Orienting Transversals and Transition Polynomials of Multimatroids

TL;DR

This paper advances the theory of multimatroids by providing new evaluations of transition polynomials, connecting them to classical polynomials like the Tutte-Martin and interlace polynomials, and establishing structural theorems for binary tight 3-matroids.

Contribution

It introduces uniform, matroid-theoretic evaluations of transition polynomials and extends known results from graphs and delta-matroids to multimatroids.

Findings

Efficient evaluation of the Tutte-Martin polynomial for isotropic systems.

Generalization of transition polynomial evaluations from 4-regular graphs to multimatroids.

An excluded-minor theorem for binary tight 3-matroids.

Abstract

Multimatroids generalize matroids, delta-matroids, and isotropic systems, and transition polynomials of multimatroids subsume various polynomials for these latter combinatorial structures, such as the interlace polynomial and the Tutte-Martin polynomial. We prove evaluations of the Tutte-Martin polynomial of isotropic systems from Bouchet directly and more efficiently in the context of transition polynomials of multimatroids. Moreover, we generalize some related evaluations of the transition polynomial of 4-regular graphs from Jaeger to multimatroids. These evaluations are obtained in a uniform and matroid-theoretic way. We also translate the evaluations in terms of the interlace polynomial of graphs. Finally, we give an excluded-minor theorem for the class of binary tight 3-matroids (a subclass of multimatroids) based on the excluded-minor theorem for the class of binary delta-matroids from Bouchet.