Maximum minimum degree at fixed metric dimension

Determine the maximum possible minimum degree of a graph having metric dimension k, and in particular prove or disprove the conjecture that this maximum equals 3^k−1.

Background

The paper cites a known upper bound of 3k−1 for the maximum possible minimum degree of a graph with metric dimension k. The authors conjecture that the upper bound is attained, but matching constructions are known only for k ≤ 3, so the problem remains unresolved for general k.

References

We note that there is a related open problem [12] which remains unsolved. Specifically, what is the maximum possible minimum degree of a graph of metric dimension k? 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 Conclusion, Section 9

We note that there is a related open problem [12] which remains unsolved. Specifically, what is the maximum possible minimum degree of a graph of metric dimension k?

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