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Chip-firing and Parking Functions on Hypergraphs

Algebra & Discrete Mathematics

Speaker: Timothy Blanton, UC Davis
Location: 2112 MSB
Start time: Tue, Mar 5 2024, 1:10PM

Classical chip-firing is a game played on a graph $G$, where chips are placed on the vertices of $G$ and distributed according to simple rules. The dynamics of chip-firing has found applications in many areas of mathematics and physics, and give rise to $G$-parking functions as well as the critical group of $G$. We extend these constructions to the setting of a hypergraph $H$ and define a notion of superstable configurations (and hence parking functions). We discuss how one can recover such configurations via chip firing on digraphs, and show that maximal superstable configurations are in bijection with certain acyclic orientations of $H$. We employ a version of Dhar's burning algorithm for hypergraphs that leads to notions of spanning trees that generalize other constructions from the literature. 

This is joint work with Anton Dochtermann, Isabelle Hong, Suho Oh, and Zhan Zhan