103, Math Section 11, Tuesday, April 5, 2011
Homework to be discussed in class on Tuesday, April 5 , Chapter 5, Page 191,
Problems 1,2 5, and 11
Homework for Chapter 5 is due when we finish Chapter 5.
Problems 20,26,30,34,38,42 and 46 will be collected and graded.
Homework for Friday, April 8. Page 192 Exercises 9, 10, 18, 19, 21.
Unicursal tracing- trace each drawing without lifting the pencil or retracing any of the
lines.
Courtesy of damianspain of the San Francisco Bay area.
Most students find the standard applications of Euler circuits and paths
(routing garbage pick-up, mail deliveries, security guard patrols, parades,
etc.) not too interesting or compelling. But here is a new application
that I think is very modern and cool: Google Street View. In order to
map an entire city at the street view level, Google carves out the
metropolitan area into sections. For each section, a car with a special
-degree camera mounted on the roof is assigned to travel along the
streets of that section. The goal is to sweep through the section covering
every block. When a block has to be covered more than once, the camera
is turned off to eliminate the need to edit out duplications. Obviously
you want to design the routes with the minimum number of
duplications. This is clearly a eulerization problem.
Peter Tannenbaum, California State UniversityFresno
Routing problems- finding ways to deliver goods
Existence and how to find the route
We will discuss Euler circuit problems-exhaustive routes
Named after Leonard Euler 1701-1783 (pronounced Leonard Oiler)
We will use the tools of Graph theory to solve these problems.
What is a graph? Dots are called vertices of the graph and lines are called the edges of
the graph
A graph is a set of points called vertices and a list of lines called edges, connecting
some of the vertices.
It is possible for an edge to connect a vertex back to itself. These types of edges are
called loops. That is, an edge from a vertex to itself is called a loop.
It is possible for 2 edges to connect the same pair of vertices. These edges are called
multiple edges.
Here is a graph. The vertex set V={A,B,C,D,E}
The edge set E={AC,AD,BD,CC,CA,CD,CE,DA,DB,DC,DE,DE}
Figure 1
The edge CC is a loop.
The edges DE and DE are multiple edges.
2 vertices in a graph are adjacent if they are joined by an edge.
Vertices A and C are adjacent. A and B are not adjacent.
2 edges are adjacent if they share a common vertex.
.
Figure 1 is a connected graph.
How can you get from A to B?
Figure 2 is a disconnected graph.
An Euler path in a connected graph is a path that travels through all the edges of the
graph. Every edge of the graph is traveled once and only once.
An Euler circuit is a circuit travels that through every edge of a connected graph. The
trip must end where it started and travel along every edge of the graph once and only
once.
A connected graph cannot have both an Euler path and an Euler circuit.-it can have one
or the other or neither, but not both.
In the following 3 graphs, one has an Euler path, one has an Euler circuit, and one has
neither an Euler circuit nor an Euler path.
Which is which?
Practice Exercise 1.
a. For the graph shown above, what is the edge set?
b. Consider the graph from problem 1; what is the vertex set ?
c. Consider the graph from problem 1; how many paths are there from A to B?
Practice Exercise 2.
a. For the graph shown above, what is the degree of vertex D?
b Which edges are adjacent to edge AD?
c. How many paths of size three are there which start at vertex A and end at vertex B?
Answers to Practice Exercises.
1. a. AB,AD,BC,BD,CC,CD
b. V={A,B,C,D}
c. There are 4 paths. path #1 is A,B; path #2 is A,D,B; path #3 is A,D,C,B
A,D,C,C,B
2. a. degree=4
b. Edges adjacent to AB, AC, AE, BD, CD, ED
c. 5 paths. They are:
Path 1 is A,D,E,B
Path 2 is A,C,D,B
Path 3 is A,E,D,B
Path 4 is A,E,B,B
Path 5 is A,D,B,B
path #4 is
Solutions to homework: Exercises 1,2 5,11 on pages 191-192
Graph Modeling-use graph theory graphs to represent word problems
Modeling Exercise 1
Exercise 17 on page 192
Below is a map of the downtown area of the hamlet of Kingsburg, with the Kings River running
through the downtown area and the 3 islands (A,B, and C) connected to each other and both
banks by 7 bridges. You have been hired by the Kingsburg Chamber of Commerce to organize
the annual downtown parade. Part of your job is to plan the route for the parade.
Draw a graph that models the layout of Kingsburg. Let the vertices represent land masses and the
edges represent bridges.
Modeling Exercise 2.
Modeling Exercise 3.
Draw a graph which models the practical problem of the mail carrier who
must walk along each side of a street which has houses on it, hence
twice along blocks which have houses on both sides of the street.
Modeling Exercise 4. Homework
4. Four visitors to Rutgers meet for lunch. Each visitor shakes hands with every other visitor
once and only once.
a. Using people as vertices and handshakes as edges, construct a graph which models this
situation.
b. How many handshakes (edges) are there?
Modeling Exercise 5. Homework
5. Five visitors to Rutgers meet for lunch. Each visitor shakes hands with every other visitor
once and only once.
a. Using people as vertices and handshakes as edges, construct a graph which models this
situation.
b. How many handshakes (edges) are there?
Modeling Exercise 1.
Modeling Exercise 2
Modeling Exercise 3