the. Pages in category theorems in graph theory the following 52 pages are in this category, out of 52 total.
Pdf we present a short topological proof of the 5color theorem using only the nonplanarity of k6. A simpler proof of the four color theorem is presented. The crossreferences in the text and in the margins are active links. Apr 09, 2014 through a considerable amount of graph theory, the four color theorem was reduced to a nite, but large number 8900 of special cases. An introduction to graph theory basics and intuition with applications to scheduling, coloring, and even sexual promiscuity. The four color theorem 9 april 2014 4 color theorem 9 april 2014. Reinhard diestel graph theory electronic edition 2000 c springerverlag new york 1997, 2000 this is an electronic version of the second 2000 edition of the above springer book, from their series graduate texts in mathematics, vol. May 11, 2018 5 color theorem proof using mathematical induction method graph theory lectures discrete mathematics graph theory video lectures in hindi for b. A mobius strip requires six colors tietze 1910 as do 1planar graphs graphs. In proceedings of the thirtythird annual acm symposium on theory. Now of course, we could color this graph with five colors, every node could just get its own color.