Tetrachromagea

TL;DR

This paper introduces a novel mathematical framework involving weak Hamiltonians and mutation to study 4-colorings of planar cubic graphs, providing insights into the structure of colorings and supporting the four-color theorem.

Contribution

It constructs a moduli space of 4-colorings via weak Hamiltonians and mutation, offering a new perspective on graph coloring and the four-color theorem.

Findings

Defines weak Hamiltonians as a generalization of Hamiltonian cycles.

Introduces the Weak Hamiltonian graph and the chromatic graph.

Provides a heuristic argument supporting four-color sufficiency.

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.