Asymptotic hamiltonicity of Hamming hypercubes

Determine whether the sequence of n-dimensional Hamming hypercubes is asymptotically hamiltonian.

Background

Each Hamming hypercube is known to be Hamiltonian, but ordinary Hamiltonicity does not provide the uniform coarse control required by asymptotic hamiltonicity. The paper explicitly states that this asymptotic question is not known to the author.

References

It is well-known and easy to see that $Q_n$ is hamiltonian, but I do not know how to answer the following: Is $(Q_n)$ asymptotically\ hamiltonian?

— Asymptotic connectivity, hamiltonicity and toughness  (2610.09992 - Georgakopoulos, 7 Oct 2026) in Problem, Section 1, subsection “The main problems”