Lifting a prescribed group of automorphisms of graphs

TL;DR

This paper studies how to construct graph coverings that lift specific automorphism groups, enabling precise control over symmetries for applications in graph theory and related fields.

Contribution

It provides a comprehensive method for constructing regular coverings of graphs that lift prescribed automorphism subgroups while controlling the full automorphism group of the cover.

Findings

Method for lifting specific automorphism groups of graphs.

Examples illustrating selective lifting of automorphisms.

Framework applicable to various graph symmetry problems.

Abstract

In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the correct statements of our results. Let $P$ be the Petersen graph, say, and let $\wp:\tilde{P}\to P$ be a regular covering projection. With the current covering machinery, it is straightforward to find $\wp$ with the property that every subgroup of $\Aut(P)$ lifts via $\wp$. However, for constructing peculiar examples and in applications, this is usually not enough. Sometimes it is important, given a subgroup $G$ of $\Aut(P)$, to find $\wp$ along which $G$ lifts but no further automorphism of $P$ does. For instance, in this concrete example, it is interesting to find a covering of the Petersen graph lifting the alternating group $A_5$ but not the whole symmetric group $S_5$. (Recall that $\Aut(P)\cong S_5$.) Some other time it is important, given a subgroup $G$ of $\Aut(P)$, to find $\wp$ with the property that $\Aut(\tilde{P})$ is the lift of $G$. Typically, it is desirable to find $\wp$ satisfying both conditions. In a very broad sense, this might remind wallpaper patterns on surfaces: the group of symmetries of the dodecahedron is $S_5$, and there is a nice colouring of the dodecahedron (found also by Escher) whose group of symmetries is just $A_5$. In this paper, we address this problem in full generality.