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{\bf Heidi Gebauer}
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{\bf Enumerating all Hamilton Cycles and Bounding the Number of Hamilton Cycles in 3-Regular Graphs}
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We describe an algorithm which enumerates all Hamilton cycles of
a given 3-regular $n$-vertex graph in time $O(1.276^{n})$,
improving on Eppstein's previous bound.
The resulting new upper bound of $O(1.276^{n})$ for the maximum number of
Hamilton cycles in 3-regular $n$-vertex graphs gets close to the
best known lower bound of $\Omega(1.259^{n})$. Our method differs
from Eppstein's in that he considers in each step a new graph and
modifies it, while we fix (at the very beginning) one Hamilton cycle $C$
and then proceed around $C$, successively producing partial Hamilton cycles.

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