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Tensor product formulas for the Bollobás-Riordan and Krushkal polynomials

Iain Moffatt, Maya Thompson

TL;DR

The paper develops a unified framework to extend Brylawski's tensor product formula from the classical Tutte polynomial to topological Tutte polynomials of graphs embedded in pseudo-surfaces, notably the Bollobás--Riordan and Krushkal polynomials. By introducing arrow presentations and their packaged/vertex-partitioned variants, the authors define 2-sums and tensor products in a setting that accommodates non-cellular embeddings and deletion–contraction constructions via a general polynomial $Q$ (and its specialization $Z$). A central result is a multivariate tensor-product theorem (Theorem mainmv) that expresses $Q$ of a tensor product in terms of transformed edge-parameters obtained from solving a system of equations, thereby unifying BR, Krushkal, and related polynomials and enabling their recovery as special cases. The framework recovers known results for the Bollobás--Riordan polynomial, the topological transition polynomial, and the Tutte polynomial, providing both computational tools and conceptual cohesion for topological graph polynomials across embeddings.

Abstract

Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the ribbon graph polynomial and transition polynomials of graphs embedded in surfaces, as well as for the Bollobás-Riordan polynomial in some special cases. We define the tensor product of graphs embedded in pseudo-surfaces and use this to generalize and unify all of the above results, providing Brylawski-style formulas for both the Bollobás-Riordan and Krushkal polynomials.

Tensor product formulas for the Bollobás-Riordan and Krushkal polynomials

TL;DR

The paper develops a unified framework to extend Brylawski's tensor product formula from the classical Tutte polynomial to topological Tutte polynomials of graphs embedded in pseudo-surfaces, notably the Bollobás--Riordan and Krushkal polynomials. By introducing arrow presentations and their packaged/vertex-partitioned variants, the authors define 2-sums and tensor products in a setting that accommodates non-cellular embeddings and deletion–contraction constructions via a general polynomial (and its specialization ). A central result is a multivariate tensor-product theorem (Theorem mainmv) that expresses of a tensor product in terms of transformed edge-parameters obtained from solving a system of equations, thereby unifying BR, Krushkal, and related polynomials and enabling their recovery as special cases. The framework recovers known results for the Bollobás--Riordan polynomial, the topological transition polynomial, and the Tutte polynomial, providing both computational tools and conceptual cohesion for topological graph polynomials across embeddings.

Abstract

Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the ribbon graph polynomial and transition polynomials of graphs embedded in surfaces, as well as for the Bollobás-Riordan polynomial in some special cases. We define the tensor product of graphs embedded in pseudo-surfaces and use this to generalize and unify all of the above results, providing Brylawski-style formulas for both the Bollobás-Riordan and Krushkal polynomials.

Paper Structure

This paper contains 15 sections, 18 theorems, 61 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Arrow presentations and packaged arrow presentations
  3. Arrow presentations
  4. 2-sums and tensor products for arrow presentations
  5. Packaged arrow presentations
  6. 2-sums and tensor products for packaged arrow presentations
  7. Vertex partitioned arrow presentations
  8. Tensor product formulas for topological Tutte polynomials
  9. The Krushkal polynomial
  10. A review of the Krushkal polynomial
  11. The tensor product formula
  12. Some special cases
  13. The Bollobás--Riordan polynomial
  14. The topological transition and ribbon graph polynomials
  15. The Tutte polynomial

Key Result

Lemma 2.1

Let $\mathbb{G}$ be an arrow presentations with an edge $f$, let $\mathbb{H}$ be an arrow presentation with distinct edges $e$ and $g$, and let $\mathbb{K}$ an arrow presentation with an edge $h$. In addition let $\varphi_{f,e}$ be a coupling of $f$ and $e$, and $\varphi_{g,h}$ be a coupling of $g$

Figures (9)

  • Figure 1: An arrow presentation, its corresponding ribbon graph, and the results of ribbon graph operations.
  • Figure 2: The 2-sum of $\mathbb{G}$ and $\mathbb{H}$ with respect to the coupling $\varphi$.
  • Figure 3: Two arrow presentations, $\mathbb{G}$ and $\mathbb{H}$, with a coupling $\varphi$, and the 2-sum $\mathbb{G}\oplus_{\varphi} \mathbb{H}$.
  • Figure 4: Contracting and merge-deleting the edge $f$.
  • Figure 5: Naming conventions used in Definition \ref{['d.big2sum']}.
  • ...and 4 more figures

Theorems & Definitions (38)

  • Definition 1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark
  • Lemma 2.3
  • proof
  • Definition 5
  • ...and 28 more