Arbitrarily frequent oscillations of hyperbolic sphericity

Construct a family of graphs whose hyperbolic sphericity oscillates arbitrarily often between two values as the ball radius increases.

Background

The paper constructs a graph whose hyperbolic sphericity is two for small and large radii but at least three on an intermediate interval. It suggests obtaining arbitrarily many oscillations by taking disjoint unions of suitable variants of this construction, each with a distinct invalid interval for two-dimensional representations. The authors note that additional work is needed to control overlaps between the invalid and valid radius regions of the component graphs.

References

Finally, we conjecture that it is possible to construct a family of graphs where the hyperbolic sphericity oscillates arbitrarily often between two values as the radius increases. To achieve this, we can take the disjoint union of different variations of the graph $\tilde C$ from \cref{thm:jump_example}.

— Hyperbolic Sphericity  (2610.01336 - Bläsius et al., 1 Oct 2026) in Section 5, Open Problems; following Figure 9