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Tutte polynomials of matroids as universal valuative invariants

Luis Ferroni, Benjamin Schröter

TL;DR

This work classifies inclusion-maximal, dual- and minor-closed families of matroids for which the Tutte polynomial is a universal valuative invariant, identifying four principal families: elementary split matroids, uniform matroids with added loops and coloops, graphic Schubert matroids, and a class containing all sparse paving matroids together with certain partition matroids. Using the universal Derksen–Fink $\mathcal{G}$-invariant and rank-based criteria, the authors show Tutte universality on each class and derive integral decompositions of Tutte polynomials in terms of cuspidal matroids or graphic Schubert matroids, yielding new relations among Tutte polynomials. They establish that every matroid's Tutte polynomial can be uniquely expressed as an integral combination of polynomials from the cuspidal family or from the graphic Schubert family, depending on the decomposition, and provide a comprehensive proof of the main characterization by analyzing excluded minors. The results illuminate the structure of valuative invariants in matroid theory, suggest possible extensions to other invariants, and open questions about the robustness of universality under relaxing dual-closedness assumptions and about the combinatorial meaning of the graphic Schubert coefficients.

Abstract

We provide a full classification of all families of matroids that are closed under duality and minors, and for which the Tutte polynomial is a universal valuative invariant. There are four inclusion-wise maximal families, two of which are the class of elementary split matroids and the class of graphic Schubert matroids. As a consequence of our framework, we derive new relations among Tutte polynomials of matroids. For example, we show that the Tutte polynomial of every matroid can be expressed uniquely as an integral combination of Tutte polynomials of graphic Schubert matroids.

Tutte polynomials of matroids as universal valuative invariants

TL;DR

This work classifies inclusion-maximal, dual- and minor-closed families of matroids for which the Tutte polynomial is a universal valuative invariant, identifying four principal families: elementary split matroids, uniform matroids with added loops and coloops, graphic Schubert matroids, and a class containing all sparse paving matroids together with certain partition matroids. Using the universal Derksen–Fink -invariant and rank-based criteria, the authors show Tutte universality on each class and derive integral decompositions of Tutte polynomials in terms of cuspidal matroids or graphic Schubert matroids, yielding new relations among Tutte polynomials. They establish that every matroid's Tutte polynomial can be uniquely expressed as an integral combination of polynomials from the cuspidal family or from the graphic Schubert family, depending on the decomposition, and provide a comprehensive proof of the main characterization by analyzing excluded minors. The results illuminate the structure of valuative invariants in matroid theory, suggest possible extensions to other invariants, and open questions about the robustness of universality under relaxing dual-closedness assumptions and about the combinatorial meaning of the graphic Schubert coefficients.

Abstract

We provide a full classification of all families of matroids that are closed under duality and minors, and for which the Tutte polynomial is a universal valuative invariant. There are four inclusion-wise maximal families, two of which are the class of elementary split matroids and the class of graphic Schubert matroids. As a consequence of our framework, we derive new relations among Tutte polynomials of matroids. For example, we show that the Tutte polynomial of every matroid can be expressed uniquely as an integral combination of Tutte polynomials of graphic Schubert matroids.
Paper Structure (19 sections, 24 theorems, 28 equations, 2 figures)

This paper contains 19 sections, 24 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elementary split and cuspidal matroids
  4. Valuative invariants and universality
  5. A reformulation of universality for valuations
  6. Four families for which Tutte polynomials are universal
  7. Elementary split matroids and Tutte universality
  8. A second class of matroids containing all sparse paving matroids
  9. Two more classes of matroids
  10. Uniform matroids with additional loops and coloops
  11. Minimal matroids with additional loops and coloops
  12. Universality for the two classes $\mathcal{U}$ and $\mathcal{T}$
  13. Proof of our characterization
  14. Final remarks and some open problems
  15. Analogs for other valuative invariants
  16. ...and 4 more sections

Key Result

Theorem A

Let $\mathcal{C}$ be an inclusion-wise maximal family of matroids closed under taking duals and minors, and for which the Tutte polynomial is valuatively universal. Then, one of the following four cases applies:

Figures (2)

  • Figure 1: The lattice path representation of the matroid $\mathsf{\Lambda}_{4,7,8,16}$.
  • Figure 2: The underlying graph of a minimal matroid $\mathsf{T}_{5,8}$

Theorems & Definitions (48)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem 2.1: bercziferroni-schroterferroni-schroter-mw
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • ...and 38 more