
Planar graphs
Let be a simple, connected, planar graph with vertices, edges and faces. Assume that every vertex of has degree at least .
a) Show that and use Euler's formula to deduce that .
b) Assume now that all faces of have degree either or . Let and the number of faces of degree and , respectively, so . Show that . Use this and the previous part of the question to show in this case.
To bring you up to speed with what I know, I have no idea how to show in part a), but I do know that , so I'm on the cusp of deducing the other thing in part a), but I just can't quite get it.
For part b) I know that (where the 's are the faces in , so again I'm kind of halfway there, I just can't quite get what the question's wanting (Worried) cheers for any help in advance.