Babai-like quadratic diameter conjecture for S_n

Prove that the diameter of every Cayley graph of S_n is O(n^2), uniformly over all choices of generators.

Background

The paper places the LRX diameter problem in the broader theory of diameters of Cayley graphs. It records a Babai-like conjecture asserting a quadratic diameter bound for S_n independently of the generating set, as a concrete instance of the broader Babai conjecture for finite simple groups.

References

In the case of the symmetric group $S_n$ there are two open conjectures which are easy to formulate, but somewhat representative for the field.

— CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs  (2502.18663 - Chervov et al., 25 Feb 2025) in Section 2.6, “Cayley graphs: applications, diameters, random walks, open conjectures”