planar region

Question

• How many planar regions?
• How many closed regions? and how many are unbounded?
• How many of then are bounded by a cycle of length $4$ ?
• Now, for example (a different question, not related to above diagram ) a question says, In a connected 3 regular graph, every planar region is bounded by exactly 5 edges, then count no of edges?

Please explain the last QS with the help of Euler's equation.  

Answer 1

A) As the given graph is a planar graph and secondly we have only one connected component , so we can find the number of regions of the graph as  : 

       r   =   e - n + 2

==> r   =   12 - 9 + 2

==> r   =   5

Out of these 4 will be closed regions and 1 as unbounded region..

Now as clear from the graph , all of 4 closed regions are cycles of 4 edges..

Now coming to your last question,

Say number of regions are "r" , then 

No of edges which bound the region =  5r

But to remove double counting , we do no of edges  =  5r / 2

Also say we have n vertices ,and the graph is 3 regular , so sum of degree  = 3n which is also = 2e  =  5r  according to Handshaking Theorem..

So as the graph is planar so we have :

          r = e - n + 2

  ==> 3n / 5  =  (3n / 2) - n + 2

  ==>  (3n / 5) - (n / 2)   =   2

  ==>  n  =  20 vertices

  So no of edges = 3n/2  =  30 edges 

Comments

Ya your doubt is also genuine..I am also not able to digest this thing..:)..But this is done mathematically

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degree of a region is the number of boundary edges touching a region.

And In a simple connected planar graph with minimum degree of a region $k$.

e <= k(V-2)/(k-2) and $kr <= 2e$

And in graph in question, $k = 4$

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Please explain ::::

Say number of regions are "r" , then

No of edges which bound the region =  5r