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The r-Dynamic Chromatic Number is Bounded in the Strong 2-Coloring Number

Miriam Goetze, Torsten Ueckerdt

TL;DR

This work addresses the problem of bounding the $r$-dynamic chromatic number $\chi_r(G)$ in terms of sparsity parameters rather than maximum degree. It proves a central bound: if $G$ has strong 2-coloring number at most $k$, then $\chi_r(G) \le (k-1)\,r + 1$ for all $r \ge 1$, via an inductive, prefix-extension coloring method that preserves a strongly proper coloring and a weakly $r$-dynamic property while tightly managing forbidden colors. Consequently, for graph classes of bounded expansion, $\chi_r(G)$ grows linearly with $r$, and the authors provide explicit bounds for treewidth and row-treewidth, including planar graphs. The results connect structural sparsity notions (strong coloring numbers, row-treewidth) to dynamic coloring guarantees, enabling practical linear-in-$r$ bounds across broad graph families and highlighting the nuanced behavior under subdivisions and nowhere-dense settings.

Abstract

A proper vertex-coloring of a graph is $r$-dynamic if the neighbors of each vertex $v$ receive at least $\min(r, \mathrm{deg}(v))$ different colors. In this note, we prove that if $G$ has a strong $2$-coloring number at most $k$, then $G$ admits an $r$-dynamic coloring with no more than $(k-1)r+1$ colors. As a consequence, for every class of graphs of bounded expansion, the $r$-dynamic chromatic number is bounded by a linear function in $r$. We give a concrete upper bound for graphs of bounded row-treewidth, which includes for example all planar graphs.

The r-Dynamic Chromatic Number is Bounded in the Strong 2-Coloring Number

TL;DR

This work addresses the problem of bounding the -dynamic chromatic number in terms of sparsity parameters rather than maximum degree. It proves a central bound: if has strong 2-coloring number at most , then for all , via an inductive, prefix-extension coloring method that preserves a strongly proper coloring and a weakly -dynamic property while tightly managing forbidden colors. Consequently, for graph classes of bounded expansion, grows linearly with , and the authors provide explicit bounds for treewidth and row-treewidth, including planar graphs. The results connect structural sparsity notions (strong coloring numbers, row-treewidth) to dynamic coloring guarantees, enabling practical linear-in- bounds across broad graph families and highlighting the nuanced behavior under subdivisions and nowhere-dense settings.

Abstract

A proper vertex-coloring of a graph is -dynamic if the neighbors of each vertex receive at least different colors. In this note, we prove that if has a strong -coloring number at most , then admits an -dynamic coloring with no more than colors. As a consequence, for every class of graphs of bounded expansion, the -dynamic chromatic number is bounded by a linear function in . We give a concrete upper bound for graphs of bounded row-treewidth, which includes for example all planar graphs.
Paper Structure (5 sections, 13 theorems, 5 equations, 1 figure)

This paper contains 5 sections, 13 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounding the r-chromatic number in the strong 2-coloring number
  4. Applying Theorem 5 and explicit bounds
  5. Discussion

Key Result

Theorem 1

For $r \geq 8$ and every planar graph $G$, we have $\chi_r(G) \leq 2r+16$.

Figures (1)

  • Figure 1: Left: A $2$-dynamic $5$-coloring of the graph $K'_5$ obtained from $K_5$ (large vertices) by adding a subdivision vertex (small) on every edge. Right: Adding one universal vertex allows for a $2$-dynamic $3$-coloring.

Theorems & Definitions (14)

  • Theorem 1: Song and Lai song_planar_2018
  • Theorem 2: Loeb et al. loeb_genus_2018
  • Theorem 3
  • Theorem 4: Zhu zhu_colouring_2009
  • Theorem 5
  • Corollary 6
  • Lemma 7
  • Corollary 9
  • Lemma 10: Van den Heuvel and Wood vandenheuvel_improper_arxiv_2018
  • Corollary 11
  • ...and 4 more