Recently, Gross, Mansour and Tucker introduced the partial-dual genus polynomial of a ribbon graph as a generating function that enumerates the partial duals of the ribbon graph by genus. It is analogous to the extensively-studied polynomial in topological graph theory that enumerates by genus all embeddings of a given graph. To investigate the partial-dual genus polynomial one only needs to focus on bouquets, i.e. ribbon graphs with only one vertex. In this paper, we shall further show that the partial-dual genus polynomial of a bouquet essentially depends on the signed intersection graph of the bouquet rather than on the bouquet itself. That is to say the bouquets with the same signed intersection graph will have the same partial-dual genus polynomial. We then prove that the partial-dual genus polynomial of a bouquet contains non-zero constant term if and only if its signed intersection graph is positive and bipartite. Finally we consider a conjecture posed by Gross, Mansour and Tucker. that there is no orientable ribbon graph whose partial-dual genus polynomial has only one non-constant term, we give a characterization of non-empty bouquets whose partial-dual genus polynomials have only one term by consider non-orientable case and orientable case separately.
On uniform large genus asymptotics of Witten's intersection numbers
University of Science and Technology of China（中国科学技术大学）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；International Centre for Theoretical Physics（国际理论物理中心）
The Sitting Closer to Friends than Enemies Problem in Trees
度量空间$\mathcal{T}$是\emph{real tree}，如果对于任何一对点$x, y \in \mathcal{T}$，段$[0,1]$到$\mathcal{T}$的所有拓扑嵌入$σ$，使得$σ(0)=x$和$σ(1)=y$具有相同的图像（这是从$x$到$y$的测地线段）。\emph{signed graph}是每条边都有正号或负号的图。树中的\emph{Sitting Closer to Friends than Enemies}问题具有作为输入的带符号图$S$。目的是确定是否存在从$V(S)$到实树的点的内射映射（称为\emph{valid distance drawing}），使得对于每个$u \in V(S)$，对于$u$的每个正邻居$v$和$u$的每个负邻居$w$，$v$和$u$之间的距离小于$w$和$u$之间的距离。在这项工作中，我们证明了一个完全的有符号的图有一个有效的距离图在一个真正的树当且仅当它的子图组成的所有（且仅）它的积极的边缘有一个相交表示的单位球在一个真正的树。此外，作为一个工具性结果，我们证明了一个图在实树中具有单位球的交表示当且仅当它在实树中具有真球的交表示，以及当且仅当它在实树中具有任意球的交表示。
