On covers of graphs by Cayley graphs

TL;DR

This paper proves that certain highly symmetric, planar, vertex-transitive graphs can cover all graphs locally similar to them, and explores whether this property extends to all finitely presented Cayley graphs.

Contribution

It establishes a covering property for vertex-transitive, planar, 1-ended graphs and raises questions about its generality for finitely presented Cayley graphs.

Findings

Vertex-transitive, planar, 1-ended graphs cover graphs with large local isomorphisms.

The covering property holds for sufficiently large radius balls.

Open questions about the property for all finitely presented Cayley graphs.

Abstract

We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented Cayley graphs, as well as further related questions.