Question

37. Not many neighbors. Show that every map has at least one country (oceans count as countries) with five or fewer neighborsâ€”that is, countries that share a border with it.

Put a vertex in the center of each country and join two vertices if the two countries share a border. This procedure produces a graph with one vertex for each country. The number of edges emanating from a vertex equals the number of neighboring countries. Suppose each country has six or more neighbors. What's the smallest number of edges the graph would have? How many faces would there have to be to make the Euler Characteristic hold?

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