I love the exercises 2.1.63 below (on page 78) in the West's textbook Introduction to Graph Theory.
2.1.63. Prove that every $n$-vertex graph with $n+1$ edges has girth at most $\lfloor(2 n+2) / 3\rfloor$. For each $n$, construct an example achieving this bound.
I am considering an $n$-vertex graph with $n+k$ edges where $k\leq n$; would there also be a general result? More generally, if there are $m$ edges ($m\ge 1$), would there be a more general result?
For the second question, the link may have something. But if $m$ is not very big, e.g., $m= n+\mathcal{O}(1)$ or $m =\mathcal{O}(n)$ do we have a similar result?