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

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

14. The Traveling Salesman Problem (TSP)

15. Solve this TSP

16. Platonic Solids

17. Further Study Graph theory Discrete mathematics