Characterize when the trivial girth bound is tight for degrees 3 and 4

Determine, for each Δ∈{3,4}, the greatest integer g≥3 such that the parameter (Δ,g) equals its trivial upper bound g.

Background

The inequality (Δ,g)≤g holds trivially for all Δ≥2 and g≥3. The paper shows that equality holds for Δ=3 and g∈{3,4,5}, while (3,10)<10, restricting the possible greatest value to {5,6,7,8,9}.

For Δ=4, the paper proves (4,3)=3, (4,6)<6, and consequently restricts the possible greatest value to {3,4,5}. The open problem asks for the exact threshold in each degree case.

References

Therefore, we have the following problem as it can be seen from Fig. \ref{table:Results} that $(\Delta, 3)=2<3$ for all $\Delta\geq 6$ and to determine such $g$ for $\Delta=5$ is equivalent to solving our Conjecture \ref{conj:1}. For $\Delta=3$ or $4$, what is the greatest integer $g\geq 3$ such that $(\Delta,g)=g$?

— 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), Problem after the discussion of the trivial bound