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. This also opens possibilities for extensions to higher surfaces since we can find such flat parts in any large-width triangulation 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. To efficiently handle a linear number of reducible configurations, we need them to be sufficiently robust to be useful in other applications. All our reducible configurations are what is known as D-reducible.

翻译：我们针对平面图提出了一种近线性时间的4-染色算法，改进了Robertson等人1996年提出的二次时间算法。该算法无法通过已知的四色定理(4CT)证明实现。技术层面上，我们展示了以下4CT的重要推广：每个平面三角剖分中包含线性多对互不接触的可约构型，或长度至多为5的互不交叉阻碍环（这些均允许进行有效4-染色约简）。已知的4CT证明仅能说明上述陈述中存在单个可约构型或阻碍环。该存在性通过基于组合曲率的放电法证明，在局部邻域具有正组合曲率的区域识别可约构型。我们的结果显著强化了已知4CT证明，表明在曲率为零的大面积"平坦"区域中亦可找到约简，且实际上可在给定平面图的几乎任意位置进行约简。这为向更高曲面推广开辟了可能性，因为我们在任意固定曲面的任意大宽度三角剖分中都能找到此类平坦区域。从计算视角看，旧证明允许我们对规模减少某常量的子问题应用归纳法，归纳步需线性时间导致总时间复杂度为二次。利用线性数量的可约构型或阻碍环，我们可将问题规模按常数因子缩减。归纳步耗时$O(n\log n)$，最终以$O(n\log n)$总时间完成4-染色。为高效处理线性数量的可约构型，需确保其具有足够鲁棒性以适用于其他场景。我们所有可约构型均属于已知的D-可约类型。