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{\bf Karola M\'esz\'aros}
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{\bf On the Number of Genus One Labeled Circle Trees}
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A genus one labeled circle tree is a tree with its vertices on a
circle, such that together they can be embedded in a surface of genus
one, but not of genus zero. We define an e-reduction process whereby
a special type of subtree, called an e-graph, is collapsed to an
edge. We show that genus is invariant under e-reduction. Our main
result is a classification of genus one labeled circle trees through
e-reduction. Using this we prove a modified version of a conjecture of
David Hough, namely, that the number of genus one labeled circle trees
on $n$ vertices is divisible by $n$ or $n/2$. Moreover, we explicitly
characterize when each of these possibilities occur.
\bye