Question

Show that in a connected planner linear graph with 6
vertices and 12 edges, each of the regions is bounded by

3 edges.
​

Answer 1

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Answer 2

Answer:

You should have learned about Euler's formula

p−q+f=2 p - vertices q - edges f - faces

You can use that result to show each face has 3 edges, look at 7.4.2Since 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 three edges.