Sandpiles, Spanning Trees, and Plane Duality

Chan, Melody; Glass, Darren B.; Macauley, Matthew; Perkinson, David; Werner, Caryn; Yang, Qiaoyu

doi:10.1137/140982015

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…Motivated by a question of Ellenberg [19], [28], Baker and Wang proved that if the ribbon structure is planar, then all Bernardi torsors are isomorphic and they respect plane duality [9]. Similar results were proven for rotor routing by Chan et al [15], [16] (see [27] for background on rotor routing). In fact, Baker and Wang showed that the torsors induced by Bernardi bijections are isomorphic to the torsors induced by rotor routing if the ribbon structure is planar.…”

Section: Introductionmentioning

confidence: 75%

Geometric bijections between spanning trees and break divisors

Yuen

2017

Journal of Combinatorial Theory, Series A

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The Jacobian group Jac(G) of a finite graph G is a group whose cardinality is the number of spanning trees of G. G also has a tropical Jacobian which has the structure of a real torus; using the notion of break divisors, An et al. obtained a polyhedral decomposition of the tropical Jacobian where vertices and cells correspond to elements of Jac(G) and spanning trees of G, respectively. We give a combinatorial description of bijections coming from this geometric setting. This provides a new geometric method for constructing bijections in combinatorics. We introduce a special class of geometric bijections that we call edge ordering maps, which have good algorithmic properties. Finally, we study the connection between our geometric bijections and the class of bijections introduced by Bernardi; in particular we prove a conjecture of Baker that planar Bernardi bijections are "geometric". We also give sharpened versions of results by Baker and Wang on Bernardi torsors. arXiv:1509.02963v2 [math.CO]

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Section: Introductionmentioning

confidence: 75%

Geometric bijections between spanning trees and break divisors

Yuen

2017

Journal of Combinatorial Theory, Series A

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“…In the guise of "abelian sandpile model", chip-firing games play an important role in the study of self-organized criticality in statistical physics [4,20]. The chip firing game introduced by Baker and Norine is relevant for classical combinatorial problems about graphs, relating to spanning trees [16], the uniqueness of graph involutions [6], and potential theory on electrical network graphs [7].…”

Section: Known Resultsmentioning

confidence: 99%

Recognizing Hyperelliptic Graphs in Polynomial Time

Bodewes

Bodlaender

Cornelissen

et al. 2018

Graph-Theoretic Concepts in Computer Science

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Based on analogies between algebraic curves and graphs, Baker and Norine introduced divisorial gonality, a graph parameter for multigraphs related to treewidth, multigraph algorithms and number theory. Various equivalent definitions of the gonality of an algebraic curve translate to different notions of gonality for graphs, called stable gonality and stable divisorial gonality. We consider so-called hyperelliptic graphs (multigraphs of gonality 2, in any meaning of graph gonality) and provide a safe and complete set of reduction rules for such multigraphs. This results in an algorithm to recognize hyperelliptic graphs in time O (m + n log n), where n is the number of vertices and m the number of edges of the multigraph. A corollary is that we can decide with the same runtime whether a two-edge-connected graph G admits an involution σ such that the quotient G/ σ is a tree.

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“…Similarly, define σ * by orienting each simple cycle of the dual graph G * clockwise and composing with the natural bijection between oriented cuts of G and oriented cycles of G * . By [38,Theorem 15], the simply transitive action of Jac(G) on T (G) afforded by Theorem 1.1.2 in this case coincides with the "Bernardi torsor" defined in [8] and a posteriori with the "rotor-routing torsor" defined in [12,13]. In particular, we get a new "geometric" proof of the bijectivity of the Bernardi map.…”

Section: Then the Map T → [O(t )] Is A Bijection Between T (G) And G(g)mentioning

confidence: 89%

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

Backman

Baker

Yuen

2019

Forum of Mathematics, Sigma

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Let M be a regular matroid. The Jacobian group Jac(M ) of M is a finite abelian group whose cardinality is equal to the number of bases of M . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) Jac(G) of a graph G (in which case bases of the corresponding regular matroid are spanning trees of G).There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph Jac(G) and spanning trees. However, most of the known bijections use vertices of G in some essential way and are inherently "non-matroidal". In this paper, we construct a family of explicit and easy-todescribe bijections between the Jacobian group of a regular matroid M and bases of M , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of M admits a canonical simply transitive action on the set G(M ) of circuit-cocircuit reversal classes of M , and then define a family of combinatorial bijections β σ,σ * between G(M ) and bases of M . (Here σ (resp. σ * ) is an acyclic signature of the set of circuits (resp. cocircuits) of M .) We then give a geometric interpretation of each such map β = β σ,σ * in terms of zonotopal subdivisions which is used to verify that β is indeed a bijection.Finally, we give a combinatorial interpretation of lattice points in the zonotope Z; by passing to dilations we obtain a new derivation of Stanley's formula linking the Ehrhart polynomial of Z to the Tutte polynomial of M . arXiv:1701.01051v3 [math.CO] 23 Apr 20191 If e and e * are dual edges of G and G * , respectively, then given an orientation for e * we orient e by rotating the orientation of e * clockwise locally near the crossing of e and e * .

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Sandpiles, Spanning Trees, and Plane Duality

Cited by 6 publications

References 9 publications

Geometric bijections between spanning trees and break divisors

Geometric bijections between spanning trees and break divisors

Recognizing Hyperelliptic Graphs in Polynomial Time

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

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