Find the smallest degree forcing decomposition number two at arbitrarily large girth

Determine the smallest integer Δ such that (Δ,g)=2 for arbitrarily large values of g.

Background

The parameter (Δ,g) is always at least 2 for non-acyclic oriented graphs. The paper proves that (Δ,g)=2 for every g≥3 when Δ≥1238, showing that sufficiently large maximum degree eliminates any improvement over the trivial lower bound, regardless of directed girth.

The authors leave open whether 1238 is close to optimal and ask for the smallest degree at which this phenomenon occurs for arbitrarily large directed girth.

References

We known that for every $g\geq 3$, $(\Delta,g)=2$ for all $\Delta\geq 1238$. It would interesting to determine the smallest $\Delta$ such that $(\Delta,g)=2$ for arbitrarily large $g$. What is the smallest $\Delta$ such that $(\Delta,g)=2$ for arbitrarily large $g$?

— Feedback Arc Sets and Feedback Arc Set Decompositions in Weighted and Unweighted Oriented Graphs  (2501.06935 - Gutin et al., 12 Jan 2025) in Section 5 (Conclusion), final Problem