Signed Gyárfás–Sumner conjecture

Let $\widehat{K}_2^- = \widehat{(K_2,-)}$ and let $\widetilde{G}$ denote the signed graph obtained from a graph $G$ by replacing each edge with a digon. A finite set of signed graphs is a GS set when its forbidden induced-subgraph class has bounded balanced chromatic number. Signed Gyárfás–Sumner conjecture. For every forest $F$ and complete graph $K_t$, the set

$$\{\widehat{(K_2,-)},\widetilde{K}_t,\widetilde{F}\}$$

is a GS set.

Balanced coloring extends ordinary proper coloring: $\chi_b(\widetilde{G})=\chi(G)$. Thus this is a direct signed-graph restatement of the classical Gyárfás–Sumner conjecture, and it remains open.