Coloring Trivalent Graphs: A Defect TFT Approach

TL;DR

This paper introduces a topological field theory framework with defects to interpret graph coloring, connecting combinatorial problems with quantum and topological concepts, and providing new insights into the structure of graph colorings.

Contribution

It develops a novel TFT-based approach to graph coloring using defects, linking it to quantum theory, bundle sections, and higher category cobordism, offering a new perspective on classical combinatorial problems.

Findings

TFT with defects computes Tait-Colorings of planar trivalent graphs.

Defects generalize group actions in the context of graph coloring.

Interprets the word problem as a cobordism problem in higher categories.

Abstract

We show that the combinatorial matter of graph coloring is, in fact, quantum in the sense of satisfying the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with defects gives meaning to it. This TFT has the property that when evaluated on a planar trivalent graph, it provides the number of Tait-Coloring of it. Defects can be considered as a generalization of groups. With the Klein-four group as a 1-defect condition, we reinterpret graph coloring as sections of a certain bundle, distinguishing a coloring (global-sections) from a coloring process (local-sections.) These constructions also lead to an interpretation of the word problem, for a finitely presented group, as a cobordism problem and a generalization of (trivial) bundles at the level of higher categories.