# Determine the number of edges and the number of faces G in terms of n

Note: Preparation for exam, not homework/assignment

Let G be a connected planar graph on n>= 5 vertices. Suppose G has an embedding where every face boundary is a cycle of length exactly 5. Determine the number of faces of G in terms of n and the number of edges.

Looking at this problem. I know that the degree of every face is 5. So by the Faceshake lemma, I know that 5f = 2q where f is the number of faces and q is the number of edges. But I"m unsure of how to apply this to eulers formula in order to actually get values...

You have $5f=2e$ and Euler's formula $v+f=e+2$. Here $v=n\geq5$ is given. Now solve for $f$ and $e$. Certain values $n$ will be forbidden.