Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Arboricity Polynomial

Felix Breuer, Caroline J Klivans

TL;DR

The paper studies the enumerative invariant $\mathcal{A}_M(k)$, counting independent covers of a matroid $M$ into at most $k$ parts, and proves that it is a polynomial in $k$ of degree $|E|$ by embedding the problem in the scheduling framework and Ehrhart theory. It recasts arboricity as a scheduling problem, linking to the Braid arrangement and NCQSym, and shows $\mathcal{A}_M(k)=\sum_i f_i\binom{k}{i}$ with integer coefficients, where each $f_i$ is a multiple of $i!$. The authors compute the cycle-graph case explicitly as $\mathcal{A}_{M(C_n)}(k)=k^n-k$, and present key properties: (i) the leading term is $n!\binom{k}{n}$, (ii) $\mathcal{A}_M(k)$ is not a Tutte invariant, (iii) it is not generally contraction/deletion or valuative with respect to matroid base polytopes. These insights connect matroid partitioning to scheduling and Ehrhart theory, offering a geometry-driven approach to arboricity and highlighting fundamental limitations of classical matroid invariants in this context.

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.

The Arboricity Polynomial

TL;DR

The paper studies the enumerative invariant , counting independent covers of a matroid into at most parts, and proves that it is a polynomial in of degree by embedding the problem in the scheduling framework and Ehrhart theory. It recasts arboricity as a scheduling problem, linking to the Braid arrangement and NCQSym, and shows with integer coefficients, where each is a multiple of . The authors compute the cycle-graph case explicitly as , and present key properties: (i) the leading term is , (ii) is not a Tutte invariant, (iii) it is not generally contraction/deletion or valuative with respect to matroid base polytopes. These insights connect matroid partitioning to scheduling and Ehrhart theory, offering a geometry-driven approach to arboricity and highlighting fundamental limitations of classical matroid invariants in this context.

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.
Paper Structure (8 sections, 4 theorems, 40 equations, 4 figures)

This paper contains 8 sections, 4 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Arboricity
  4. Scheduling
  5. Arboricity as Scheduling
  6. Cycle Graphs
  7. Non Contraction/Deletion
  8. Non Valuative

Key Result

Theorem 3.5

Scheduling Given a scheduling problem $S$ on $n$ items, the scheduling counting function, $\chi_S(k)$ is a polynomial in $k$ of degree at most $n$, the scheduling polynomial of $S$. where the coefficients $f_1,\ldots,f_n$ are non-negative integers counting the number of ordered set partitions $\Phi$ with $i$ non-empty blocks such that $S(\Phi)$ holds.

Figures (4)

  • Figure 1: On the left, a graph decomposed into $3$ forests. On the right, the same graph decomposed into $2$ forests.
  • Figure 2: Scheduling Geometry. Suppose the cubes shown have side length $k$. The $k$-schedules of the described scheduling problems are the integer points inside the cube that do not lie on any of the shaded regions.
  • Figure 3: The graphical zonotope of the complete graph $K_3$ and the corresponding graphical arrangement.
  • Figure 4: Matroid base polytope of $U_{2,4}$

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • ...and 8 more