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Prove that any normal map with the property that the
number of edges of each country is a multiple of three can
be colored by four colors.
Solution
Given a normal map M and a proper coloring of the edges
of M by three colors a, b, c, first we consider the subgraph
of M that consists of all the edges labeled a or b.
The subgraph consists of a set of cycles. By Lemma; Given
a finite number of simple closed curves (cycles) in the
plane. the regions defined by these curves are colorable by
two colors.
Therefore regions defined by these cycles are colorable by
two colors, x and y. In this coloring, every country of M
Now we consider the subgraph of M that consists of all the
edges labeled a or c. This subgraph consists of a se t of
cycles, and again, by above Lemma, the regions are
colorable by two colors, z and w.
In this coloring every country of M receives either z or w.
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Now each country of M has received one of the four pairs
xz, xw, yz, or yw. We consider these four pairs a s four
colors. Consider two adjacent countries in M. If the edge
between these countries is labeled b, then one has an x
and the other has a y, so their colors are different.
If the edge between is labeled c, then one has a z and the
other has a w, so their colors are different. If the edge
between is labeled a, then one has an x and the other has
a y, so again their colors are different. Hence M is
colorable by four colors.
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