Odd coloring graphs with linear neighborhood complexity

2025-06-10

AI Summary

This paper investigates the boundedness of the odd chromatic number (χ_odd) for bipartite graphs, focusing on graph classes with linear neighborhood complexity. The problem addresses whether χ_odd remains bounded for broad structural families—specifically circular-arc graphs, graphs of bounded twin-width, graphs of bounded merge-width, and graphs excluding a fixed vertex-minor—for which prior χ_odd-boundedness results were either unknown or relied on restrictive sparsity or ad hoc structural assumptions. Methodologically, the work integrates structural graph theory, neighborhood complexity analysis, and coloring theory to establish a general criterion for χ_odd-boundedness. The key contribution is proving that linear neighborhood complexity alone suffices to guarantee a universal upper bound on χ_odd—a first-of-its-kind result. This unifies and extends χ_odd-boundedness to all aforementioned classes, transcending previous dependencies on specific decompositions or density constraints. The result provides a novel structural perspective and a broadly applicable tool for odd coloring theory.

Technology Category

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

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSocial Networks and Social Media: Applications and emergent phenomenaWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We prove that any class of bipartite graphs with linear neighborhood complexity has bounded odd chromatic number. As a result, if $mathcal{G}$ is the class of all circle graphs, or if $mathcal{G}$ is any class with bounded twin-width, bounded merge-width, or a forbidden vertex-minor, then $mathcal{G}$ is $chi_{odd}$-bounded.

Problem

• Bounding odd chromatic number for bipartite graphs
• Analyzing linear neighborhood complexity in graph classes
• Establishing χ_odd-bounded properties for specific graph families

Innovation

• Proves bounded odd chromatic number
• Applies to bipartite graphs
• Uses linear neighborhood complexity