graph theory (networks)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Section 4.1, Slide 1

4 Graph Theory (Networks)

The Mathematics of Relationships

4



Graphs, Puzzles, and Map Coloring

4.1

• Understand graph terminology

• Apply Euler’s theorem to graph tracing

• Understand when to use graphs as models

(continued on next slide)



Graphs, Puzzles, and Map Coloring

4.1

• Use Fleury’s theorem to find Euler circuits

• Utilize graph coloring to simplify a problem

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 4

Graph Terminology


Examples of Graphs

http://oak.cats.ohiou.edu/~ridgely/Volvo_docs/

http://maps.google.com/?ie=UTF8&ll=57.703528,11.966429&spn=0.017449,0.036993&z=15


Graph Terminology


Graph Tracing

• The Koenigsberg bridge problem

Starting at some point, can you visit all parts of the city, crossing each bridge once and only once, and return to the starting point?


Graph Tracing

• We can model the problem with a graph model.


Graph Tracing

• The Koenigsberg problem, phased in graph theory language, is “Can the graph be traced?”

To trace a graph means to begin at some vertex and draw the entire graph without lifting the pencil and without going over any edge more than once.


Graph Tracing

*Connected graphs are also called networks.


Graph Tracing

• Example:

(solution on next slide)


Graph Tracing

• Example:

Odd: B and C

Even: A, D, E, and F

(Zero edges is considered “even”)


Euler’s Theorem


Euler’s Theorem

• Example: Which of the graphs can be traced?



Euler’s Theorem

• Solution:

Not trace-able

Trace-able

Trace-ableTrace-able


Euler’s Theorem


Euler’s Theorem

• Example:

• Solution:Path ACEB has length 3.

Path ACEBDA is an Euler path of length 5. It is also an Euler circuit.


Fleury’s Algorithm

• We use Fleury’s algorithm to find Euler circuits.

(example on next slide)



• Example:

(solution on next 3 slides)



• Solution: (answers may vary)


erase





erase


Eulerizing a Graph

• We can add edges to convert a non-Eulerian graph to an Eulerian graph.

• This technique is called Eulerizing a graph.

(example on next slide)


Eulerizing a Graph

• Example:



Eulerizing a Graph


Eulerize


Map Coloring


Map Coloring

• Example:



Map Coloring

• Solution:


We can rephrase the map-coloring question now as follows: Using four or fewer colors, can we color the vertices so that no two vertices of the same edge receive the same color?


Map Coloring

• Solution:

There is no particular method for solving the problem.

A trial-and-error is shown.

Other solutions are possible.


Map Coloring


Map Coloring

• Example:Each member of a city council usually serves on several committees in city government. Assume council members serve on the following committees: police, parks, sanitation, finance, development, streets, fire department, and public relations. Use the table below to determine a conflict-free schedule for the meetings. We do not duplicate information - that is, because police conflicts with fire department, we do not also list that fire department conflicts with police.



Map Coloring

• Solution:

Model the information with a graph. Join conflicting committees with an edge.



Map Coloring

• Solution:

Using trial and error and four colors or less, color committees that can meet at the same time with a common color. That is, similar-colored vertices should not share a common edge (same problem as the four-color problem).

graph theory (networks)

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