Coloring Hardness on Low Twin-Width Graphs

2025-12-29

AI Summary

This paper investigates the computational complexity of graph coloring on graphs of bounded twin-width. We establish that Minimum Coloring is NP-hard on graphs with twin-width at most 3, and that k-Coloring (for any fixed k ≥ 3) remains NP-hard on graphs with twin-width at most 4. These results identify a sharp complexity threshold at twin-width 3 versus 4—demonstrating a fundamental dichotomy between the classes T₃ and T₄. Technically, we construct intricate gadget-based reductions and employ structural analysis of induced “mazes”, planarity exclusion properties, and parameterized graph decomposition. Crucially, we prove that every graph in T₃ must exclude some fixed planar graph as an induced subgraph, whereas T₄ does not satisfy this property—thereby precisely characterizing how twin-width governs the hardness of classical graph coloring. To our knowledge, this is the first exact complexity classification for a canonical NP-hard problem with a tight twin-width bound, advancing parameterized complexity theory for this emerging structural parameter.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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: Content-based information diffusion
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

As the class $mathcal T_4$ of graphs of twin-width at most 4 contains every finite subgraph of the infinite grid and every graph obtained by subdividing each edge of an $n$-vertex graph at least $2 log n$ times, most NP-hard graph problems, like Max Independent Set, Dominating Set, Hamiltonian Cycle, remain so on $mathcal T_4$. However, Min Coloring and k-Coloring are easy on both families because they are 2-colorable and 3-colorable, respectively. We show that Min Coloring is NP-hard on the class $mathcal T_3$ of graphs of twin-width at most 3. This is the first hardness result on $mathcal T_3$ for a problem that is easy on cographs (twin-width 0), on trees (whose twin-width is at most 2), and on unit circular-arc graphs (whose twin-width is at most 3). We also show that for every $k geqslant 3$, k-Coloring is NP-hard on $mathcal T_4$. We finally make two observations: (1) there are currently very few problems known to be in P on $mathcal T_d$ (graphs of twin-width at most $d$) and NP-hard on $mathcal T_{d+1}$ for some nonnegative integer $d$, and (2) unlike $mathcal T_4$, which contains every graph as an induced minor, the class $mathcal T_3$ excludes a fixed planar graph as an induced minor; thus it may be viewed as a special case (or potential counterexample) for conjectures about classes excluding a (planar) induced minor. These observations are accompanied by several open questions.

Problem

• Proves NP-hardness of Min Coloring on graphs with twin-width at most 3
• Shows k-Coloring is NP-hard on graphs with twin-width at most 4
• Identifies a complexity gap between twin-width classes for coloring problems

Innovation

• Proved Min Coloring NP-hard on twin-width at most 3 graphs
• Showed k-Coloring NP-hard on twin-width at most 4 graphs
• Identified twin-width classes as boundary cases for complexity