Let $Q_n$ be the $n$-dimensional hypercube graph. How many vertex cycle covers exist on $Q_n$? (Presumably the best we can hope for are upper and lower bounds.) To be clear, a single "vertex cycle cover" is a set of cycles in $Q_n$ such that each vertex is a member of one and only one cycle.

To fix some notation, like $N=2^n$ and let $C_n$ be the count of the number of vertex cycle covers on $Q_n$.

Note that there are about $(n/e)^N$ Hamiltonian cycles in $Q_n$ (cf Feder and Subi, 2008 for more precise upper and lower bounds), which provides a lower bound on $C_n$. Per a comment of Jon Noel's below, it may be worth mentioning that there are about $\sqrt{(n/e)^N}$ perfect matchings on $Q_n$ (this is also referenced in Feder and Subi's paper).

I am considering $Q_n$ with labelled vertices (i.e., I am counting vertex cycle covers with "no symmetries"), and considering only "proper" cycles, which on $Q_n$ means of length $\geq 4$. So, for example, there is only one vertex cycle cover on $Q_2$, namely the full 4-cycle.