Centered colorings and weak coloring numbers in minor-closed graph classes

TL;DR

This paper determines bounds on centered colorings and weak coloring numbers for minor-closed graph classes, improving previous bounds and providing a unified framework for these parameters.

Contribution

It introduces a general framework to bound centered colorings and weak coloring numbers in minor-closed classes, improving existing bounds and extending to fractional treedepth parameters.

Findings

Bound the $q$-centered chromatic number for $K_t$-minor-free graphs as $O(q^{t-1})

Achieve bounds within an $ ext{O}(q)$-factor for classes excluding certain minors

Provide bounds for fractional treedepth fragility rates

Abstract

Let $\mathcal{C}$ be a proper minor-closed class of graphs. Given the minors excluded in $\mathcal{C}$, we determine the maximum $q$-centered chromatic number and the maximum $q$th weak coloring number of graphs in $\mathcal{C}$ within an $\mathcal{O}(q)$-factor. Moreover, when $\mathcal{C}$ excludes a planar graph, we determine it within a constant factor. Our results imply that the $q$-centered chromatic number of $K_t$-minor-free graphs is in $\mathcal{O}(q^{t-1})$, improving on the previously known $\mathcal{O}(q^{h(t)})$ bound with a large and non-explicit function $h$. We include similar bounds for another family of parameters, the fractional treedepth fragility rates. All our bounds are proved via the same general framework.