Erdős Problem #65 — Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots$ be the lengths of cycles in $G$.

Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots$ be the lengths of cycles in $G$. Is it true that $\sum\frac{1}{a_i}\gg \log k?$ Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?