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arXiv · 2511.16959

Generating the symmetric group by three prefix reversals

Saúl A. Blanco·Mikhail P. Golubyatnikov·Elena V. Konstantinova·Natalia V. Maslova·Luka A. Nikiforov
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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 $\mathrm{Sym}_n$. 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 also considered. Computational results on the diameter and the girth of some cubic pancake graphs are presented, and conjectures for future research are formulated.

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Saúl A. Blanco, Mikhail P. Golubyatnikov, Elena V. Konstantinova, Natalia V. Maslova, Luka A. Nikiforov. 2025-11-21. Generating the symmetric group by three prefix reversals. https://arxiv.org/abs/2511.16959

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