Tightness of the higher-genus asymmetric-depth bound

Determine whether the bound $d(F)\le 2H(g)-2$ for graphs of orientable genus $g>0$ is tight by establishing whether infinitely many genera admit a graph attaining equality, or whether the maximal asymmetric depth is at least of order $\sqrt g$.

Background

The paper extends the planar asymmetric-depth bound to graphs embedded on orientable surfaces, obtaining d(F)≤2H(g)−2d(F)\le 2H(g)-2, where H(g)H(g) is the Heawood number. It explicitly states that the bound is not known to be tight; the available constructions achieve arbitrary depth but have genus growing too rapidly to match the bound.

References

It is unclear whether the bound for graphs of higher genus is tight. We have a construction of asymmetric graphs using columns of hypercubes described in that yields a graph of asymmetric depth $d$ for any $d \ge 0$, with increasing genus but the genus grows too fast to match the bound. We record this as an open problem.

— Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes  (2609.02585 - Pastorek, 2 Sep 2026) in Question 2, Section 7, “Concluding remarks”