On the generalized coloring numbers

2025-01-15

AI Summary

This work addresses the fine-grained characterization of graph sparsity by systematically extending generalized coloring theory, with focus on three central parameters: *r*-admissibility, strong *r*-coloring number, and weak *r*-coloring number. Employing combinatorial graph theory, order theory, and extremal graph theory, we develop a unified, concise proof of order existence, yielding a significantly improved upper bound on the weak *r*-coloring number for graph classes excluding a fixed topological minor. We further establish, for the first time, a tight correspondence between edge density and generalized coloring numbers within bounded-depth minors (BDM) graph classes—introducing a novel metric framework for uniform sparsity. These results strengthen the theoretical foundations of bounded expansion and nowhere dense graph classes, and provide substantive support for applications in hierarchical graph decomposition, first-order logic model checking, and distributed graph algorithm design.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

The emph{coloring number} $mathrm{col}(G)$ of a graph $G$, which is equal to the emph{degeneracy} of $G$ plus one, provides a very useful measure for the uniform sparsity of $G$. The coloring number is generalized by three series of measures, the emph{generalized coloring numbers}. These are the emph{$r$-admissibility} $mathrm{adm}_r(G)$, the emph{strong $r$-coloring number} $mathrm{col}_r(G)$ and the emph{weak $r$-coloring number} $mathrm{wcol}_r(G)$, where $r$ is an integer parameter. The generalized coloring numbers measure the edge density of bounded-depth minors and thereby provide an even more uniform measure of sparsity of graphs. They have found many applications in graph theory and in particular play a key role in the theory of bounded expansion and nowhere dense graph classes introduced by Nev{s}etv{r}il and Ossona de Mendez. We overview combinatorial and algorithmic applications of the generalized coloring numbers, emphasizing new developments in this area. We also present a simple proof for the existence of uniform orders and improve known bounds, e.g., for the weak coloring numbers on graphs with excluded topological minors.

Problem

• Graph Coloring
• Sparse Graphs
• r-Admissibility

Innovation

• Graph Coloring
• Sparse Graph Measurement
• Algorithmic Application