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### Paper:

TR08-087 | 31st July 2008 00:00

#### Nearly Tight Bounds on the Number of Hamiltonian Circuits of the Hypercube and Generalizations (revised)

TR08-087
Authors: Tomas Feder, Carlos Subi
Publication: 19th September 2008 20:45
Keywords:

Abstract:

It has been shown that for every perfect matching $M$ of the $d$-dimensional
$n$-vertex hypercube, $d\geq 2, n=2^d$, 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 a generalization of a special case of this
result when there are two dimensions that do not get used by $M$.
It is known that the number $M_d$ of perfect matchings of the $d$-dimensional
hypercube satisfies $M_d={(\frac{d}{e}(1+o(1)))}^{n/2}$ and,
in particular, ${(2d/n)}^{n/2} (n/2)!\leq M_d\leq {(d!)}^{n/(2d)}$.
It has also been shown that the number $H_d$ of Hamiltonian circuits of the
hypercube satisfies
$1\leq \lim_{d\rightarrow\infty} (\log H_d)/(\log M_d)\leq 2$.
We finally strenthen this result to a nearly tight bound
${((d\log 2/(e\log\log d))(1-o(1)))}^n\leq H_d\leq {(d!)}^{n/(2d)}{((d-1)!)}^{n/(2(d-1))}/2$
proving that $\lim_{d\rightarrow\infty} (\log H_d)/(\log M_d)=2$.
The proofs are based on a result for 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.

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