Séminaire Lotharingien de Combinatoire, 95B.105 (2026), 12 pp.
Timothy Blanton, Anton Dochtermann, Isabelle Hong, Suho Oh and Zhan Zhan
Parking Functions and Chip-Firing on Hypergraphs
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 has 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.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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