Sandpiles, Spanning Trees, and Plane Duality
Persistent URL
Author(s)
Chan, Melody
Glass, Darren
Macauley, Matthew
Perkinson, David
Werner, Caryn
Yang, Qiaoyu
Date Issued
March 10, 2015
Abstract
Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it; i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.
Journal
SIAM Journal on Discrete Mathematics
Department
Mathematics
Women's, Gender, & Sexuality Studies
Citation
Chan, M., et al. 2015. "Sandpiles, Spanning Trees, and Plane Duality." SIAM Journal on Discrete Mathematics 29, no. 1: 461-471.
Publisher
Society for Industrial and Applied Mathematics
Version of Article
Preprint
DOI
10.1137/140982015
ISSN
0895-4801
1095-7146
Rights
© 2015, Society for Industrial and Applied Mathematics
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Werner 2015 Sandpiles Preprint.pdf
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