Universal upper bound of three

Establish whether every finite loopless digraph has fractional majority coloring number at most 3, and thereby determine whether the bound is sharp for all digraphs.

Background

The paper proves the universal upper bound 523/140 < 3.736 for the fractional majority coloring number of finite loopless digraphs. The directed triangle has fractional majority coloring number three, so an upper bound of three would be best possible.

The authors identify the universal bound of three as the central unresolved extremal issue. They note that a terminal strongly connected component isomorphic to a directed triangle forces the lower bound three, establishing sharpness if the proposed universal upper bound holds.

References

The central extremal question remains whether $(D)\le3$ for every digraph $D$. Such a bound would be sharp: a terminal strongly connected component isomorphic to a directed triangle forces $(D)\ge3$, since every majority-stable set contains at most one of its vertices.

— Fractional majority coloring of digraphs: bounds and inapproximability  (2610.11997 - Ben-Ameur et al., 8 Oct 2026) in Section 5, Concluding remarks