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arXiv 2407.04588math.COcs.DM
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次闭图类的弱着色数

Weak coloring numbers of minor-closed graph classes

Jędrzej Hodor, Hoang La, Piotr Micek, Clément Rambaud

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52 pages, 17 figures, revision
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AI总结：

研究排除固定图作为次图的图的弱着色数增长率，在$\mathcal{O}(r \log r)$因子内确定多项式，将指数与$2$-treedepth联系，改进前人界并在有界treewidth平面图情形给出最优界。

摘要：

我们研究排除固定图作为次图的图的弱着色数增长率。Van den Heuvel等人（European J. of Combinatorics, 2017）证明，对于固定图$X$，$X$-minor-free图的第$r$弱着色数最大值是关于$r$的多项式。我们在$\mathcal{O}(r \log r)$因子内确定了该多项式。此外，我们将多项式指数与$X$的结构属性，即$2$-treedepth联系起来。结果表明，对于固定图$X$和$X$-minor-free图$G$，$\mathrm{wcol}_r(G)= \mathcal{O}(r^{\mathrm{td}(X)-1}\mathrm{log}\ r)$，改进了Dujmović等人（SODA, 2024）给出的界$\mathrm{wcol}_r(G) = \mathcal{O}(r^{g(\mathrm{td}(X))})$，其中$g$为指数函数。在有界treewidth的平面图情形中，我们证明第$r$弱着色数最大值属于$\mathcal{O}(r^2\mathrm{log}\ r)$，这是最优的。

英文摘要：

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 Dujmović 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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