论文标题
表面上图的非回收计数
Non-recursive Counts of Graphs on Surfaces
论文作者
论文摘要
地图枚举的问题涉及计算连接的空间图，并具有指定的数字$ j $顶点，可以将其嵌入属属$ g $的紧凑表面，以使其补体产生表面的细胞分解。因此，这个问题在于低维拓扑和图理论中组合研究的跨道路。根据$ g $和$ j $的封闭经典组合功能，与递归处方相反的封闭经典组合功能的确定是一个长期存在的问题，与仅以$ g $非常低的值而闻名。在本文中，我们得出具有任意数量的偶数顶点的地图计数的封闭形式表达式，并嵌入了任意属的表面中。特别是，我们在文献中尚未出现过4个价值地图的许多较高属示例。
The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of the surface. As such this problem lies at the cross-roads of combinatorial studies in low dimensional topology and graph theory. The determination of explicit formulae for map counts, in terms of closed classical combinatorial functions of $g$ and $j$ as opposed to a recursive prescription, has been a long-standing problem with explicit results known only for very low values of $g$. In this paper we derive closed-form expressions for counts of maps with an arbitrary number of even-valent vertices, embedded in surfaces of arbitrary genus. In particular, we exhibit a number of higher genus examples for 4-valent maps that have not appeared prior in the literature.
