Parking functions and chip-firing on hypergraphs

TL;DR

This paper extends the concept of parking functions and chip-firing from graphs to hypergraphs, establishing new combinatorial and algebraic properties, including characterizations via acyclic orientations and spanning trees.

Contribution

It introduces $H$-parking functions for hypergraphs, characterizes them through acyclic orientations, and connects them to superstable configurations in a novel chip-firing model.

Findings

$H$-parking functions are counted by $q$-rooted spanning trees.

Maximal $H$-parking functions correspond to acyclic orientations.

Superstable configurations can be recovered via chip-firing on associated digraphs.

Abstract

For a connected graph $G$ with sink vertex $q$, a $G$-parking function is a vector of nonnegative integers whose entries are determined by cut-sets in $G$. Such objects also arise as the superstable configurations in the context of chip-firing. The set of all $G$-parking functions have various algebraic and combinatorial properties; for instance they relate to evaluations of the Tutte polynomial and in particular are counted by spanning trees of $G$. We extend these constructions to the setting of hypergraphs, where edges can have multiple vertices. For a hypergraph $H$ with sink $q$, we define $H$-parking functions in terms of cuts in $H$ and prove that the maximal such sequences are characterized by certain acyclic orientations of $H$. We introduce a notion of a $q$-rooted spanning tree for $H$, and prove that the set of all such objects are counted by $H$-parking functions. We also show how $H$-parking functions can be recovered as the superstable configurations in a version of chip-firing on $H$, where chips have a choice of where to go when fired. We prove that one can recover such configurations via chip-firing on a family of digraphs associated to $H$.