Tetrachromagea

Abstract: We construct a moduli space of four colorings on planar cubic graphs. More precisely, we introduce the notion of weak Hamiltonian, a generalization of Hamiltonian cycles, and relate it to 4-colorings. Weak Hamiltonians have a form of deformation, which we call mutation, which gives them a graph structure, the Weak Hamiltonian graph. This graph encodes the different colorings as 3 vertex cliques. Identifying vertices on these cliques, we obtain a new graph, the chromatic graph, whose vertices are exactly the colorings of the original graph. Also, this construction gives a heuristic argument on why 4 colors are sufficient to color planar maps.