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Centered colorings in minor-closed graph classes

Jędrzej Hodor, Hoang La, Piotr Micek, Clément Rambaud

TL;DR

This work proves that for every fixed $t\ge 2$, there is a constant $c=c(t)$ such that every $K_t$-minor-free graph $G$ admits a $p$-centered coloring using at most $c \, p^{t-1}$ colors for all positive integers $p$. The authors introduce $(p,c)$-good colorings and develop a two-step framework: first, establish a centered-Helly-type property for $K_t$-minor-free graphs by exploiting a layered RS-decomposition, yielding a $(p,c)$-good coloring with $p+1$ colors; second, lift this to a $p$-centered coloring by partitioning $G$ into parts and coloring the quotient $G/\\mathcal{P}$ with a polynomial-in-$p$ bound, then combining with a local coloring on parts. The combination uses an ISW22-inspired lifting that confines the dependence on $t$ to a multiplicative constant, resulting in the explicit $O(p^{t-1})$ bound. The approach reduces reliance on the graph minor structure theorem to achieve a more explicit polynomial degree and tightens the known bounds between weak coloring numbers and centered colorings in $K_t$-minor-free graphs, with implications for sparsity-based algorithmic design.

Abstract

A vertex coloring $\varphi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$, either $\varphi$ uses more than $p$ colors on $H$, or there is a color that appears exactly once on $H$. We prove that for every fixed positive integer $t$, every $K_t$-minor-free graph admits a $p$-centered coloring using $\mathcal{O}(p^{t-1})$ colors.

Centered colorings in minor-closed graph classes

TL;DR

This work proves that for every fixed , there is a constant such that every -minor-free graph admits a -centered coloring using at most colors for all positive integers . The authors introduce -good colorings and develop a two-step framework: first, establish a centered-Helly-type property for -minor-free graphs by exploiting a layered RS-decomposition, yielding a -good coloring with colors; second, lift this to a -centered coloring by partitioning into parts and coloring the quotient with a polynomial-in- bound, then combining with a local coloring on parts. The combination uses an ISW22-inspired lifting that confines the dependence on to a multiplicative constant, resulting in the explicit bound. The approach reduces reliance on the graph minor structure theorem to achieve a more explicit polynomial degree and tightens the known bounds between weak coloring numbers and centered colorings in -minor-free graphs, with implications for sparsity-based algorithmic design.

Abstract

A vertex coloring of a graph is -centered if for every connected subgraph of , either uses more than colors on , or there is a color that appears exactly once on . We prove that for every fixed positive integer , every -minor-free graph admits a -centered coloring using colors.

Paper Structure

This paper contains 5 sections, 12 theorems, 49 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries
  4. From good colorings to centered colorings
  5. Constructing good colorings

Key Result

Theorem 1

Let $t$ be an integer with $t \geqslant 2$. There exists an integer $c$ such that for every $K_t$-minor-free graph $G$ and for every positive integer $p$,

Figures (10)

  • Figure 1: Finding a $\zeta$-center of $V(H)$ in $P_H$.
  • Figure 2: 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 $\mathrm{torso}_G(W_x)-A_x$ 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 $\mathrm{torso}_G(W_x) - A_x$, and so, it is contained in a single bag of $\mathcal{D}_x$ and in at most two layers of $\mathcal{L}_x$.
  • Figure 3: After pigeonholing pairwise disjoint members of $\mathcal{F}$, we obtain a situation as in the figure. Here, $t = 5$. The model of $K_t$ is constructed in blue.
  • Figure 4: The pink vertices depict the set $Z$. The yellow pieces are components of $G - R - Z$. In the figure, $i=2$, that is, $C$ does not have a neighbor in $R_2$. The set $Z_C$ consists of two vertices $z_1$ and $z_2$ obtained by contracting the orange parts. In order to apply induction, we replace $R_2$ with $\{z_1,z_2\}$.
  • Figure 5: In this example, $G$ is an $8$-vertex path. $\mathcal{P} = \{P_1,P_2\}$ and $\mathcal{Q} = \{Q_1,Q_2,Q_3,Q_4\}$. One can check that $(\mathcal{V},\mathcal{Q})$ refines $(\mathcal{U},\mathcal{P})$ which is witnessed by $f$ and $g$. Note that $f$ is a function on $V(S)$ but for the readability reasons we depict it as it acted on bags of $\mathcal{V}$.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2: PS19
  • Lemma 3
  • proof : Proof of \ref{['thm:main_Kt_minor_free']}
  • Lemma 4: GM5
  • Lemma 5
  • Theorem 6: Dujmovi2017
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 15 more