Bipartite hereditary union-covering conjecture

Establish whether, for every hereditary graph class G consisting of bipartite graphs, there exists a function f such that every bipartite host graph H satisfies the induced union-covering bound \(\icn{u}{G}{H} \leq f(\cn{u}{G}{H})\).

Background

The paper leaves several boundedness cases open when the guest or host class has bounded chromatic number. It reduces four of these cases to a single hypothesis concerning hereditary bipartite guest classes and bipartite host graphs. If established, the hypothesis would imply boundedness results for hereditary guest classes on host classes of bounded chromatic number, including corresponding union-, local-, and folded-induced covering numbers.

References

Yet, many cases (six in total) where~$H$ or~$G$ have bounded chromatic number remain open. In four of these (corresponding to the cells marked in $\textcolor{openColorCond}\blacksquare$ in \cref{tab:overview}) we obtain $(\icn{x}{G}{},\cn{x}{G}{})$-boundedness if the following hypothesis holds:

— Boundedness and Separation Between Induced and Non-Induced Covering Numbers  (2609.09873 - Goetze et al., 9 Sep 2026) in Hypothesis 5.1, Section 5, subsection “Boundedness and Separation”

Can this result be extended to the induced setting?

— Boundedness and Separation Between Induced and Non-Induced Covering Numbers  (2609.09873 - Goetze et al., 9 Sep 2026) in Section 5, subsection “Algorithms and Complexity”