Weak coloring numbers of minor-closed graph classes

TL;DR

This work analyzes the growth of weak coloring numbers $ ext{wcol}_r(G)$ in minor-closed graph classes, tying the exponent to refined structural parameters. The authors introduce rooted $2$-treedepth $ ext{rtd}_2$ and show it tightly controls the asymptotics of $ ext{wcol}_r$ via bounds that connect to $ ext{td}(X)$ and the obstruction graphs $G_{r,t}$, with $ ext{rtd}_2(X)-2 leq f(X) leq ext{rtd}_2(X)-1$. They prove the main bounds $ ext{wcol}_r(G) = ext{O}igl(r^{ ext{td}(X)-1} ext{log} rigr)$ and $ ext{wcol}_r(G) = ext{O}igl(( ext{tw}(G)+1) ext{r}^{ ext{td}(X)-2} ext{log} rigr)$ for $X$-minor-free graphs, with a tight $ ext{O}(r^2 ext{log} r)$ bound for planar graphs of bounded treewidth. The analysis uses a hitting-set framework, notions of $ ext{F}$-rich models, and universal constructions $G_{r,t}$ to characterize obstructions, together with a detailed study of rooted $2$-treedepth and tree partitions to enable inductions. The results sharpen previous exponential-type bounds and have implications for sparsity notions and algorithmic applications in minor-closed graph classes. In the planar- and bounded-treewidth regime, the bounds are tight up to logarithmic factors, highlighting a structural dichotomy between general minor exclusions and planar-like graphs.

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 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.

Figures (17)

• Figure 1: Connections of $f$ to other graph parameters. An arrow from a parameter $p$ to a parameter $q$ indicates that there is a function $\alpha$ such that $p(X) \leqslant \alpha(q(X))$ for every graph $X$. We show that $f$ is tied to $\operatorname{td}_2$ and $\operatorname{rtd}_2$ but not to $\operatorname{tw}$, $\operatorname{pw}$, $\operatorname{td}$, $\mathop{\mathrm{vc}}\nolimits$, or $|V|$. The results marked in the figure (top-to-bottom) are in vdHetal17, vandenHeuvel2018, DHHJLMMRW24, and Grohe15 respectively.
• Figure 2: An construction of $L_2(B,H,u)$, where $B$ is a triangle and $H$ is a path on three vertices with $u$ being one of its endpoints.
• Figure 3: The pink vertices correspond to the set $S$. The vertices in $S$ highlighted blue are in $\mathop{\mathrm{WReach}}\nolimits_3[G,S,\sigma,u]$.
• Figure 4: On the left-hand side, we depict an $\mathcal{F}$-rich model of $X$, where $X$ is a cycle on $8$ vertices and $\mathcal{F}$ is the family of all connected subgraphs of $G - \{u\}$ containing a neighbor of $u$ in $G$. On the right-hand side, we show how to construct, given an $\mathcal{F}$-rich model of $X$, a model of $K_1 \oplus X$.
• Figure 5: An example of an eliminating ordering of a complete binary tree of height $3$.

Theorems & Definitions (66)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• Theorem 8
• Lemma 9
• Lemma 17: \ref{['lemma:E-P_in_Kk_minor_free_graphs_outline']} restated