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TR07-063 | 2nd July 2007 00:00
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#### Nearly Tight Bounds on the Number of Hamiltonian Circuits of the Hypercube and Generalizations

**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.