Minimum-subpath cubic graphs

Prove that, for every even n in the specified range, the cubic graph L_n defined as the path-like structure formed from copies of K_4-e and terminated by the prescribed pendant blocks is the unique graph minimizing the subpath number among all cubic graphs on n vertices.

Background

The paper introduces a family of cubic graphs L_n whose construction depends on whether n is congruent to 2 or 0 modulo 4. These graphs consist of a path-like chain of K_4-e blocks with pendant terminal blocks arranged to produce a cubic graph.

Based on their investigation, the authors conjecture that this explicitly constructed graph is the unique minimizer of the subpath number among cubic graphs of order n. The statement is presented as a conjecture rather than proved in the paper.

References

We have the following conjecture. In the class of cubic graphs on $n$ vertices, the graph $L_{n}$ is the only graph which minimizes the subpath number.

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