Colorful Mathematics: Part II
2. Insights into coloring planar graphs
A knowledge of the Jordan Curve Theorem provides a way to tell easily, even for very convoluted polygons. Starting at a point clearly outside the polygon, draw a ray to the (blue) point that avoids the vertices of the polygon, as shown below.
V + F - E = 2
For example, for the map in Figure 1:
The idea of a list-coloring takes a while to get used to. If one has a plane graph isomorphic to that of the regular dodecahedron (a plane 3-valent graph made up of 20 vertices and 12 5-gon faces) and one specifies the colors 1, 2, 3 for the first vertex, colors 4, 5, 6 for the second vertex, etc., then the coloring of the plane graph one gets is a coloring with 12 different colors, though each list has three colors. On the other hand, if one specifies the list 1, 2, 3 as the potential colors for every vertex, then Thomassen's Theorem concerning plane graphs of girth 5 means that one can find a vertex-coloring of the dodecahedron with 3 colors.
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