Graph theory

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An undirected graph.

Graph theory is a field of mathematics about graphs. A graph is an abstract representation of: A number of points are connected by lines. Each point is usually called a vertex (more than one are called vertices), and the lines are called edges. Graphs are a tool for modelling relationships. They are used to find answers to a number of problems.

Some of these questions are:

  • What is the best way for a mailman to get to all of the houses in the area in the least amount of time? The points could represent street corners and lines could represent the houses along the street. (see Chinese postman problem)
  • A salesman has to visit different customers, but wants to keep the distance traveled as small as possible. The problem is to find a way so they can do it. This problem is known as Travelling Salesman Problem (and often abbreviated TSP). It is among the hardest problems to solve. If a commonly believed conjecture is true (described as PNP), then an exact solution requires one to try all possible routes to find which is shortest.
  • How many colors would be needed to color a map, if countries sharing a border are colored differently? The points could represent the different areas and the lines could represent that two areas are neighboring. (look at the Four color theorem)
  • Can a sketch be drawn in one closed line? The lines of the drawing are the lines of the graph and when two or more lines collide, there is a point in the graph. The task is now to find a way through the graph using each line one time. (look at Seven Bridges of Königsberg)

Different kinds of Graphs[change | edit source]

A directed graph.
  • Graph theory has many aspects. Graphs can be directed or undirected. An example of a directed graph would be the system of roads in a city. Some streets in the city are one way streets. This means, that on those parts there is only one direction to follow.
  • Graphs can be weighted. An example would be a road network, with distances, or with tolls (for roads).
  • The nodes (the circles in the schematic) of a graph are called vertices. The lines connecting the nodes are called edges. There can be no line between two nodes, there can be one line, or there can be multiple lines.

History[change | edit source]

The Königsberg Bridge problem, on a city map
The same problem, but using a graph

Leonhard Euler used to live in a city that was called Königberg for some time. (The city has been called Kaliningrad since 1946). The city is situated on the river Pregel. There is an island in the river, and there are a number of bridges across the river. Euler loved to go for walks, so he asked himself, whether it was possible to do a walk, use each of the bridges once, and come back to the point where he started. In 1736, he published a scientific article where he showed that this was not possible. Today, this problem is known as the Seven Bridges of Königsberg. The article is seen as the first paper in the history of graph theory.[1]

This article, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula was about the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy[2] and L'Huillier,[3] and is at the origin of topology.

Over a century after Euler, Cayley studied a special class of graphs, called the trees. A tree is a graph where there is only one path between two vertices. Cayley tried to solve a problem from differential calculus. His solution influenced the development of theoretical chemistry. The technique he used mainly focused on listing all graphs which had certain properties. Today, this is called Enumerative graph theory. It developed from Cayleys results, a scientific publication done by Pólya between 1935 and 1937 and the generalization of these by De Bruijn in 1959. Cayley linked his results on trees with the contemporary studies of chemical composition.[4]

The fusion of the ideas coming from mathematics with those coming from chemistry is at the origin of a part of the standard terminology of graph theory. In particular, the term "graph" was introduced by Sylvester in an article published in 1878 in Nature.[5]

One of the most famous and productive problems of graph theory is the four color problem: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by Francis Guthrie in 1852 and its first written record is in a letter of De Morgan addressed to Hamilton the same year. Many incorrect proofs have been proposed, including those by Cayley, Kempe, and others. The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. Tait's reformulation generated a new class of problems, the factorization problems, particularly studied by Petersen and Kőnig. The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, extremal graph theory.

The four color problem remained unsolved for more than a century. A proof produced in 1976 by Kenneth Appel and Wolfgang Haken,[6][7] which involved checking the properties of 1,936 configurations by computer, was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas.[8]

The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist Gustav Kirchhoff, who published in 1845 his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits.

Graph theory in perspective[change | edit source]

Graph theory is an important part of mathematics and computer science. To many such problems, exact solutions do exist. Many times however, they are very hard to calculate. Therefore, very often, approximations are used. There are two kinds of such approximations, Monte-Carlo algorithms and Las-Vegas algorithms.

Graphs are normally represented by two different sets, typically set a graph G would be represented as the collection of the sets V and E. The set V is a discrete set containing all vertices of the graph. The set E is a binary set, whose pairwise elements are elements of set V. Each pair in set E represents an edge connecting two vertices.

If any two nodes have an edge between them, then the graph is called the complete graph.

References[change | edit source]

  1. Biggs, N.; Lloyd, E. and Wilson, R. (1986). Graph Theory, 1736-1936. Oxford University Press.
  2. Cauchy, A.L. (1813). "Recherche sur les polyèdres - premier mémoire". Journal de l'Ecole Polytechnique 9 (Cahier 16): 66–86.
  3. L'Huillier, S.-A.-J. (1861). "Mémoire sur la polyèdrométrie". Annales de Mathématiques 3: 169–189.
  4. Cayley, A. (1875). "Ueber die Analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen". Berichte der deutschen Chemischen Gesellschaft 8: 1056–1059. doi:10.1002/cber.18750080252.
  5. Sylvester, J.J. (1878). "Chemistry and Algebra". Nature 17: 284. doi:10.1038/017284a0.
  6. Appel, K. and Haken, W. (1977). "Every planar map is four colorable. Part I. Discharging". Illinois J. Math. 21: 429–490.
  7. Appel, K. and Haken, W. (1977). "Every planar map is four colorable. Part II. Reducibility". Illinois J. Math. 21: 491–567.
  8. Robertson, N.; Sanders, D.; Seymour, P. and Thomas, R. (1997). "The four color theorem". Journal of Combinatorial Theory Series B 70: 2–44. doi:10.1006/jctb.1997.1750.