Generating the symmetric group by three prefix reversals

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 $\mathrm{Sym}_n$. As the largest prefix reversal of length $n$ is always included in a triple, we give a complete solution of the problem when any of the two smallest or the two largest lengths but $n$ are included in a triple of prefix reversals. Moreover, some conditions implying a triple of prefix reversals does not generate $\mathrm{Sym}_n$ are considered. Computational results on the diameter and the girth of some cubic pancake graphs are presented, and conjectures for future research are formulated.

26 pages, 4 tables, 3 figures, 27 references