Weak coloring numbers of minor-closed graph classes
<|reference_start|>Weak coloring numbers of minor-closed graph classes: 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.<|reference_end|>
@article{hodor2024weak,
    title={Weak coloring numbers of minor-closed graph classes},
    author={Jk{e}drzej Hodor, Hoang La, Piotr Micek, Cl'ement Rambaud},
    journal={arXiv preprint arXiv:2407.04588},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.04588},
    primaryClass={math.CO cs.DM}
}