On the Number of Hamiltonian Cycles in a Boolean Cube

Abstract: It is shown that, as $n\to\infty$, the logarithm of the number of decompositions into cycles of the $n$-dimensional Boolean cube $E<sup>n$ is [ 2^n(\ln n-1+o(1)), ] and the logarithm of the number of Hamiltonian cycles in $E<sup>n$ is at least [ 2^{n-1}(\ln n-1+o(1)). ] It is proved that, in $E<sup>n$, every perfect matching whose edges belong to at most $k$ directions can be extended to a Hamiltonian cycle for every $n\geq n_0(k)$.