Every planar triangulation has linearly many pairwise non-touching reducible configurations or non-crossing obstructing cycles of length at most 5, yielding an O(n log n) 4-coloring algorithm.
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The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring
Every planar triangulation has linearly many pairwise non-touching reducible configurations or non-crossing obstructing cycles of length at most 5, yielding an O(n log n) 4-coloring algorithm.