Determine (3,6) and (4,4)

Determine the exact values of the feedback arc set decomposition parameters (3,6) and (4,4), where the first argument bounds maximum degree and the second bounds directed girth.

Background

The paper proves (3,g)=g for gāˆˆ{3,4,5} and obtains the bounds (3,6)āˆˆ{5,6}. It also establishes (4,3)=3 and determines that (4,4)āˆˆ{3,4}.

Thus, these two problems ask for the first unresolved exact values immediately beyond the cases settled by the paper's main theorems.

References

As the first step towards determining more values, we would like to pose the following problem for which we have showed that $(3,6)\in {5,6}$ and $(4,4)\in {3,4}$. What is $(3,6)$ and $(4,4)$?

— 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), Problem following Conjecture 1