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arXiv · 2602.16184

Restricted Chip-Firing: Toric Toppling Ideals, Picard Groups, and Cellular Resolutions

Rahul Karki
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Abstract

We study certain groups and ideals arising from the chip-firing game on a generalisation of graphs called pargraphs. Several well-known families of toric ideals arise as toppling ideals of pargraphs. These include ideals defining rational normal curves, binomial edge ideals of complete graphs, and toric ideals of certain Fano polytopes. As our central result, we provide several equivalent conditions for the Picard group of a pargraph to be free, which equivalently characterise when its toppling ideal is toric. Furthermore, we construct a Gr\"obner basis for the toppling ideal, a minimal cellular free resolution for a distinguished initial ideal known as the $G$-parking function ideal, and establish Cohen-Macaulay property for these ideals.

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Rahul Karki. 2026-02-18. Restricted Chip-Firing: Toric Toppling Ideals, Picard Groups, and Cellular Resolutions. https://arxiv.org/abs/2602.16184

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