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.

翻译：我们给出了一种平面图的近线性时间四着色算法，改进了Robertson等人1996年提出的二次时间算法。已知的四色定理证明方法无法实现此类算法。从技术角度而言，我们证明了四色定理的以下重要推广：每个平面三角剖分要么包含线性数量互不相触的可约构型，要么包含线性数量长度不超过5且互不交叉的障碍环（两者均能实现高效的四着色约简）。上述结论中，已知的四色定理证明仅能证明存在单个可约构型或障碍环。该存在性基于组合曲率的放电法证明，通过识别局部邻域具有正组合曲率区域中的可约构型。我们的结果显著强化了已知的四色定理证明，表明在曲率为零的大范围"平坦"区域中同样能发现约简，且实际上可在平面图的几乎任意位置进行约简。有趣的是，这类大面积平坦区域也存在于任意固定曲面的三角剖分中。从计算视角看，旧证明允许我们对规模减小某个加法常数的子问题应用归纳法，其中归纳步骤耗时线性，导致总时间为二次。而通过线性数量的可约构型或障碍环，我们可将问题规模缩减常数因子。归纳步骤耗时$O(n\log n)$，从而在$O(n\log n)$总时间内完成四着色。为高效处理线性数量的可约构型，需确保这些构型具备特定鲁棒性——这一特性在其他应用中同样具有价值。本文所有可约构型均属于D-可约类型。