The Babylonian Graph

TL;DR

This paper investigates the properties of the Babylonian graph, which connects integers based on Pythagorean triples, exploring its structure, subgraphs, and growth patterns, and presents initial results on its planarity and related conjectures.

Contribution

It provides the first results on the structure and properties of the Babylonian graph, including planarity thresholds and questions about its connectivity and subgraphs.

Findings

Identified the threshold where the graph becomes non-planar

Explored the growth rate of edges and triangles in the graph

Presented initial results and open questions about the graph's properties

Abstract

The Babylonian graph B has the positive integers as vertices and connects two if they define a Pythagorean triple. Triangular subgraphs correspond to Euler bricks. What are the properties of this graph? Are there tetrahedral subgraphs corresponding to Euler tesseracts? Is there only one infinite connected component? Are there two Euler bricks in the graph that are disconnected? Do the number of edges or triangles in the subgraph generated by the first n vertices grow like of the order n W(n), where n is the product log? We prove here some first results like the threshold where B(n) becomes non-planar. In an appendix, we include handout from a talk on Euler cuboids given in the year 2009.