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Take a Tour with Euler

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  1. Elementary Graph Theory – Euler Circuits and Hamiltonian Circuits Amro Mosaad – Middlesex County Academy Take a Tour with Euler

  2. Leonhard Euler (1707-1783) • Swiss – also worked in Russia and Germany • Considered one of greatest and most prolific mathematicians ever; contributed greatly to Number Theory Calculus Geometry Trigonometry Algebra • Father of Graph Theory

  3. Named after Leonhard Euler • Euler's number (e) • Euler's formula • Euler's identity • Euler's theorem • Euler numbers • Euler approximations • Euler-Mascheroni constant • Euler path • Euler circuit

  4. Euler's Bridges of Konigsberg Problem

  5. Basic Graph Theory • Vertex (or node) - represented by a dot • Edge - segment connecting two vertices

  6. An Euler Circuit • A path that • (a) visits each edge exactly once, and • (b) starts and ends at the same vertex • Find an Euler circuit for the graph to the right

  7. Bridges of Konigsberg Problem • The key is to represent the map as a graph with vertices and edges - • each land mass is a vertex, and • each bridge is an edge

  8. A graph has an Euler path if there are no more than two vertices of odd degree. Euler Circuits • A graph has an Euler circuit if and only if all vertices have an even degree.

  9. Chinese Postman Problem • To find the shortest circuit of a graph that visits each edge (with some edges possibly visited more than once). It is called 'eulerizing' a graph.

  10. Hamiltonian Circuits • To visit each vertex of a graph exactly once and return to the starting vertex. • Named after Sir William Rowan Hamilton (1805-1865) – Irish physicist, astronomer, and mathematician

  11. The Icosian Game • Invented by Hamilton • The idea is to wrap the string around each of twenty pegs exactly once and return to the starting vertex

  12. Find a Hamiltonian Circuit

  13. Find a Hamiltonian Circuit

  14. The Traveling Salesman Problem(TSP)

  15. Solve this TSP

  16. Platonic Solids

  17. Further Study • Graph theory • Discrete • mathematics