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arXiv · 2603.13097

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

Jędrzej Hodor·Hoang La·Piotr Micek·Clément Rambaud
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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.

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Jędrzej Hodor, Hoang La, Piotr Micek, Clément Rambaud. 2026-03-13. Centered colorings and weak coloring numbers in minor-closed graph classes. https://arxiv.org/abs/2603.13097

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