## Planar graphs

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

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

To bring you up to speed with what I know, I have no idea how to show $2m\geq3n$ in part a), but I do know that $n-m+r=2$, 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 $\sum^{r}_{i=1}deg(F_i)=2m$ (where the $F_i$'s are the faces in $G$, 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.