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On the compatible sets expansion of the Tutte polynomial

Laura Pierson

Abstract

Kochol (2021) gave a new expansion formula for the Tutte polynomial of a matroid using the notion of \emph{compatible sets}, and asked how this expansion relates to the internal-external activities formula. Here, we provide an answer, which is obtained as a special case of a generalized version of the expansion formula to Las Vergnas's trivariate Tutte polynomials of matroid perspectives. The same generalization to matroid perspectives and bijection with activities have been independently proven by Kochol (2022 and 2023) in parallel with this work, but using different methods. Kochol proves both results recursively using the contraction-deletion relations, whereas we give a more direct proof of the bijection and use that to deduce the compatible sets expansion formula from Las Vergnas's activities expansion.

On the compatible sets expansion of the Tutte polynomial

Abstract

Kochol (2021) gave a new expansion formula for the Tutte polynomial of a matroid using the notion of \emph{compatible sets}, and asked how this expansion relates to the internal-external activities formula. Here, we provide an answer, which is obtained as a special case of a generalized version of the expansion formula to Las Vergnas's trivariate Tutte polynomials of matroid perspectives. The same generalization to matroid perspectives and bijection with activities have been independently proven by Kochol (2022 and 2023) in parallel with this work, but using different methods. Kochol proves both results recursively using the contraction-deletion relations, whereas we give a more direct proof of the bijection and use that to deduce the compatible sets expansion formula from Las Vergnas's activities expansion.
Paper Structure (2 sections, 5 theorems, 30 equations, 1 figure, 1 table)

This paper contains 2 sections, 5 theorems, 30 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Proof of the bijection and the expansion formula

Key Result

Theorem 4

The Tutte polynomial $T_M(x,y)$ for a matroid $M=(E,\mathcal{B})$ is given by

Figures (1)

  • Figure 1: Sample graphical matroid perspective $(M, M')$ used in Example \ref{['ex:bijection_example']}, along with $(M')^*.$

Theorems & Definitions (27)

  • Definition 1
  • Definition 2: Crapo Crapo
  • Definition 3: Kochol Kochol21
  • Theorem 4: Kochol Kochol21
  • Definition 5
  • Example 6
  • Theorem 7: Las Vergnas LasVerg
  • Remark 1
  • Definition 8: Kochol Kochol22
  • Theorem 9: independently proven by Kochol in Kochol23
  • ...and 17 more