Lifting graph automorphisms along solvable regular covers

TL;DR

This paper studies how solvable graph covers can be broken down into simpler abelian covers and how automorphisms lift through these layers, with applications to classifying specific graph covers.

Contribution

It introduces a method to decompose solvable covers into abelian layers and analyzes automorphism lifts within this framework, advancing understanding of graph symmetries.

Findings

Decomposition of solvable covers into abelian covers.

Automorphism lifts can be layered through abelian covers.

Classification of metacyclic covers of the tetrahedron.

Abstract

A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of lifts of the automorphism in the layers of the abelian covers. This procedure is applied to classify metacyclic covers of the tetrahedron branched at face-centers.