The 8/3 bound without a directed-triangle terminal component

Determine whether every digraph with no terminal strongly connected component isomorphic to a directed triangle has fractional majority coloring number at most 8/3.

Background

The proposed 8/3 bound refines the universal bound of three for digraphs that lack the specific obstruction of a terminal strongly connected component isomorphic to a directed triangle.

The paper states that 8/3 would be sharp and supplies a five-vertex digraph with arcs a→ba\to b, b→cib\to c_i, and ci→ac_i\to a for i=1,2,3i=1,2,3, whose fractional majority coloring number is 8/3.

References

We ask whether every digraph with no such component satisfies $(D)\le8/3$. This constant would also be sharp: the digraph on ${a,b,c_1,c_2,c_3}$ with arcs $a\to b$, $b\to c_i$ and $c_i\to a$ for $i=1,2,3$ has fractional majority coloring number $8/3$.

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