For any matroid M realizable over Q , we give a combinatorial interpretation of the Tutte polynomial T M (x, y) which generalizes many of its known interpretations and specializations, including Tutte's coloring and flow interpretations of T M (1t, 0), T M (0, 1t); Crapo and Rota's finite field inte
The Tutte polynomial
โ Scribed by Dominic Welsh
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 151 KB
- Volume
- 15
- Category
- Article
- ISSN
- 1042-9832
No coin nor oath required. For personal study only.
โฆ Synopsis
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.
๐ SIMILAR VOLUMES
Following Crapo [2], let `(x, y)(M)=x r(M) y r(M\*) , where K=Z[x, y]. Lemma 1. `(x, y) &1 =`(&x, &y).
We present two splitting formulas for calculating the Tutte polynomial of a matroid. The first one is for a generalized parallel connection across a 3-point line of two matroids and the second one is applicable to a 3-sum of two matroids. An important tool used is the bipointed Tutte polynomial of a
W. T. Tutte conjectured that the coefficients \(t_{i, j}\) of his dichromate form unimodal sequences in \(i\) and \(j\) separately. P. D. Seymour and D. J. A. Welsh conjectured more generally that the same holds for the coefficients of the Tutte polynomial of an arbitrary matroid. We show, by an exa
For each pair of algebraic numbers (x, y) and each field F, the complexity of computing the Tutte polynomial T(M; x, y) of a matroid M representable over F is determined. This computation is found to be \*P-complete except when (x&1)( y&1)=1 or when |F| divides (x&1)( y&1) and (x, y) is one of the s