Sharpness of the minimum-degree upper bound

Prove or disprove that the upper bound 3^k − 1 for the maximum possible minimum degree of a graph with metric dimension k is sharp by constructing matching lower-bound examples or establishing a smaller extremal value.

Background

The known upper bound for the maximum possible minimum degree at metric dimension k is 3k − 1. The authors explicitly conjecture that this bound is attained, but matching constructions are known only for k ≤ 3. Thus, the unresolved problem is to settle the conjecture for all k.

References

An upper bound of 3k-1 was obtained in [12]. We conjecture that this is sharp, but matching lower bound constructions have only been found [12] for k ≤ 3.

— Fault tolerance for metric dimension and its variants  (2502.02731 - Geneson et al., 4 Feb 2025) in Section 9, Conclusion