Diameter of consecutive 4-cycle Cayley graphs

Determine the diameter, antipodes, and asymptotic mean and mode for the Cayley graph of S_n generated by consecutive 4-cycles and their inverses, according to the stated formulas indexed by n modulo 3.

Background

The paper considers Cayley graphs generated by consecutive 4-cycles and their inverses, stating that the diameter had not previously been discussed in the literature. Direct BFS verifies the proposed formulas only for small n, while AI-based path-finding provides additional non-proof evidence for larger n.

References

Although these are rather obvious generators, to the best of our knowledge their diameter has not been discussed in the literature. We propose the following conjecture for this set of generators.

— 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”