Sandpiles, Spanning Trees, and Plane Duality
Project Author
Issue Date
2015-03-10
Authors
Chan, Melody
Glass, Darren
Macauley, Matthew
Perkinson, David
Werner, Caryn
Yang, Qiaoyu
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Keywords
sandpiles , chip-firing , rotor-router model , ribbon graphs , planarity
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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.
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Department
Mathematics
Women's, Gender, & Sexuality Studies
Women's, Gender, & Sexuality Studies
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License
© 2015, Society for Industrial and Applied Mathematics
Citation
Chan, M., et al. 2015. "Sandpiles, Spanning Trees, and Plane Duality." SIAM Journal on Discrete Mathematics 29, no. 1: 461-471.
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Preprint
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Society for Industrial and Applied Mathematics