The twist number of a plane graph associated with a link’s checkerboard coloring is defined. The twist number of any planar pseudograph is shown to be determined by the Tutte polynomial. This allows the definition of twist number to be extended to non-planar graphs, for which the twist number is also determined by the Tutte polynomial. As a corollary, the twist number of an alternating link is ...

The notion of activities with respect to spanning trees in graphs was introduced by W.T. Tutte, and generalized to activities with respect to bases in matroids by H. Crapo. We present a further generalization, to activities with respect to arbitrary subsets of matroids. These generalized activities provide a unified view of several different expansions of the Tutte polynomial and the chromatic ...

For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate nspecialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n ≤ 2, we define the different one variable n-specializations of the dichromatic polynomials, a...

Tutte paths are one of the most successful tools for attacking Hamiltonicity problems in planar graphs. Unfortunately, results based on them are non-constructive, as their proofs inherently use an induction on overlapping subgraphs and these overlaps hinder to bound the running time to a polynomial. For special cases however, computational results of Tutte paths are known: For 4-connected plana...

We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully purs...

This is a close approximation to the content of my lecture. After a brief survey of well known properties, I present some new interpretations relating to random graphs, lattice point enumeration, and chip firing games. I then examine complexity issues and concentrate in particular, on the existence of randomized approximation schemes. © 1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 210–...

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arise...

We show how the Tutte polynomial of a matroid M can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of M is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks. They furthermore generalize similiar statements for perfect matroid designs and near d...