Planar graph has a minimum girth of 6.
What is the best upper bound on the number of edges in this graph?
Say $e$ = number of edges and $n$ = number of vertices.
I know that for planar graphs: $e \le 3n - 6.$
So I think I should subtract $6$ from $n$, to get number of vertices not apart of the cycle that the girth is calculated from. This cycle will have $6$ edges, since all cycles have $n$ edges.
Then I would get for my final answer $$e \le 3(n - 6) - 6 + 6 \implies e <= 3n - 18$$ But if we test this for n = 6, which means all vertices are part of the girth, then it would say that there cannot be any edges, which is obviously wrong.
I don't see an upper bound. Imagine two concentric regular $n-$gons with $n$ a multiple of $3$. Connect every third vertex of the inner one to the matching vertex of the outer one. This gives $\frac 73n$ edges.