On the number of genus one labeled circle trees

Details On the number of genus one labeled circle trees On the number of genus one labeled circle trees

2005-09-18

arxiv.org/abs/math/0509407v1

Mathematics

Combinatorics

32 pages, 30 figures

Scientific paper

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 if it is not divisible by $n$ then it is divisible by $n/2$. Moreover, we explicitly characterize when each of these possibilities occur.

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