Hopf algebras and Tutte polynomials

TL;DR

This paper develops a unified framework linking Hopf algebras to Tutte polynomials, encompassing various graph polynomials and revealing their shared properties and new relations.

Contribution

It introduces a canonical association of Tutte polynomials with combinatorial objects via Hopf algebras, unifying and extending known graph polynomial theories.

Findings

Several graph polynomials are derived from the Hopf algebra framework.

Shared properties like deletion-contraction, universality, and duality are established.

New relations and properties for existing graph polynomials are obtained.

Abstract

By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the literature arise from this framework. These polynomials include the classical Tutte polynomial of graphs and matroids, Las Vergnas' Tutte polynomial of the morphism of matroids and his Tutte polynomial for embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal polynomial, and the Penrose polynomial. We show that our Tutte polynomials of Hopf algebras share common properties with the classical Tutte polynomial, including deletion-contraction definitions, universality properties, convolution formulas, and duality relations. New results for graph polynomials from the literature are then obtained as examples of the general results. Our results offer a framework for the study of the Tutte polynomial and its analogues in other settings, offering the means to determine the properties and connections between a wide class of polynomial invariants.