Extremality of Sheveleva2 and Koltsov3involutions

Establish that the Sheveleva2 generators and Koltsov3involutions generators yield maximal or near-maximal diameters for infinitely many n in the directed and standard undirected cases, respectively.

Background

The paper defines two explicit families of permutation generators derived from the square-with-whiskers pattern. Sheveleva2 consists of two generators for directed Cayley graphs, while Koltsov3involutions consists of three involutions for undirected Cayley graphs.

References

Generators "Koltsov3involutions" and "Sheveleva2" provide maximal (or near) diameters for standard undirected and directed cases respectively (at least for infinite number of $n$).

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 7, subsection “New generators with large diameter: ‘Sheveleva2’ and ‘Koltsov3involutions’”