Optimal density radius for expander cycles

Determine the asymptotic behavior of the function R(c) giving the least radius such that every finite c-expander has an R(c)-dense cycle.

Background

If the weak-hamiltonicity question for expander families has a positive answer, the paper asks for the optimal quantitative dependence of the density radius on the expansion constant. The function is defined as an infimum, with value infinity when no uniform radius exists.

References

Determine the asymptotics of the function $$R(c):= \inf{ R \in \cup {\infty} \mid \text{ every $c$-expander has an $R$-dense cycle}}.$$

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