Diameter bounds for directed and Schreier graphs

Determine the sharp quadratic coefficients and establish the corresponding linear-error diameter bounds for non-inverse-closed Cayley graphs, binary Schreier coset graphs of S_n, and circular-permutation Cayley graphs.

Background

The paper gives conjectural bounds of the form Cn2+O(n) for directed Cayley graphs, Schreier coset graphs, and circular permutations, while explicitly stating uncertainty about the constants. The authors particularly suspect a coefficient of 1/2 for the binary Schreier coset graphs.

References

We expect variations of the conjecture to be true, but currently we are less sure about the constants in these cases:

— 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”