Determine the value of (5,3)

Prove the conjecture that the parameter (5,3), defined as the minimum feedback arc set decomposition number over oriented graphs of maximum degree at most 5 and directed girth at least 3, equals 3.

Background

The paper studies the parameter (Δ,g), defined as the minimum value of fasd(D) over oriented graphs D with maximum degree at most Δ and directed girth at least g. The authors determine (Δ,3) for every Δ≥3 except Δ=5.

They establish (4,3)=3 and cite the previously known result (5,3)=1/3 for the corresponding feedback arc set ratio. The conjectured equality (5,3)=3 would unify and generalize these results in the decomposition framework.