Generating the symmetric group by three prefix reversals

TL;DR

This paper investigates the generation of the symmetric group using three prefix reversals, providing partial characterizations and computational insights into the structure of related cubic pancake graphs.

Contribution

It offers a partial characterization of generating sets of three prefix reversals containing specific elements, advancing understanding of symmetric group generation.

Findings

Characterized generating sets with at least one of r_2, r_3, r_{n-2}, r_{n-1}.

Provided computational results on diameter and girth of cubic pancake graphs.

Extended knowledge on symmetric group generation by specific prefix reversals.

Abstract

The cubic pancake graphs are Cayley graphs over the symmetric group $\mathrm{Sym}_n$ generated by three prefix reversals. There is the following open problem: characterize all the sets of three prefix reversals that generate $\mathrm{Sym}_n$. We present a partial answer to this problem, in particular, we characterize all generating sets of three elements that contain at least one of the prefix reversals $r_2, r_3, r_{n-2}$, and $r_{n-1}$. We also give some computational results relating to the diameter and the girth of some cubic pancake graphs.