On the Algebraic Complexity of Some Families of Coloured Tutte Polynomials

Complexity of the Bollobás-Riordan Polynomial

The coloured Tutte polynomial by Bollobás and Riordan is, as a generalisation of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contractiondeletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines. We establish a similar...

Tutte polynomials of wheels via generating functions

We find an explicit expression of the Tutte polynomial of an $n$-fan. We also find a formula of the Tutte polynomial of an $n$-wheel in terms of the Tutte polynomial of $n$-fans. Finally, we give an alternative expression of the Tutte polynomial of an $n$-wheel and then prove the explicit formula for the Tutte polynomial of an $n$-wheel.

Tutte Polynomials of Flower Graphs

Counting planar maps, coloured or uncoloured

We present recent results on the enumeration of q-coloured planar maps, where each monochromatic edge carries a weight ν. This is equivalent to weighting each map by its Tutte polynomial, or to solving the q-state Potts model on random planar maps. The associated generating function, obtained by Olivier Bernardi and the author, is differentially algebraic. That is, it satisfies a (nonlinear) di...

Tutte polynomials of hyperplane arrangements and the finite field method

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a h...