论文标题
图形枚举和彩色玻璃窗，1：矩形网格
Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids
论文作者
论文摘要
当多边形的边缘以均匀间隔点标记，每对点由一条线连接时，对枚举问题的调查引起的枚举问题。这些问题中的一些已经解决了，一个经典的例子是k_n形成的图形k_n时，当普通的n gon的所有顶点与和弦结合在一起时，由Poonen和Rubinstein在1998年进行了分析。但是，这些问题中的大多数都无法解决，但是，这些两部分的文章从此类问题上提供了许多效率的镜子，这些镜子经常出现，这些镜子经常出现。所考虑的多边形包括矩形，空心矩形（或框架），三角形，五角星，五角星，十字架等，以及通过绘制在线上连接相同空间点的半圆形形成的数字。 ％的论文以简要讨论如何为这些图表设计美学上令人愉悦的色彩的问题进行了简要讨论。第一部分讨论了矩形网格。 Legendre和Griffiths研究了1 x n网格，或同样的图k_ {n+1，n+1}，在这里我们研究了具有给定数量的边缘数量的细胞数量以及具有给定程度的节点的数量。我们只为M X N矩形有部分结果，包括节点和细胞数量的上限。
A survey of enumeration problems arising from the study of graphs formed when the edges of a polygon are marked with evenly spaced points and every pair of points is joined by a line. A few of these problems have been solved, a classical example being the the graph K_n formed when all pairs of vertices of a regular n-gon are joined by chords, which was analyzed by Poonen and Rubinstein in 1998. Most of these problems are unsolved, however, and this two-part article provides data from a number of such problems as well as colored illustrations, which are often reminiscent of stained glass windows. The polygons considered include rectangles, hollow rectangles (or frames), triangles, pentagons, pentagrams, crosses, etc., as well as figures formed by drawing semicircles joining equally-spaced points on a line. %The paper ends with a brief discussion of the problem of how to %design aesthetically pleasing colorings for these graphs. This first part discusses rectangular grids. The 1 X n grids, or equally the graphs K_{n+1,n+1}, were studied by Legendre and Griffiths, and here we investigate the number of cells with a given number of edges and the number of nodes with a given degree. We have only partial results for the m X n rectangles, including upper bounds on the numbers of nodes and cells.
