On the number of Hamiltonian cycles in a Boolean cube

We show that as $n\to\infty$ the logarithm of the number of partitions of an $n$-dimensional Boolean cube $E^n$ into cycles is equal to $2^n(\ln n-1+o(1))$ and the logarithm of the number of Hamiltonian cycles in $E^n$ is at least $2^{n-1}(\ln n-1+o(1))$. We prove that each perfect matching in $E^n$ in which the edges have at most $k$ directions can be complemented to a Hamiltonian cycle for any $n\geqslant n_0(k)$.