On Leighton's graph covering theorem

TL;DR

This paper reviews Leighton's graph covering theorem and its proofs, discusses a generalization to symmetry-restricted graphs, and explores the applicability of Bass's Conjugation Theorem in this broader context.

Contribution

It provides clear expositions of existing proofs and investigates a new generalization to symmetry-restricted graphs, establishing some cases where it holds.

Findings

Leighton's theorem is explained through two proofs.

A generalization to symmetry-restricted graphs is proposed.

Bass's Conjugation Theorem holds in the symmetry-restricted context for some cases.

Abstract

We give short expositions of both Leighton's proof and the Bass-Kulkarni proof of Leighton's graph covering theorem, in the context of colored graphs. We discuss a further generalization, needed elsewhere, to "symmetry-restricted graphs." We can prove it in some cases, for example, if the "graph of colors" is a tree, but we do not know if it is true in general. We show that Bass's Conjugation Theorem, which is a tool in the Bass-Kulkarni approach, does hold in the symmetry-restricted context.