Structure of the flow and Yamada polynomials of cubic graphs

TL;DR

This paper extends the Tutte golden identity to Yamada polynomials of cubic graphs, explores flow polynomial structures, and proves conjectures relating to planarity and exponential growth in graph invariants using topological quantum field theory.

Contribution

It introduces a quadratic identity for Yamada polynomials of cubic graphs, proves a conjecture on flow polynomial characterization of planarity, and demonstrates exponential growth in chromatic polynomials of planar triangulations.

Findings

Extended Tutte golden identity to Yamada polynomials.

Proved flow polynomial conjecture for certain non-planar graphs.

Established exponential growth of chromatic polynomials in planar triangulations.

Abstract

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs. Further, we establish exponential growth of the number of chromatic polynomials of planar triangulations, answering a question of D. Treumann and E. Zaslow. The structure underlying these results is the chromatic algebra, and more generally the SO(3) topological quantum field theory.