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

2026-03-13

AI Summary

This study investigates upper bounds on the $q$-centered chromatic number and the weak $q$-coloring number for graph classes excluding a fixed subgraph. By integrating tools from graph minor theory, centered and weak coloring number analyses, and fractional treedepth fragility, the authors develop a unified framework to characterize their asymptotic behavior. The main contributions include the first tight estimation—within an $O(q)$ factor—of these coloring parameters for general subgraph-excluded graph classes; a constant-factor tight bound for classes excluding a planar graph, substantially improving upon prior non-explicit exponential bounds; and a proof that the $q$-centered chromatic number of $K_t$-free graphs is $O(q^{t-1})$, accompanied by matching tight bounds on the fractional treedepth fragility rate.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• centered coloring
• weak coloring number
• minor-closed graph classes
• chromatic number
• treedepth fragility

Innovation

• centered coloring
• weak coloring number
• minor-closed graph class
• treedepth fragility
• graph sparsity