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Explorations in Topology by David Gay

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4

Map Data and Map Coloring

Using Euler’s formula and counting arguments from Chapter 3, this chapter develops an inequality involving only the numbers of countries with given numbers of borders or edges. One consequence of this is that every map on the sphere, all vertices of order three, must have a country with five or fewer edges. With a reduction theorem that relates maps, all of whose vertices are of order three. To all maps whose vertices are of order three or more, this fact is then used to show that every map can be colored in six or fewer colors. This argument is subsequently adapted to prove that every map can be colored in five or fewer colors. These are the first theorems about map coloring applicable to all maps on the sphere. Chapters ...

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