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

2025-01-23

AI Summary

This paper addresses the problem of establishing tight upper bounds on the $r$-dynamic chromatic number $chi_r^d(G)$, particularly for graph classes with bounded expansion (e.g., planar graphs). To overcome the looseness of classical bounds, we introduce a novel quantitative connection between $r$-dynamic coloring and strong 2-coloring: if a graph admits a strong 2-coloring using at most $k$ colors, then $chi_r^d(G) leq (k-1)r + 1$, a bound that is both tight and attainable. As a consequence, all bounded-expansion graph classes satisfy $chi_r^d(G) = O(r)$, and specifically, planar graphs satisfy $chi_r^d(G) leq 3r + 1$. Our method integrates structural graph theory, neighborhood-constrained modeling, and analysis of strong coloring numbers—departing from prior approaches reliant on local structural arguments or inductive constructions. This yields a unified, concise, and structurally grounded framework for deriving upper bounds on dynamic chromatic numbers.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• r-dynamic coloring
• upper bound
• planar graphs

Innovation

• Dynamic Coloring
• Graph Theory
• Linear Function Bound