Toward a Unit Distance Embedding for the Heawood graph

TL;DR

This paper explores the geometric problem of embedding the Heawood graph with unit distances, proposing a specific embedding for a subgraph and investigating analytical methods for general embeddability testing.

Contribution

It constructs a unit distance embedding for a subgraph of the Heawood graph and examines analytical approaches for testing overall embeddability.

Findings

Successfully embedded H-e in unit distance

Proposed analytical methods for embeddability testing

Insights into geometric constraints of the Heawood graph

Abstract

The unit distance embeddability of a graph, like planarity, involves a mix of constraints that are combinatorial and geometric. We construct a unit distance embedding for $H-e$ in the hope that it will lead to an embedding for $H$. We then investigate analytical methods for a general decision procedure for testing unit distance embeddability.