Can you find a way to cross every bridge only once? The History: In Königsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once. A crude map of the center of Königsberg might look like this: Can you find a way to cross every bridge only once? Leonard Euler created a way of looking at the problem using vertices and edges. He found that the path suggested was impossible.

Euler and Hamilton Paths/Circuits Day 1

Definitions A __________ is a figure made up of points (vertices) connected by non-intersecting edges (Also, known as vertex-edge graphs) A ___________is the ___________ of two edges. A _______ is _____ if it is connected to an _____________________________ A _______ is ______ if it is connected to an _____________________________ network vertex intersection vertex odd odd number of edges vertex even even number of edges

More Definitions vertices An ______ joins any two___________. It can be _________ or ___________. edge curved straight Odd vertex Even vertex edge

Euler Paths A graph has an Euler path if it can be traced in 1 sweep without lifting the pencil from paper AND without tracing the same edge more than once. Vertices may be passed through more than once. The starting and ending points are not the same. 5

Euler Circuits A circuit is similar to a Euler path, except the starting and ending points must be the same.

Can you find an Euler Path or Circuit for the following networks? Neither Euler Circuit Euler Path

Complete the exploration on the relationship between the nature of the vertices and the kind of graph in your notes. Conclusions: Based on the observations of your table: A graph with all vertices being even contains an Euler _________ A graph with ______ odd vertices and _________________ contains an Euler ________. A graph with more than 2 _______ vertices does not contain an Euler _______________ circuit 2 some even vertices path odd path or circuit

To name a path or circuit you list the vertices in order Example 1: Name a Euler circuit A B C D E F One possible solution is D,E,F,A,D,C,A,B,D b) Can you find another one?

Example 2: Given A,B,E,F,B,C,D,F,E,D is this a Euler path or circuit or neither? How can you tell? Explain your answer Neither , touches EF twice A B C F E D Find a Euler circuit if possible, if not list a Euler path 2 odd vertices so has to be a path, starting at E or C 1 Possible solution: EBADEFDCBFC

Hamilton Paths and Circuits A ______________ is a continuous path that passes through every _________ once and only once. A _______________ is a Hamilton path that begins and ends at the same vertex. (the starting/end vertex will be the only vertex touched twice vertex Hamilton Circuit How is a Hamilton Path different from a Euler path or Circuit?

Finding a Hamilton Path Remember: In a Hamilton Path you only have to touch each vertex once, you don’t have to traverse each edge!!! S M There are many Hamilton paths for this network. One path name would be: MATHROKS (of course!) K A O T R H

Summary: HW: Homework #1 WS Where are paths and circuits used in real life?

Extra Materials/Practice http://mathworld.wolfram.com/Graph.html http://www.cut-the-knot.org/Curriculum/Combinatorics/GraphPractice.shtml http://mathforum.org/library/topics/graph_theory/?keyid=2145639&start_at=51&num_to_see=50