1. Discrete math
Objectives: To learn vocabulary, vertex/edge patterns in discrete math.
Vertex Edge Graph - A collection of points some of which are joined by line segments or curves.
This graph has 6 vertices and 7 edges
Each point is a vertex and each line is an edge

2. Degrees of a vertex
The degree of a vertex in a graph is the number of edges that touch it. 2 3
Each vertex is labeled with its degree 4 3 3 3 2
A graph is regular if every vertex has the same degree. 2 2 2

3. Euler Graph (pronounced oilier)
An Eulerian path in a graph is a path that travels along every edge of the graph exactly once. An Eulerian path might pass through individual vertices of the graph more than once.
A Euler path is a snowplow problem where a snow plow needs to plow every street once.
Start and finish
Euler circuits is a path that ends at the same vertex it started

4. 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.

5. Implication (for a connected graph)
Use this chart to see if a Euler path or circuit may be drawn
# of ODD Vertices
Implication (for a connected graph)
There is at least one Euler Circuit. 1
THIS IS IMPOSSIBLE! Can’t be drawn 2
There is no Euler Circuit but at least 1 Euler Path.
more than 2
There are no Euler Circuits or Euler Paths.

6. Vocabulary:
Hamiltonian Circuit/Paths:
A Hamiltonian path in a graph is a path that passes through every vertex in the graph exactly once. A Hamiltonian path does not necessarily pass through all the edges of the graph, however.
A Hamiltonian path which ends in the same place in which it began is called a Hamiltonian circuit.

7. 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.

8. N 6 5 2 3 4
Bank
Pharmacy
Post Office
Dry Cleaners
Grocery Store
Home Sweet Home
Video Store
Find the quickest route from home to the pharmacy
A weighted edge has a value assigned to the edge (i.e.- miles, time, gallons of gas, any unit of measure)

9. Review:
Euler Graphs
Passes over edge exactly once. May pass through a vertex more than once.
Hamiltonian
Graphs
Passes through every vertex exactly once but not necessarily over every edge.
The path ends at the same vertex it started.
Circuits