Edge colouring models for the Tutte polynomial and related graph invariants

TL;DR

This paper introduces new edge q-colouring models that connect the Tutte polynomial, q-flows, and vertex colourings, providing a unified framework and new evaluation methods for graph invariants.

Contribution

It develops novel edge colouring models for the Tutte polynomial and related invariants, extending their applicability and linking them to proper edge colourings on k-regular graphs.

Findings

Derived edge q-colouring models for the Tutte polynomial on hyperbola H_q.

Established models for symmetric weight enumerators of q-flows.

Presented non-symmetric models for proper edge k-colourings on Pfaffian graphs.

Abstract

For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley's symmetric monochrome polynomial. In the second half of the paper we exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q >= k each evaluate the number of proper edge k-colourings of G when G is Pfaffian.