Séminaire Lotharingien de Combinatoire, 86B.73 (2022), 12 pp.
Ankan Ganguly and Alex McDonough
Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs
Abstract.
A sandpile torsor algorithm is a map which associates each plane graph (i.e., planar embedding) with a free transitive action of its sandpile group on its spanning trees. We define a notion of consistency, which requires a torsor algorithm to be preserved with respect to a certain class of contractions and deletions. We then show that the rotor-routing sandpile torsor algorithm (which is equivalent to the Bernardi algorithm) is consistent. Furthermore, we demonstrate that there are only three other consistent algorithms, which all have the same structure as rotor-routing. This proves a conjecture of Klivans. The paper corresponding to this extended abstract can be found at
arχiv:2203.15079.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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