Show that in a connected planner linear graph with 6 vertices and 12 edges, each of the regions is bounded by 3 edges.
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Since G is planar we can use Euler's Identity, n − m + r = 2, where n = 6 and m = 12. Thus 6 − 12 + r = 2 implies that r = 8. By The First Theorem of Graph Theory the sum of all the degrees in G is 2m = 2(12) = 24. Since the number of regions is r = 8 we know that each region is bounded by 24/8 = 3 edges.
If G is a connected plane graph with at least three edges, then the boundary of every region of G has at least three edges. In this particular problem it turns out that the boundary of every region of G has 3 edges.
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