Three-Colorings of Cubic Graphs and Tensor Operators

TL;DR

This paper explores the algebraic representation of cubic graphs with free ends through tensor operators, building on Penrose's connection between edge 3-colorings and tensor algebras.

Contribution

It introduces an algebraic framework for cubic graphs with free ends, extending Penrose's work on tensor algebras and graph colorings.

Findings

Developed algebraic representations for cubic graphs with free ends

Extended Penrose's tensor algebra approach to non-planar graphs

Provided new tools for analyzing graph colorings algebraically

Abstract

Penrose's work \cite{8} established a connection between the edge 3-colorings of cubic planar graphs and tensor algebras. We exploit this point of view in order to get algebraic representations of the category of cubic graphs with free ends.