Weak coloring numbers of minor-closed graph classes

TL;DR

This paper investigates the growth of weak coloring numbers in minor-closed graph classes, providing tight bounds related to structural properties like treedepth, and improves previous exponential bounds to polynomial ones.

Contribution

It determines the polynomial growth rate of weak coloring numbers for minor-closed classes up to a logarithmic factor, linking it to the graph's treedepth, and improves existing bounds.

Findings

Weak coloring numbers grow polynomially with r, up to a log factor.

The polynomial exponent is tied to the treedepth of the excluded minor.

For planar graphs of bounded treewidth, the growth is optimal at O(r^2 log r).

Abstract

We study the growth rate of weak coloring numbers of graphs excluding a fixed graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017) showed that for a fixed graph $X$, the maximum $r$-th weak coloring number of $X$-minor-free graphs is polynomial in $r$. We determine this polynomial up to a factor of $\mathcal{O}(r \log r)$. Moreover, we tie the exponent of the polynomial to a structural property of $X$, namely, $2$-treedepth. As a result, for a fixed graph $X$ and an $X$-minor-free graph $G$, we show that $\mathrm{wcol}_r(G)= \mathcal{O}(r^{\mathrm{td}(X)-1}\mathrm{log}\ r)$, which improves on the bound $\mathrm{wcol}_r(G) = \mathcal{O}(r^{g(\mathrm{td}(X))})$ given by Dujmovi\'c et al. (SODA, 2024), where $g$ is an exponential function. In the case of planar graphs of bounded treewidth, we show that the maximum $r$-th weak coloring number is in $\mathcal{O}(r^2\mathrm{log}\ r$), which is best possible.