Map Coloring to Graph Coloring

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Map Coloring to Graph Coloring

• Part of a unit on discrete mathematics.

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Discrete math What is it?

• It is mathematics which studies phenomena which
  are not continuous, but happens in small, or
  discrete, chunks.
• Some areas include graph theory (networks),
  counting techniques, coloring theory, game
  theory, and more.
• It is the mathematics of computers.
• It is finite, rather than infinite. It handles
  distinct chunks of information. You might talk
  about the schedule of flights leaving Chicago,
  but not the acceleration of one of those planes.

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Maps versus Graphs

• A map is a two-dimensional drawing with regions
  to be colored.

• A graph is a mathematical object made of dots
  connected by lines.

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Graphs in Computer Science

• The term graph is used in a different sense in
  mathematics to mean a chart displaying numerical
  data, such as a bar graph, but the graphs that
  computer scientists use are not related to these.
• In computer science, graphs are drawn using
  circles or large dots, technically called
  "nodes," to denote objects.
• Lines between nodes indicate some sort of
  relationship between the objects.

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For example, we can change one of our maps from
class into a graph
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We can represent the map of South America using a
graph

• Each circle represents a country.
• Countries which share a common border on the map
  are connected by lines called edges on the graph.

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We have been studying map coloring in class.
Recall the rules of map coloring

• Regions (or countries) which share a border must
  be colored different colors.
• Regions which touch at only one point at a time
  may be colored the same color.
• You must use the least amount of colors possible.

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We can study graph coloring as well

• Dots (or nodes) represent the regions, and a line
  between two nodes indicates that those two
  regions share a border.
• On the graph, the coloring rule is that no
  connected nodes should be colored the same color.
• Unlike a map, there is no limit to the number of
  colors that a general graph may require.
• The "graph coloring problem" is to find the
  minimum number of colors that are needed for a
  particular graph.
• This minimum number of colors needed to color a
  graph is called the chromatic number of the graph.

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Try coloring our Lands map using a graph
The graph can be colored with two colors, just
like the map! The chromatic number of this
graph is 2.
Now try coloring the South America map using a
graph!
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Here is another graph

• A line between two subjects indicates that at
  least one student is taking both subjects, and so
  they should not be scheduled for the same period.
  Using this representation, the problem of finding
  a workable timetable using the minimum number of
  periods is equivalent to the coloring problem,
  where the different colors correspond to
  different periods.

What is the chromatic number of this graph?
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What are some other ways graphs and graph
coloring are used to organize information?

• Round-Robin Tournaments
• Precedence Graphs (graphs which determine the
  order in which to complete tasks)
• Logic Puzzles
• Television and Radio Frequencies
• See this example on the Web
• Fish and Fish Tanks also on the Web
• Many more applications can be found by following
  this link!

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Now its time for a quiz!

• Click below for a fun and invigorating quiz show
  game!

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