Centered colorings in minor-closed graph classes

2024-11-04

arXiv.org

AI Summary

This paper studies the $p$-centered coloring problem on $K_t$-minor-free graphs: a coloring requires that every connected subgraph $H$ using at most $p$ colors contains at least one color appearing exactly once in $H$. Prior results relied on parameters such as treewidth or depth, limiting their applicability. We establish, for the first time, a tight upper bound of $O(p^{t-1})$ on the $p$-centered chromatic number for $K_t$-minor-free graphs, revealing an exact power-law relationship between minor structure and coloring complexity; this bound is proven tight for $t = 2$ and $t = 3$. Technically, our approach integrates structural graph theory, hierarchical decompositions, recursive construction, and probabilistic analysis—departing from existing paradigms—and yields a unified, optimal theoretical framework for centered colorings on sparse graph classes.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• 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: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Studies p-centered colorings in minor-closed graphs
• Proves K_t-minor-free graphs need O(p^{t-1}) colors
• Ensures unique colors for connected subgraphs

Innovation

• p-centered coloring for minor-free graphs
• uses O(p^(t-1)) colors
• ensures unique color per subgraph