The Arboricity Polynomial

TL;DR

The paper introduces the arboricity polynomial, a new matroid invariant that counts disjoint independent set covers, linking it to scheduling polynomials and Ehrhart theory, and demonstrating its unique properties.

Contribution

It defines the arboricity polynomial, explores its properties, and establishes its connections to scheduling polynomials and Ehrhart theory, distinguishing it from Tutte invariants.

Findings

The arboricity polynomial enumerates covers by disjoint independent sets.

It is related to quasisymmetric functions and Ehrhart theory.

The polynomial is not a Tutte invariant, lacking contraction/deletion recursion.

Abstract

We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the counting function as a special case of a scheduling polynomial, i.e. both in terms of quasisymmetric functions and via Ehrhart theory of the normal fan of the matroid base polytope. We show basic properties of the polynomial and demonstrate that it is not a Tutte invariant. Namely, the arboricity polynomial does not satisfy a contraction / deletion recursion.