Bounded cyclic-host gap

Determine whether the ratio η_cyc(G)/η(G), where η_cyc(G) is the minimum host order over cyclic groups, is bounded over all finite graphs.

Background

The paper finds that restricting hosts to cyclic groups is harmless for most of its examples but costly for some: the Petersen graph has cyclic optimum 36 versus η=16, and the Frucht graph has cyclic optimum 59 versus η=27. The authors ask whether this ratio is universally bounded and note that it is unknown whether sharing a prime among invariant factors is necessary or sufficient for a strict gap.