A topological quantum field theory approach to graph coloring

TL;DR

This paper introduces a topological quantum field theory framework to define new homology theories for surfaces, which can be used to analyze graph colorings and offers novel insights into the four color theorem.

Contribution

It develops a TQFT-based method to create homology theories that relate to graph coloring and provides new invariants and approaches to classical conjectures.

Findings

Homology dimensions count face colorings with no adjacent same colors.

New graph invariants derived from TQFT.

Potential constructive approach to the four color theorem.

Abstract

In this paper, we use a topological quantum field theory (TQFT) to define families of new homology theories of a $2$-dimensional CW complex of a smooth closed surface. The dimensions of these homology groups can be used to count the number of ways that each face of the CW complex can be colored with one of $n$ colors so that no two adjacent faces have the same color. We use these homologies to define new invariants of graphs, give new characterizations of well-known polynomial invariants of graphs, and rephrase and offer new approaches to famous conjectures about graph coloring. In particular, we show that the TQFT has the potential to generate $4$-face colorings of a bridgeless planar graph, leading to a constructive approach to the four color theorem. The TQFT has ramifications for the study of smooth surfaces and provides examples of new types of Frobenius algebras.