## Planar graphs

Let $\displaystyle G$ be a simple, connected, planar graph with $\displaystyle n\geq3$ vertices, $\displaystyle m$ edges and $\displaystyle r$ faces. Assume that every vertex of $\displaystyle G$ has degree at least $\displaystyle 3$.

a) Show that $\displaystyle 2m\geq3n$ and use Euler's formula to deduce that $\displaystyle 2\leq r-\frac{m}{3}$.
b) Assume now that all faces of $\displaystyle G$ have degree either $\displaystyle 5$ or $\displaystyle 6$. Let $\displaystyle p$ and $\displaystyle h$ the number of faces of degree $\displaystyle 5$ and $\displaystyle 6$, respectively, so $\displaystyle r=p+h$. Show that $\displaystyle 2m=5p+6h$. Use this and the previous part of the question to show $\displaystyle p\geq12$ in this case.

To bring you up to speed with what I know, I have no idea how to show $\displaystyle 2m\geq3n$ in part a), but I do know that $\displaystyle n-m+r=2$, 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 $\displaystyle \sum^{r}_{i=1}deg(F_i)=2m$ (where the $\displaystyle F_i$'s are the faces in $\displaystyle G$, so again I'm kind of halfway there, I just can't quite get what the question's wanting cheers for any help in advance.