Let "$k$-necklace" denote the (multi)graph obtained from a cycle of length $k$ by duplicating every edge.

Note that the number of cycles in $k$-necklace is at least $2^k.$

**Question :** Suppose a simple graph $G$ contains at least $n^k$ cycles where $n$ is the number of vertices of $G$. Is it possible
to obtain a $\Omega(k)$-necklace from $G$ by a sequence of deletion and contraction of edges?

I would be equally happy if one can guarantee $\sqrt k$-necklace (or $k^\alpha$ necklace for some $\alpha > 0$) instead of $k$-necklace.