The six-color theorem can be proved as a map-coloring problem without using the dual graph….

The six-color theorem can be proved as a map-coloring problem without using the dual graph. Instead of creating the dual graph, put nodes at the intersections of boundaries and straighten the boundaries of regions so that the problem of coloring the map shown in figure (a) is represented by the problem of coloring the enclosed regions of the graph in figure (b). First assume that no country has a hole in it. Then the graph will be loop-free, planar, and connected. Also, every node will have degree at least 3.

The six-color theorem can be proved as a map-coloring problem without using the dual graph. Instead...

a. Show that the graph can be assumed to be simple by proving that if six colors are sufficient for coloring a simple graph, they are sufficient for a graph with parallel arcs as well. (Hint: Use temporary small countries at nodes.)

b. Prove that in a simple, connected, planar graph with R enclosed regions, n – a + R = 1.

c. Consider a simple, connected, planar graph and assume that every enclosed region has at least six edges adjacent to it. Show that 2a ' 3n – 3.

d. Now consider a simple, connected, planar graph where every node has degree at least 3. Show that such a graph has at least one enclosed region with no more than five adjacent edges.

e. Prove that six colors are sufficient to color any planar map where no country has a hole in it.

f. Prove that six colors are sufficient to color any planar map. (Hint: Cut some temporary slits in the map.)

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