A note on the number of edges in a Hamiltonian graph with no repeated cycle length

Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markstr"{o}m asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such a graph can have at most $n+\sqrt{2n}+1$ edges. Using difference sets in ${\mathbb{Z}}_n$, we show that for infinitely many $n$, there is an $n$-vertex Hamiltonian graph with $n+\sqrt{n-3\mathrm{/}4}-3\mathrm{/}2$ edges and no repeated cycle length.