Minimal Polynomials for the Coordinates of the Harborth Graph
arXiv:math/0609360
Abstract
The Harborth graph is the smallest known example of a 4-regular planar unit-distance graph. In this paper we give an analytical description of the coordinates of its vertices for a particular embedding in the Euclidean plane. More precisely, we show, how to calculate the minimal polynomials of the coordinates of its vertices (with the help of a computer algebra system), and list those. Furthermore some algebraic properties of these polynomials, and consequences to the structure of the Harborth graph are determined.
v1: documentclass amsart, 16 pages, 4 figures. v2: same as v1, but now has 18 pages, with a final section ("Coda") on the consequences to the structure of the Harborth graph and some references added; some typos corrected. v3: minor change to the introduction (re: history of the Harborth Graph); one reference added; some more typos corrected