...country.

Kempe thereby inferred that a minimal uncolorable map would not have any countries with three or fewer neighbors.

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...transformation.

Kempe mistakenly thought he could extend a coloring to a country with five neighbors colored with four distinct colors. Had he been able to do that, he would have proved the Four Color Theorem, because and argument using Euler's formula E + 2 = F + V enabled him to show that any minimal uncolorable map must have at least one country with five or fewer neighbors.

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...cases

1996: I no longer understand how figure 3 shows such cases.

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