The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring

2026-03-25

AI Summary

This work proposes a near-linear-time algorithm for four-coloring planar graphs, overcoming the limitation of classical approaches to the Four Color Theorem that rely on individual reducible configurations. By integrating the discharging method with the identification of D-reducible configurations, the authors establish—for the first time—that every planar triangulation contains either a linear number of pairwise non-adjacent reducible configurations or a set of non-crossing obstruction cycles. Leveraging this structural insight, they design a divide-and-conquer recursive strategy combined with efficient graph reduction techniques to achieve an O(n log n) time complexity for four-coloring, significantly improving upon the previous best-known O(n²) bound and extending applicability to arbitrary planar graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT). Technically speaking, we show the following significant generalization of the 4CT: every planar triangulation contains linearly many pairwise non-touching reducible configurations or pairwise non-crossing obstructing cycles of length at most 5 (which all allow for making effective 4-coloring reductions). The known proofs of the 4CT only show the existence of a single reducible configuration or obstructing cycle in the above statement. The existence is proved using the discharging method based on combinatorial curvature. It identifies reducible configurations in parts where the local neighborhood has positive combinatorial curvature. Our result significantly strengthens the known proofs of 4CT, showing that we can also find reductions in large ``flat" parts where the curvature is zero, and moreover, we can make reductions almost anywhere in a given planar graph. An interesting aspect of this is that such large flat parts are also found in large triangulations of any fixed surface. From a computational perspective, the old proofs allowed us to apply induction on a problem that is smaller by some additive constant. The inductive step took linear time, resulting in a quadratic total time. With our linear number of reducible configurations or obstructing cycles, we can reduce the problem size by a constant factor. Our inductive step takes $O(n\log n)$ time, yielding a 4-coloring in $O(n\log n)$ total time. In order to efficiently handle a linear number of reducible configurations, we need them to have certain robustness that could also be useful in other applications. All our reducible configurations are what is known as D-reducible.

Problem

• Four Color Theorem
• planar graphs
• reducible configurations
• near-linear time coloring
• obstructing cycles

Innovation

• Four Color Theorem
• near-linear time algorithm
• reducible configurations
• discharging method
• planar graph coloring