Bounding strong coloring numbers by shallow bramble numbers

Determine whether, for every graph class with polynomial expansion, there exists a polynomial function f such that, for every integer r≥0 and every graph G in the class, the strong r-coloring number of G is at most f evaluated at r and the depth-r bramble number of G.

Background

The paper introduces bounded-radius bramble number as one of several shallow connectivity parameters that are polynomially bounded in r on any fixed graph class with polynomial expansion.

The authors explain that a positive answer to this question would suffice to establish polynomial bounds on strong coloring numbers for classes with polynomial expansion, thereby potentially resolving the preceding open problem.

References

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 Introduction, Question labeled \ref{quest:equiv}