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== <u></u>'''Example 1: Square''' == | == <u></u>'''Example 1: Square''' == | ||
| − | '''A basic example of how to use the | + | '''A basic example of how to use the theorems is a simple two-coloring of a square. <br>''' |
| + | |||
| + | '''Step 1: Look at all the different possible combinations of colorings:''' | ||
| + | |||
| + | |||
| + | |||
| + | {| width="200" border="1" cellspacing="1" cellpadding="1" | ||
| + | |- | ||
| + | | gggg | ||
| + | | gggr | ||
| + | | ggrg | ||
| + | | rggg | ||
| + | |- | ||
| + | | grgg | ||
| + | | ggrr | ||
| + | | rgrg | ||
| + | | rrgg | ||
| + | |- | ||
| + | | grrg | ||
| + | | rgrr | ||
| + | | grrr | ||
| + | | rrgr | ||
| + | |- | ||
| + | | rrrg | ||
| + | | rrrr | ||
| + | | grrg | ||
| + | | rggr | ||
| + | |} | ||
| + | |||
| + | |||
== '''Definitions:''' == | == '''Definitions:''' == | ||
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http://math.berkeley.edu/~mbaker/Tucker.pdf | http://math.berkeley.edu/~mbaker/Tucker.pdf | ||
| − | http://www.whitman.edu/mathematics/SeniorProjectArchive/2012/Huisinga.pdf | + | http://www.whitman.edu/mathematics/SeniorProjectArchive/2012/Huisinga.pdf |
| − | + | ||
| + | <br> | ||
<br> [[2014 Spring MA 375 Walther|Back to MA375 Spring 2014]] | <br> [[2014 Spring MA 375 Walther|Back to MA375 Spring 2014]] | ||
[[Category:MA375Spring2014Walther]] [[Category:Math]] [[Category:Project]] | [[Category:MA375Spring2014Walther]] [[Category:Math]] [[Category:Project]] | ||
Revision as of 12:32, 20 April 2014
We discuss in class colorings of graphs, where adjacent vertices have different colors. Suppose you took the graph to be a polygon and allowed the graph to be reflected and rotated. How many different colorings do you get?
Contents
Outline/Title?
Introduction
In graph theory, it is sometimes necessary to find the number of ways to color the vertices of a polygon. Two theorems that work together to solve this problem are the Polya theorem and Burnside theorem.
Example 1: Square
A basic example of how to use the theorems is a simple two-coloring of a square.
Step 1: Look at all the different possible combinations of colorings:
| gggg | gggr | ggrg | rggg |
| grgg | ggrr | rgrg | rrgg |
| grrg | rgrr | grrr | rrgr |
| rrrg | rrrr | grrg | rggr |
Definitions:
- Burnside
- Polya
Formula:
- show formula
- breakdown of each element
- relate back to example 1
Proof:
References and Additional Information
For further reading on the Polya theorem:
http://arxiv.org/pdf/1001.0072.pdf
http://math.berkeley.edu/~mbaker/Tucker.pdf
http://www.whitman.edu/mathematics/SeniorProjectArchive/2012/Huisinga.pdf
