The Four Color Theorem meets Shapes of Polyhedra

We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,\mathrm{.}\mathrm{.}\mathrm{.},6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $\tau$ is parametrized by the set of integer lattice points inside a $4$-dimensional rational polyhedral convex cone ${\mathcal{C}\mathcal{\text{/}}}_{\tau }$. There is an integral quadratic form $Q_{\tau }$ on ${\mathcal{C}\mathcal{\text{/}}}_{\tau }$ whose diagonal part, evaluated on a lattice point, is $3$ times the number of triangles in the corresponding triangulation. We relate this structure to the octahedral stratum of Thurston’s moduli space of flat cone structures on the sphere.