Refined Babai-type diameter bound

Prove that, for every generating set of S_n or A_n that is closed under inverses, the Cayley-graph diameter is at most n^2/2+4n.

Background

The authors propose a quantitative refinement of the Babai-type conjecture for symmetric and alternating groups. They distinguish inverse-closed generating sets from directed generating sets and suggest sharper quadratic coefficients for several related graph constructions.

References

For any choice of the generators for $S_n$ and $A_n$ the diameter of the Cayley graph is bounded by $n2/2 + 4n$ (in the standard case when the generating set includes inverses).

— 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 8, subsection “Refinement of the Babai-like conjecture for S_n”