Vector spaces spanned by Tutte polynomials

Abstract

Kung exhibited two bases for the subspace of bivariate polynomials spanned by the Tutte polynomials of matroids of size n and rank r, thereby determining its dimension, and asked what dimension subspace of bivariate polynomials is spanned by the Tutte polynomials of matroids of size n and rank r drawn from a fixed class C of matroids (such as cycle matroids of graphs or binary matroids). We give a sufficient condition for this space to be of the maximum possible dimension r(n-r)+1, i.e. the same as that spanned by the Tutte polynomials of all matroids. In particular, we produce a basis composed of the Tutte polynomials of r(n-r)+1 graphs, each of which is a series-parallel graph with a number of bridges and loops added. We also determine the dimension of the subspace of bivariate polynomials spanned by the Tutte polynomials of connected matroids of given size and rank. Again, an explicit basis is provided: it is composed only of Tutte polynomials of planar 2-connected loopless graphs, with the addition of the uniform matroid (or a small variation of it) in case n = r+2 ≥ 4 or r = 2 ≤ n-2. To obtain this, we make the intermediate step of providing an explicit basis of the subspace generated by the Tutte polynomials of all matroids of given size and rank that are connected after the removal of any loops. We additionally obtain, as a by-product, an explicit basis for the class of all loopless matroids of given size and rank.

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BibTeX RIS

BibTeX

@article{goodall2026vector,
  title = {{Vector spaces spanned by Tutte polynomials}},
  author = {Andrew Goodall and Florent Jouve and Jean-Sébastien Sereni},
  journal = {The Australasian Journal of Combinatorics},
  year = {2026},
  month = jul,
  url = {https://hal.science/hal-04485620v4},
  note = {CC BY-ND 4.0},
}

RIS

TY  - JOUR
TI  - Vector spaces spanned by Tutte polynomials
AU  - Andrew Goodall
AU  - Florent Jouve
AU  - Jean-Sébastien Sereni
PY  - 2026
DA  - 2026/07/22
JO  - The Australasian Journal of Combinatorics
UR  - https://hal.science/hal-04485620v4
AB  - Kung exhibited two bases for the subspace of bivariate polynomials spanned by the Tutte polynomials of matroids of size n and rank r, thereby determining its dimension, and asked what dimension subspace of bivariate polynomials is spanned by the Tutte polynomials of matroids of size n and rank r drawn from a fixed class C of matroids (such as cycle matroids of graphs or binary matroids). We give a sufficient condition for this space to be of the maximum possible dimension r(n-r)+1, i.e. the same as that spanned by the Tutte polynomials of all matroids. In particular, we produce a basis composed of the Tutte polynomials of r(n-r)+1 graphs, each of which is a series-parallel graph with a number of bridges and loops added. We also determine the dimension of the subspace of bivariate polynomials spanned by the Tutte polynomials of connected matroids of given size and rank. Again, an explicit basis is provided: it is composed only of Tutte polynomials of planar 2-connected loopless graphs, with the addition of the uniform matroid (or a small variation of it) in case n = r+2 ≥ 4 or r = 2 ≤ n-2. To obtain this, we make the intermediate step of providing an explicit basis of the subspace generated by the Tutte polynomials of all matroids of given size and rank that are connected after the removal of any loops. We additionally obtain, as a by-product, an explicit basis for the class of all loopless matroids of given size and rank.
SN  - 1034-4942
SN  - 2202-3518
ER  -

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