Let $g \in \mathbb{N}$ be an integer, where $g \geq 3$. Let $f(g, n)$ denote the maximum number of edges in a planar graph on $n$ vertices if the graph has no cycles of length less than $g$.
How to find an upper bound on $f(g, n)$ in terms of $g$ and $n$ by using Euler's formula?
One way to think about this: $E=n+f-2$ where $f$ is the number of faces. Now in order to maximize $E$, no two cycles share an edge. Then $f$ is at least $1+\#\text{cycles}$. The number of cycles is at least $\frac{n}{g}$. This gives us $E \le n(1+\frac{1}{g})+1$ although I think this may be improved.