Let $ \ G \ $ be a connected plane graph with $ \ v \ $ vertices and $ \ e \ $ edges so that every vertex lies on one face of size $ \ 6 \ $ , one face of size $ \ 6 \ $ and one face of size $ \ 8 \ $. Let $ \ f_4,f_6, f_8 \ $ be the number of faces of size $ \ 4,6,8 \ $ respectively. Then
(i) Express $ \ e \ $ in terms of $ \ v \ $ i.e., how many edges are incident with each vertex?
(ii) Express $ f_4,f_6,f_8 \ $ in terms of $ \ v \ $
(iii) Use (i) and (ii) to determine $ \ v \ $
(iv) Finally determine $ e,f_4,f_6,f8 \ $
Answer:
Let $ \ G \ $ be a connected graph such tha
number of edges $ \ =e \ $,
number of vertices $ \ =v \ $,
Let $ \ f= \ $ number of faces.
By Euler's formula,
$ v-e+f=2 \ $
Now I think,
$ f=f_4+f_6+f_8 \ $
But now I can not proceed to answer the questions.
I am not getting way to answer.
Can someone help me ?