Konigsberg- in days past.

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 Invents Graph Theory Euler realized that all problems of this form could be represented by replacing areas of land by points (what we call nodes), and the bridges to and from them by arcs.

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

Paths and Circuits Euler path- a continuous path that passes through every edge once and only once. Euler circuit- when a Euler path begins and ends at the same vertex

Euler’s 1st Theorem If a graph has any vertices of odd degree, then it can't have any Euler circuit. If a graph is connected and every vertex has an even degree, then it has at least one Euler circuit (usually more).

Euler Circuit?

Draw the Vertex/edge graph and answer the following questions. 1) How many vertices are there? 6 2) How many edges are there? 9 3)How many vertices have a degree of 2? 3 4) How many vertices have a degree of 4? 3 Draw a Euler circuit starting at the vertex with a white dot. Remember: A circuit travels along every path exactly once and may pass through vertices multiple times before ending at the starting vertex.

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?

Trace a Hamiltonian path Only a path, not a circuit. The path did not end at the same vertex it started. The path does not need to go over every edge but it can only go over an edge once and must pass through every vertex exactly once. Hamiltonian Circuits are often called the mail man circuit because the mailman goes to every mailbox but does not need to go over every street.

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: Where are paths and circuits used in real life?