On the Parameterized Complexity of Odd Coloring

2025-03-07

International Conference on Algorithms and Discrete Applied Mathematics

AI Summary

This paper investigates the parameterized complexity of the odd chromatic number χₒ(G), defined as the minimum number of colors required for a vertex coloring of G such that every non-isolated vertex has an odd number of neighbors in at least one color class. We establish precise complexity dichotomies with respect to classical structural parameters: the problem is fixed-parameter tractable (FPT) parameterized by vertex cover number and by feedback vertex set size, but W[1]-hard parameterized by pathwidth and by maximum degree—where hardness on pathwidth also implies NP-completeness. Methodologically, we integrate structural graph theory, tree decomposition-based dynamic programming, and intricate parameterized reductions; specifically, we design an FPT algorithm leveraging feedback vertex sets and construct tight reduction gadgets. Our main contribution is the first complete parameterized classification of odd graph coloring, revealing a complexity landscape markedly distinct from that of classical graph coloring problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine 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 graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Algorithmic accountability and transparency on the web

Problem

• Study parameterized complexity of odd coloring in graphs.
• Determine polynomial kernel existence for specific graph parameters.
• Analyze complexity on restricted graph classes like cographs.

Innovation

• Polynomial kernel with distance to clique
• Fixed-parameter tractable for several parameters
• NP-complete on specific bipartite subclasses