Abstract:
We conjecture that for every perfect matching $M$ of the $d$-dimensional $n$-vertex hypercube, $d\geq 2$, there exists a second perfect matching $M'$ such that the union of $M$ and $M'$ forms a Hamiltonian circuit of the $d$-dimensional hypercube. We prove this conjecture in the case where there are two dimensions that do not get used by $M$. As a consequence, if $M_d$ is the number of perfect matchings and $H_d$ is the number of Hamiltonian circuits of the $d$-dimensional hypercube, then $M_{d-2}^4\leq H_d\leq M_d^2/4$. By known bounds on the number of perfect matchings of the $d$-dimensional hypercube that show $M_d={(\frac{d}{e}(1+o(1)))}^{n/2}$ and, in particular, $M_d\leq {(d!)}^{n/(2d)}$ we infer that ${(\frac{d}{e}(1-o(1)))}^{n/2}\leq H_d \leq {(d!)}^{n/(2d)}{((d-1)!)}^{n/(2(d-1))}/2$. We finally strenthen this result to a nearly tight bound ${((d\log 2/(e\log\log d))(1-o(1)))}^n\leq H_d\leq {((d/e)(1+o(1)))}^n$. We extend the results to graphs that are the Cartesian product of squares and arbitrary bipartite regular graphs that have a Hamiltonian cycle. We also study a labeling scheme related to matchings.