Consecutive 4-cycle diameter

Determine the diameter of the Cayley graph generated by consecutive 4-cycles and their inverses, proving the stated parity-class formulas and the corresponding descriptions of longest elements and mean and mode diameters.

Background

The paper states that the diameter of the Cayley graph generated by consecutive 4-cycles has not been treated in the literature. Computations for n<14 support a quadratic quasi-polynomial formula, but the authors emphasize that this evidence does not constitute a proof.

References

Although diameters can be guessed already from these values of $n$ by fitting a quadratic quasi-polynomial, this might not be enough data to be confident, while direct computations for much larger $n$ are unfeasible.

— 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 16, subsection “Case study -- consecutive 4-cycles”