Dense cycles in finite tough graphs

Determine for which positive values of t there exists a radius R such that every finite t-tough graph has an R-dense cycle, and determine the dependence of the least such radius on t.

Background

This is the weak-hamiltonicity counterpart of the asymptotic toughness problem. Rather than requiring a Hamilton cycle, it asks for a cycle whose distance from every vertex is uniformly bounded in terms of the toughness parameter.

References

For which $t \in _{>0}$ is there $R\in $ such that\ every finite $t$-tough graph admits an $R$-dense cycle? How does $R$ depend on $t$?

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

Is there $R\in $ such that\ every $\frac{3}{2}$-tough plane triangulation has an $R$-dense cycle?

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