Maggie Johnson
CS103B
Handout #20
Graph Coloring
Key Topics:
* Introduction
* Coloring Algorithms
* Multicolorings
* Coloring of Planar Graphs
_____________________________________________________________________
Introduction
What is the smallest number of colors (patterns) necessary to paint the map above so that
no two adjacent countries are the same color?
We have not done a very good job of
coloring the above map…
But, like virtually every other problem in the universe, this
problem reduces to a graph problem, where nodes represent countries and edges represent
adjacency.
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coloring
of a graph
G
assigns colors to the vertices of
G
so that adjacent
vertices are given different colors.
The
chromatic number
of a graph
G
is the minimal number of colors required to
color
G
.
The chromatic number of a graph
G
is denoted c(
G
).
What is the chromatic number of a tree?
What is the chromatic number of a circuit in a graph?
Graph coloring is a useful model for many different types of problems.
For example, in
scheduling:
Consider the problem of attempting to schedule courses so that the number
of conflicts is minimized.
For example, it is probably a poor idea to schedule CIV,
freshman calculus, and freshman English for MWF at 8:00am.
(It would be funny,
though.)
If we picture every course as a node, with courses that are likely to be taken
concurrently connected by an edge, then the problem of scheduling reduces to coloring
the graph.
In this case, each color represents a (disjoint) time to offer a course.
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 Winter '08
 SAHAMI,M
 Algorithms, Graph Theory, Planar graph, Graph coloring, planar graphs, Four color theorem

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