Solving a conjecture about tessellation graphs of $\mathbb R^2$

TL;DR

This paper disproves a conjecture claiming all tessellations of the Euclidean plane with convex tiles produce non-hyperbolic graphs, revealing that some tessellations can induce hyperbolic graphs.

Contribution

The paper provides the first counterexample to the conjecture, showing that tessellations of the Euclidean plane can induce hyperbolic graphs, contrary to previous beliefs.

Findings

The conjecture is false.

Some convex-tiled tessellations induce hyperbolic graphs.

Counterexamples challenge existing assumptions.

Abstract

In the paper Planarity and Hyperbolicity in Graphs, the authors present the following conjecture: every tessellation of the Euclidean plane with convex tiles induces a non-hyperbolic graph. It is natural to think that this statement holds since the Euclidean plane is non-hyperbolic. Furthermore, there are several results supporting this conjecture. However, this work shows that the conjecture is false.