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Degrees of interior polynomials and parking function enumerators

Tamás Kálmán, Lilla Tóthmérész

TL;DR

The work establishes precise degree formulas for interior polynomials of digraphs and their matroid generalizations, showing $\deg I_G = |V|-1-\nu(G)$ where $\nu(G)$ is the minimum cardinality of a dijoin, and that the leading coefficient counts net degree vectors of minimum dijoins. By duality, these results translate to parking-function enumerators and greedoid polynomials, yielding $\deg$ of the parking-function enumerator for an Eulerian digraph as $|E|-|V|+1-\minfas(G)$ and extending to arbitrary rooted digraphs via $\mathrm{minfas}(G,r)$. The paper further generalizes to oriented regular matroids, giving analogous degree formulas and a facet description of extended root polytopes, thereby unifying graph- and matroid-theoretic perspectives. The connections illuminate how combinatorial optimizations (dijoins, cuts, and feedback arc sets) govern polynomial degrees in Ehrhart-theoretic encodings, with implications for counting generalized parking functions and analyzing greedoid polynomials across broad combinatorial structures.

Abstract

The interior polynomial of a directed graph is defined as the $h^*$-polynomial of the graph's (extended) root polytope, and it displays several attractive properties. Here we express its degree in terms of the minimum cardinality of a directed join, and give a formula for the leading coefficient. We present natural generalizations of these results to oriented regular matroids; in the process we also give a facet description for the extended root polytope of an oriented regular matroid. By duality, our expression for the degree of the interior polynomial implies a formula for the degree of the parking function enumerator of an Eulerian directed graph (which is equivalent to the greedoid polynomial of the corresponding branching greedoid). We extend that result to obtain the degree of the parking function enumerator of an arbitrary rooted directed graph in terms of the minimum cardinality of a certain type of feedback arc set.

Degrees of interior polynomials and parking function enumerators

TL;DR

The work establishes precise degree formulas for interior polynomials of digraphs and their matroid generalizations, showing where is the minimum cardinality of a dijoin, and that the leading coefficient counts net degree vectors of minimum dijoins. By duality, these results translate to parking-function enumerators and greedoid polynomials, yielding of the parking-function enumerator for an Eulerian digraph as and extending to arbitrary rooted digraphs via . The paper further generalizes to oriented regular matroids, giving analogous degree formulas and a facet description of extended root polytopes, thereby unifying graph- and matroid-theoretic perspectives. The connections illuminate how combinatorial optimizations (dijoins, cuts, and feedback arc sets) govern polynomial degrees in Ehrhart-theoretic encodings, with implications for counting generalized parking functions and analyzing greedoid polynomials across broad combinatorial structures.

Abstract

The interior polynomial of a directed graph is defined as the -polynomial of the graph's (extended) root polytope, and it displays several attractive properties. Here we express its degree in terms of the minimum cardinality of a directed join, and give a formula for the leading coefficient. We present natural generalizations of these results to oriented regular matroids; in the process we also give a facet description for the extended root polytope of an oriented regular matroid. By duality, our expression for the degree of the interior polynomial implies a formula for the degree of the parking function enumerator of an Eulerian directed graph (which is equivalent to the greedoid polynomial of the corresponding branching greedoid). We extend that result to obtain the degree of the parking function enumerator of an arbitrary rooted directed graph in terms of the minimum cardinality of a certain type of feedback arc set.
Paper Structure (12 sections, 33 theorems, 21 equations, 4 figures)

This paper contains 12 sections, 33 theorems, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Interior polynomials
  3. Greedoid polynomials and parking function enumerators
  4. Acknowledgements
  5. Graph notation
  6. The degree of the interior polynomial of a digraph
  7. Degree-minimizing orientations
  8. A generalization to regular matroids
  9. Parking function enumerators and greedoid polynomials
  10. Preliminaries on greedoids and parking functions
  11. General greedoids
  12. Arbitrary directed graphs

Key Result

Theorem 1.1

Let $G=(V,E)$ be a connected digraph. Then the degree of the interior polynomial of $G$ is equal to $|V|-1-\nu(G)$, where $\nu(G)=\min \{|K| \mid K \subseteq E \text{ is a dijoin of }G\}$.

Figures (4)

  • Figure 1: A directed graph with five disjoint directed cuts.
  • Figure 2: Illustration for Example \ref{['ex:minimizer_and_maximizer']}.
  • Figure 3: Eulerian digraph with root $s$. The non-dashed arcs form a spanning arborescence rooted at $s$.
  • Figure 4: A rooted digraph (with root $s$), and its parking functions.

Theorems & Definitions (86)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 76 more