Prove that in any simple connected planar graph e lessthanor

Prove that in any simple, connected, planar graph, e lessthanorequalto 3v - 6, if v^3 3.

Solution

Proof: The proof is very simple and here it is:

The sum of degree of al the regions in any connected graph is equal to twice the edges. SO if we have three regions
and the sum of degrees is d and number of edges is e then
      
           d = 2e
          
This above equation however holds on a condition, that graph must have degree >= 3
So we have
           2e > 3 * faces
          
According to Euler formula
v - e + f = 2 and so
f = ev+2
  
here v is number of vertices, f is number of faces

Combining
2/3 e f, f = ev+2 and v - e + f = 2
  
We got that
3 - v + 2 >=0
So 3<= v - 2
Therefore
e 3v 6.