Asymptotic hamiltonicity of asymptotically 4-connected planar graphs

Determine whether every asymptotically 4-connected sequence of finite planar graphs is asymptotically hamiltonian.

Background

Tutte’s theorem states that every finite 4-connected planar graph has a Hamilton cycle. The paper asks for the corresponding statement for sequences under its coarse notion of asymptotic 4-connectivity and asymptotic hamiltonicity.

References

Must every asymptotically\ 4-connected sequence of finite planar graphs be ?

— Asymptotic connectivity, hamiltonicity and toughness  (2610.09992 - Georgakopoulos, 7 Oct 2026) in Problem (Tutte), Section 1, subsection “Planar graphs”

Must every infinitesimally\ 4-connected Riemannian surface that embeds into $2$ be infinitesimally hamiltonian?

— Asymptotic connectivity, hamiltonicity and toughness  (2610.09992 - Georgakopoulos, 7 Oct 2026) in Problem (Tutte e), Section 1, subsection “Some fractal graph theory”