Planar Graphs Question

Question

I'm having some trouble with planar graphs. Two questions I was stuck on were:

1. Prove that each planar graph on $n \gt 3$ vertices will have a minimum of $4$ vertices of degree $5$ at most.

2. Let's say you are provided a planar graph on $17$ vertices and that there is a drawing of such a graph with $17$ faces (countries). Prove that such a graph has vertex with degree $3$ or less.

Answer 1

I ended up using a proof by contradiction. For (1), I showed multiple cases. For (2), I solved for the number of edges then assumed to the contrary that all vertices are of degree 4.

Answer 2

The tool you want is the Euler characteristic:

http://en.m.wikipedia.org/wiki/Euler_characteristic

For a planar graph, $V+F=E+2$. Don't forget to count the "outside" as a face.