Geodesic
On the number of genus one labeled circle trees
The electronic journal of combinatorics, Tome 14 (2007)

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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.
DOI : 10.37236/986
Classification : 05C30, 05C10, 05C05, 05C75, 05A99
Mots-clés : number of genus one labeled circle tree, e-reduction, e-graph 
Karola Mészáros. On the number of genus one labeled circle trees. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/986
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     author = {Karola M\'esz\'aros},
     title = {On the number of genus one labeled circle trees},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/986},
     zbl = {1159.05028},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/986/}
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