$(2,\mathcal{F})$-Avoiding Coloring and B-Coloring under Bipartite Exclusions

Let $\mathcal{F}$ be a nonempty family of connected bipartite graphs, each with at least two edges. For a graph $G$, a proper vertex coloring of $G$ is $(2,\mathcal{F})$-avoiding if no member of $\mathcal{F}$ occurs bichromatically, and $χ_{2,\mathcal{F}}(G)$ denotes the minimum number of colors in