Spanning trees and asymptotic hamiltonicity in boundedly generated graph sequences

Determine whether every asymptotically 2-connected sequence of k-generated graphs has an asymptotic spanning tree, and determine whether every such sequence is asymptotically hamiltonian.

Background

The paper introduces asymptotic spanning trees using coarse density and coarse injectivity. Its result that boundedly generated cycle spaces amplify asymptotic connectivity motivates the question whether asymptotic 2-connectivity together with k-generation guarantees either an asymptotic spanning tree or the stronger asymptotic hamiltonicity property.

References

Let $k\in $, and let (G_n) be an asymptotically\ 2-connected sequence of $k$-generated graphs. Must (G_n) have an ? Must it be asympto-tically hamiltonian?

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