Determine the asymptotic decomposition number at fixed maximum degree

Determine the limit of (Δ,g) as g tends to infinity for every fixed integer Δ≥3.

Background

The paper proves the uniform upper bound (Δ,g)≤94 for all Δ≥3 and g≥3, and obtains stronger bounds for sufficiently large Δ, including (Δ,g)=2 for all Δ≥1238 and g≥3.

For a fixed maximum degree, the asymptotic behavior as the directed girth grows remains unresolved. The proposed limit would provide an accurate large-girth upper-bound characterization for each fixed degree.