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.
This is the author's version of the work. This publication appears in Gettysburg College's institutional repository by permission of the copyright owner for personal use, not for redistribution.
Chan, Melody, Darren Glass, Matthew Macauley, David Perkinson, Caryn Werner, and Qiaoyu Yang. "Sandpiles, Spanning Trees, and Plane Duality." SIAM Journal on Discrete Mathematics 29.1 (March 2015), 461-471.
Required Publisher's Statement
Original version is available from the publisher at: http://epubs.siam.org/doi/abs/10.1137/140982015