1 / 12

Graph Theory

MATH 3220 By Patrick Curry. Graph Theory. Seven Bridges of K önigsberg. The first known work on graph theory was Leonhard's Euler's paper on The Seven Bridges of K önigsberg in 1736. The problem of the seven bridges was to traverse each bridge of Königsberg once and only once.

ban
Télécharger la présentation

Graph Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MATH 3220 By Patrick Curry Graph Theory

  2. Seven Bridges of Königsberg • The first known work on graph theory was Leonhard's Euler's paper on The Seven Bridges of Königsberg in 1736. • The problem of the seven bridges was to traverse each bridge of Königsberg once and only once. Source: Wikipedia

  3. Source: Wikipedia Seven Bridges of Königsberg (cont)‏ • Euler concluded that this was impossible and realized that only the number of vertices and paths were relevant in the problem. • Because only the vertices and lines were important, the diagram for a graph became combinations of points and the lines connecting them.

  4. A graph consists of two sets named the vertex set and the edge set. The vertex set is a nonempty set, and the edge set may be empty, but if not it contains two-element subsets of the vertex set. In drawing the graph, the location of the dots and lines are irrelevant,but how they connect to each other does matter. Also, the direction of the lines may be added resulting in a directed graph. Basic Graphs Source: Trudeau

  5. Drawing Graphs • The two above diagrams are equal graphs. • They are also complete graphs of vertex 4. • A complete graph has all possible edges of a positive integer of vertices. Source: Wikipedia, Britannica

  6. Graph Coloring • Graph coloring involves labeling each vertex of a graph with a color so that all adjacent vertices have different colors. • Another part of graph coloring involves edge coloring. In it, each edge of a graph is colored so that every adjacent edge has a different color. • The smallest possible number of colors that works for this is called the graph's chromatic number. Source: Wikipedia

  7. Source: Trudeau Planer Graphs • A planer graph is a graph that can be drawn in a plane without intersections. • When these graphs are drawn without edge-crossings, the regions that the edges form in the graph are called faces. • Using these concepts, Euler proved in 1752 that the number of vertices and faces minus the number of edges is always 2. • Put into terms: v + f – e = 2

  8. Source: Wikipedia, Chartrand and Oellermann Four Color Theorem • The four color theorem says that every vertex in a planer graph can be colored with just four different colors. • The theorem was first introduced in 1852 by Francis Guthrie. • The final proven theorem was solved in 1976 by Appel and Haken, and their proof was checked by a computer.

  9. Four Color Theorem (cont)‏ • This problem is generally drawn as a map such as a political or boundary map. • By using each country as a vertex, the problem becomes a vertex coloring problem. Source: Wikipedia

  10. Source: Wikipedia Other Subjects in Graph Theory • Platonic Graphs • Hamilton Graphs • Directed Graphs or Digraphs • Networks • Random Graph Theory • Extremal Graph Theory • Weighted Graphs • Trees

  11. Source: Wikipedia Open Problems in Graph Theory • The Erdős–Faber–Lovász conjecture (unsolved)‏ • The total coloring conjecture (unsolved)‏ • The list coloring conjecture (unsolved)‏ • Traveling Salesman Problem • Chinese Postman Problem • Three Cottage Problem

  12. References • Trudeau, Richard. Introduction To Graph Theory. New York: Dover Publications, 1993. • Wallis, W.D. A Beginner's Guide to Graph Theory. Boston: Birkhäuser, 2000. • “Graph Theory.” 26 April 2008. Hofstra University <http://people.hofstra.edu/geotrans/eng/ch1en/meth1en/ch1m2en_2ed.html> • Chartrand, Gary and Ortrud Oellermann. Applied and Algorithmic Graph Theory. New York: McGraw-Hill, 1993. • "Graph theory." Wikipedia, The Free Encyclopedia. 16 Sep 2008, 17:03 UTC. 17 Sep 2008 <http://en.wikipedia.org/ w/index.php?title=Graph_theory&oldid=238839055>.

More Related