Relating strong coloring numbers to shallow grid minors

Determine whether, for every graph class with polynomial expansion, there exists a polynomial function f that bounds the strong r-coloring number of each graph in the class in terms of r and the size of an appropriately shallow grid minor, namely the largest t for which the graph contains a t×t grid as a depth-r minor.

Background

The final remarks define the shallow grid-minor parameter as the largest integer t such that the graph contains a t×t grid as a depth-r minor. The paper observes that, for arbitrary graphs, shallow grid-minor size and strong coloring numbers do not control one another in the same way as their infinite-radius analogues.

The authors then ask whether the relationship can nevertheless persist when graphs are restricted to a fixed class with polynomial expansion. The displayed question gives the proposed polynomial dependence on r and the relevant shallow grid-minor parameter.

References

A question that remains is whether this connection does survive when restricted to classes of polynomial expansion. Is it true that for every graph class $\mathcal{C}$ with polynomial expansion, there exists a polynomial function $f$ such that for every integer $r 0$ and every graph $G \in \mathcal{C}$, \ r(G) f(r,{(r)}(G))? \

— Shallow brambles  (2502.04177 - Bousquet et al., 6 Feb 2025) in Section 5, Final remarks, Question following the paragraph beginning “A question that remains”