Maximum-subpath cubic graphs

Determine the graphs that maximize the subpath number among all cubic graphs on n vertices.

Background

The paper observes that the cubic graphs maximizing the subpath number appear to vary with the order: the Möbius strip is maximal for n=8 and the Petersen graph is maximal for n=10. For larger small orders, the observed maximizers are often symmetrical and tend to have small diameter, but no general structural characterization is established.

The authors therefore leave the cubic maximization problem unresolved and formulate it as a specific problem covering arbitrary n.

References

Hence, we propose the following problem. Find the graphs which maximize the subpath number in the class of cubic graphs on $n$ vertices.

— Invitation to the subpath number  (2503.00558 - Knor et al., 1 Mar 2025) in Concluding remarks and further work, Problem 1 (labelled Con_maxCubic)