A State Calculus for Graph Coloring

TL;DR

This paper explores new mathematical formulations for the four-color map problem, including tautological expansions and extensions of the Penrose Bracket, to better understand graph coloring in planar and cubic graphs.

Contribution

It introduces novel reformulations of the four-color problem, extending the Penrose Bracket to count colorings of cubic graphs embedded in the plane.

Findings

New tautological expansion for the coloring problem

Extension of Penrose Bracket to cubic graphs

Provides alternative formulations for graph coloring

Abstract

This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and an extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs presented as immersions in the plane.