Universal Tutte characters via combinatorial coalgebras

Clément Dupont,Luca Moci,Alex Fink

doi:10.5802/alco.35

Abstract

This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids. We also produce some new invariants along with new convolution formulae.

Highlights

The Tutte polynomial is surely the single most appreciated invariant of matroids and graphs
The Tutte polynomial satisfies interesting identities like the convolution formula of Kook–Reiner–Stanton [26], which follows from work of Etienne–Las Vergnas [18]
The present work is concerned with the many other combinatorial objects which possess invariants with properties reminiscent of these, such as matroid perspectives and their Las Vergnas polynomial [31] or delta-matroids and their Bollobás–Riordan polynomial [5], which are both matroidal frameworks for certain topological embeddings of graphs in surfaces

Summary

Introduction

The Tutte polynomial is surely the single most appreciated invariant of matroids and graphs. The present work is concerned with the many other combinatorial objects which possess invariants with properties reminiscent of these, such as matroid perspectives and their Las Vergnas polynomial [31] or delta-matroids and their Bollobás–Riordan polynomial [5], which are both matroidal frameworks for certain topological embeddings of graphs in surfaces Another example, which served as our initial motivation, is arithmetic matroids, introduced by D’Adderio and the third author [13], whose arithmetic Tutte polynomial satisfies a convolution formula (proved by Backman– Lenz [3] in a special case, and in the present paper in greater generality).

Minors systems

Norms and universal Tutte characters