The $8/5$ balanced-coloring conjecture for signed subcubic graphs

Let $G$ be a signed subcubic graph, meaning a graph of maximum degree at most $3$ whose edges have signs. Let $(K_4,-)$ denote the negatively signed complete graph on four vertices, and let $\widehat{K}_4^{\bullet}$ be the exceptional signed graph named in the source. An $(a,b)$-coloring is the signed balanced coloring notion used in the paper, and the fractional balanced chromatic number is the infimum of $a/b$ for which such a coloring exists. The $8/5$ balanced-coloring conjecture. Every signed subcubic graph not isomorphic to $(K_4,-)$ and not containing $\widehat{K}_4^{\bullet}$ admits an $(8,5)$-coloring. The conjecture would improve the bound from the preceding theorem after excluding the exceptional graph; its status is not resolved in the supplied text.