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

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.

Figures (38)

• Figure 1: An example of a graph in $\mathbf{T}(\mathcal{X})$ and a rooted tree decomposition indexed by $F$ witnessing this fact. The colored sets represent the bags of the tree decomposition.
• Figure 2: Examples of graphs in $\mathcal{S}_1=\mathcal{R}_1\subseteq \mathcal{S}_2\subseteq\mathcal{R}_2 \subseteq \mathcal{S}_3\subseteq\mathcal{R}_3$.
• Figure 3: A bag $W_x$ of $\mathcal{W}$ in a layered RS-decomposition $(\mathcal{T}, \mathcal{W}, \mathcal{A}, \mathcal{D}, \mathcal{L})$ of width at most $c$. The set $A_x$ (in red) is included in $W_x$. The graph $\mathrm{torso}_{G,\mathcal{W}}(W_x)-A_x$ has a layering $\mathcal{L}_x$ (in purple) and a tree decomposition $\mathcal{D}_x$ (in green). Note that for every $y \in V(T) - \{x\}$, $(W_x \cap W_y) - A_x$ is a clique in $\mathrm{torso}_{G,\mathcal{W}}(W_x) - A_x$, and so, it is contained in a single bag of $\mathcal{D}_x$ and it at most two layers of $\mathcal{L}_x$.
• Figure 4: On the left-hand side, we depict an $\mathcal{F}$-rich model of $X$, where $X$ is a cycle graph 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: The pink vertices correspond to the set $S$. The vertices in $S$ highlighted blue are in $\mathop{\mathrm{WReach}}\nolimits_3[G,S,\sigma,u]$.

Theorems & Definitions (217)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Corollary 1.7
• Corollary 1.8
• Theorem 2.1: Grohe15JM22Dbski2021DS_2020
• proof : Proof of \ref{['cor:general']}