Existence of graphs requiring unbounded cycle-length ceilings

Determine whether there exists a family of graphs on which no affordable fixed ceiling for per-edge cycle counting suffices to match the performance of the edge-girth descriptor.

Background

The paper compares the unbounded edge-girth-and-multiplicity descriptor with bounded per-edge dictionaries that count simple cycles only up to a prescribed length. On the Zinc-12k molecular benchmark, a dictionary capped at length eight approaches the performance of the unbounded descriptor, suggesting that removing the ceiling may offer only a modest practical advantage when the appropriate cutoff is known.

The authors explicitly leave unresolved whether there is a graph family for which every affordable fixed cutoff fails to provide sufficient structural information. They also note that constructing a benchmark isolating this phenomenon proved difficult, so the question is not settled by the reported experiments.

References

Whether a family of graphs exists on which no affordable ceiling suffices is not settled by our experiments, and constructing a benchmark that isolates it proved hard.

— Edge-Girth as a Structural Edge Feature for Graph Neural Networks  (2609.01441 - Marey et al., 1 Sep 2026) in Section 5, Discussion and Limitations, paragraph “How much the unbounded descriptor is worth.”