Solutions to the 4-color problem on sphere triangulations with degree sequence 6,6,6,6,6,6,4,4,4,4,4,4 are sorted into types parametrized by lattice points in 4D polyhedral cones, with an integral quadratic form counting triangles and related to Thurston's flat cone moduli space.
Here the regular vertices are the vertices of the partition that are contained in disk neighborhoods
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The Four Color Theorem meets Shapes of Polyhedra
Solutions to the 4-color problem on sphere triangulations with degree sequence 6,6,6,6,6,6,4,4,4,4,4,4 are sorted into types parametrized by lattice points in 4D polyhedral cones, with an integral quadratic form counting triangles and related to Thurston's flat cone moduli space.