Graph-size lower bound for a sufficient hyperbolic radius

Determine whether there exists a lower bound for a hyperbolic radius s^* depending only on graph size such that, for every graph of that size, the hyperbolic sphericity at radius s^* is at most one greater than the graph’s minimum hyperbolic sphericity.

Background

Theorem 3.4 shows that increasing the radius from s_1 to any larger radius s_2 raises hyperbolic sphericity by at most one, but it does not specify when a radius is sufficiently large relative to the graph. The open problem asks for a universal radius threshold based only on graph size. The authors suspect that stars are extremal and that the required threshold may grow logarithmically with the number of vertices.

References

It would be interesting to determine whether a lower bound for the hyperbolic radius $s\ast$ exists (dependent only on the graph size), such that the hyperbolic sphericity with radius $s\ast$ is at most $1$~larger than the minimum hyperbolic sphericity; a question already stated in.

— Hyperbolic Sphericity  (2610.01336 - Bläsius et al., 1 Oct 2026) in Section 5, Open Problems; following Theorem 3.4