Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids

TL;DR

This paper surveys enumeration problems related to graphs formed on polygons with evenly spaced points, focusing on rectangular grids, providing data, illustrations, and partial results for various polygonal shapes.

Contribution

It offers a comprehensive survey of enumeration problems on polygon-based graphs, including new data and partial results for rectangular grids and other shapes.

Findings

Analyzed the number of cells with specific edge counts in rectangular grids.

Provided partial results and upper bounds for nodes and cells in m x n rectangles.

Included colored illustrations reminiscent of stained glass windows.

Abstract

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.