Wreath product in automorphism groups of graphs

TL;DR

This paper investigates when the automorphism group of a graph is a wreath product, showing that such graphs are constructed from graphs with automorphism groups corresponding to the factors, especially in the natural imprimitive action.

Contribution

It proves that graphs with automorphism groups as wreath products in the natural action are constructed from graphs with automorphism groups matching the factors, extending classical results.

Findings

Confirmed the converse for wreath products in the natural imprimitive action.

Identified complexities and open questions for wreath products in the product action.

Extended classical results by Sabidussi and Hemminger.

Abstract

The automorphism group of the composition of graphs $G \circ H$ contains the wreath product $Aut(H) \wr Aut(G)$ of the automorphism groups of the corresponding graphs. The classical problem considered by Sabidussi and Hemminger was under what conditions $G \circ H$ has no other automorphisms. In this paper we deal with the converse. If the automorphism group of a graph (or a colored graph or digraph) is the wreath product $A \wr B$ of permutation groups, then the graph must be the result of the corresponding construction. The question we consider is whether $A$ and $B$ must be the automorphism groups of graphs involved in the construction. We solve this problem, generally in positive, for the wreath product in its natural imprimitive action (which refers to the results by Sabidussi and Hemminger). Yet, we consider also the same problems for the wreath product in its product action, which turns out to be more complicated and leads to interesting open questions involving other combinatorial structures.