亏格为1的圈树的个数
On the number of genus one labeled circle trees
摘要：
一个亏格为1的圈树是一个顶点在一个圈上的树，使得它们可以一起嵌入亏格为1的曲面，但不能嵌入亏格为0的曲面。我们定义了一个e-约简过程，其中一个特殊类型的子树，称为e-图，被折叠到一个边缘。我们证明了亏格在e-约化下是不变的。我们的主要结果是通过e-约简对亏格为1的圈树进行了分类。利用这一点，我们证明了一个修改版本的猜想的大卫霍夫，即，亏格1标记的循环树的$n$顶点的数目是由$n$整除或如果它是不可分的$n$，那么它是由$n/2$整除。此外，我们明确地描述了这些可能性中的每一种发生的时间。
英文摘要：
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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