标题： 图在平移曲面上的收缩嵌入 显示英文标题

标题： Systolic embedding of graphs on translation surfaces

摘要： 一个图在平移曲面上的嵌入被称为\emph{收缩的}，如果图的每个顶点对应一个奇点（或标记点），每个边对应平移曲面上的一条最短鞍点连接。 该嵌入被称为\emph{单元格}（分别称为\emph{必要的}）如果每个补区域是一个拓扑圆盘（分别不是拓扑圆盘）。 在本文中，我们证明了任何有限图都可以在平移曲面上进行本质-系统嵌入，并估计了这类曲面的亏格。 对于一个由$n$个圆组成的悬挂点$\Sigma_n$，$n\geq2$，我们研究了$\Sigma_n$在平移曲面上是否可以进行单元-系统嵌入，并计算了这类曲面的最小和最大亏格。 最后，我们识别出另一组丰富的图，这些图具有多个顶点，并且也可以在平移曲面上进行单元-系统嵌入。 显示英文摘要

摘要： An embedding of a graph on a translation surface is said to be \emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation surface. The embedding is said to be \emph{cellular} (respectively \emph{essential}) if each complementary region is a topological disk (respectively not a topological disk). In this article, we prove that any finite graph admits an essential-systolic embedding on a translation surface and estimate the genera of such surfaces. For a wedge $\Sigma_n$ of $n$ circles, $n\geq2$, we investigate that $\Sigma_n$ admits cellular-systolic embedding on a translation surface and compute the minimum and maximum genera of such surfaces. Finally, we have identified another rich collection of graphs with more than one vertex that also admit cellular-sytolic embedding on translation surfaces.