mgf1107lecture15

# mgf1107lecture15 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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MGF 1107 – EXPLORATIONS IN MATHEMATICS LECTURE 15 The Four Color Theorem The Four Color Theorem states that given any separation of a two dimensional area into connected regions, called a map , the regions can be colored using at most four colors so that no two adjacent regions have the same color. Note: It is important to realize that two regions are called adjacent only if they share a border segment, not just a point. The now famous theorem was first conjectured in 1852, but was not proven until 1976, by Appel and Haken. They determined that there are 1936 ways to draw a map (all others being equivalent to one of them), and that after thousands of hours of computation they had reached the conclusion that in each case only four colors are needed. Needless to say many were skeptical of this method of proof, but after the 400 pages of microfiche output were independently checked, it was declared as being valid. Ex. If we look at the map of the United States below, we see that only four colors are used to color the states.

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Note: You can use only three colors for many maps, a fourth being needed when a region has common borders with an odd number of neighboring regions. A basic example is given below.
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## This note was uploaded on 09/22/2011 for the course MAC 2311 taught by Professor Evinson during the Spring '08 term at University of Central Florida.

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mgf1107lecture15 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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