Polynomial p-centered colorings for minor-free classes with treedepth-controlled exponent
Determine whether there exists a function g such that, for every fixed graph H, every H-minor-free graph G, and every positive integer p, the graph G admits a p-centered coloring with O(r^{g(td(H))}) colors.
References
The following question was posed by Hodor et al.. \begin{problem} Does there exist a function $g$ such that for every fixed graph $H$, for every $H$-minor-free graph~$G$ and every positive integer $p$, $G$ admits a $p$-centered coloring with $O(r{g(\td(H))})$ colors? \end{problem}
Conjecture 14. There is a function d : N → N such that the following holds. Let a and b be positive integers with a ⩽ b. There exists a positive integer c such that, for every positive integer p, for every graph G, if K_{a,b} is not a topological minor of G, then χp(G) ⩽ c * pd(a). We do not know if the conjecture is true even for a = 3.