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On canonical sandpile actions of embedded graphs

Lilla Tóthmérész

TL;DR

The paper extends canonical sandpile actions beyond planar graphs by introducing a tour-rotor action for embedded Eulerian digraphs, yielding a canonical, root-independent action on compatible Eulerian tours and linking it to the rotor-routing framework. It provides a streamlined proof that rotor-routing is root-independent precisely for planar embeddings and forges a tight connection between the Jacobian of embedded graphs and the sandpile group of the medial digraph. The authors further align this Tour-Rotor framework with the Bernardi action of Baker–Ding–Kim, via Bouchet’s correspondence between medial tours and quasi-trees, and establish a canonical isomorphism Jac$(\mathcal{C}(G,\Sigma))\cong S(G^{\bowtie})$ with the two actions agreeing. Collectively, these results unify several canonical actions on combinatorial structures associated to embedded graphs, offering a cohesive picture that relates spanning trees, quasi-trees, and Eulerian tours through the lens of sandpile theory.

Abstract

The sandpile group of a connected graph is a group whose cardinality is the number of spanning trees. The group is known to have a canonical simply transitive action on spanning trees if the graph is embedded into the plane. However, no canonical action on the spanning trees is known for the nonplanar case. We show that for any embedded Eulerian digraph, one can define a canonical simply transitive action of the sandpile group on compatible Eulerian tours (a set whose cardinality equals to the number of spanning arborescences). This enables us to give a new proof that the rotor-routing action of a ribbon graph is independent of the root if and only if the embedding is into the plane (originally proved by Chan, Church and Grochow). Recently, Merino, Moffatt and Noble defined a sandpile group variant (called Jacobian) for embedded graphs, whose cardinality is the number of quasi-trees. Baker, Ding and Kim showed that this group acts canonically on the quasitrees. We show that the Jacobian of an embedded graph is canonically isomorphic to the usual sandpile group of the medial digraph, and the action by Baker at al. agrees with the action of the sandpile group of the medial digraph on Eulerian tours (which fact is made possible by the existence of a canonical bijection between Eulerian tours of the medial digraph and quasi-trees due to Bouchet).

On canonical sandpile actions of embedded graphs

TL;DR

The paper extends canonical sandpile actions beyond planar graphs by introducing a tour-rotor action for embedded Eulerian digraphs, yielding a canonical, root-independent action on compatible Eulerian tours and linking it to the rotor-routing framework. It provides a streamlined proof that rotor-routing is root-independent precisely for planar embeddings and forges a tight connection between the Jacobian of embedded graphs and the sandpile group of the medial digraph. The authors further align this Tour-Rotor framework with the Bernardi action of Baker–Ding–Kim, via Bouchet’s correspondence between medial tours and quasi-trees, and establish a canonical isomorphism Jac with the two actions agreeing. Collectively, these results unify several canonical actions on combinatorial structures associated to embedded graphs, offering a cohesive picture that relates spanning trees, quasi-trees, and Eulerian tours through the lens of sandpile theory.

Abstract

The sandpile group of a connected graph is a group whose cardinality is the number of spanning trees. The group is known to have a canonical simply transitive action on spanning trees if the graph is embedded into the plane. However, no canonical action on the spanning trees is known for the nonplanar case. We show that for any embedded Eulerian digraph, one can define a canonical simply transitive action of the sandpile group on compatible Eulerian tours (a set whose cardinality equals to the number of spanning arborescences). This enables us to give a new proof that the rotor-routing action of a ribbon graph is independent of the root if and only if the embedding is into the plane (originally proved by Chan, Church and Grochow). Recently, Merino, Moffatt and Noble defined a sandpile group variant (called Jacobian) for embedded graphs, whose cardinality is the number of quasi-trees. Baker, Ding and Kim showed that this group acts canonically on the quasitrees. We show that the Jacobian of an embedded graph is canonically isomorphic to the usual sandpile group of the medial digraph, and the action by Baker at al. agrees with the action of the sandpile group of the medial digraph on Eulerian tours (which fact is made possible by the existence of a canonical bijection between Eulerian tours of the medial digraph and quasi-trees due to Bouchet).

Paper Structure

This paper contains 18 sections, 11 theorems, 26 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs, spanning trees and spanning arborescences
  4. Ribbon graphs and Eulerian tours
  5. Sandpile group
  6. Rotor-routing
  7. Eulerian digraphs
  8. Undirected graphs
  9. A canonical action of Eulerian ribbon digraphs on compatible Eulerian tours
  10. Special case: Rotor routing in undirected ribbon graphs
  11. Basepoint independence revisited
  12. An alternative formalism for the rotor routing action
  13. Special case: The sandpile group of the medial digraph acts canonically on quasi-trees
  14. Embedded graphs, medial digraphs, and the action on quasitrees
  15. The Jacobian of an embedded graph
  16. ...and 3 more sections

Key Result

Proposition 2.5

Baker-WangBBY Suppose that $F\subseteq E(G^{\leftrightarrows})$ is an edge set. We have $\sum_{e\in F} \chi_e\sim\mathbf{0}$ if and only if $F$ is a disjoint union of directed cycles and directed elementary cuts.

Figures (8)

  • Figure 1: An Eulerian tour ${\mathcal{E}}$ compatible with the planar embedding (left panel). It corresponds to the cyclic ordering $e_1, e_4, e_3, e_2, e_8, e_9, e_5, e_6, e_7$. The right panel shows the in-arborescence $A_{e_1}({\mathcal{E}})$ obtained from ${\mathcal{E}}$ by choosing $e_1$ as the initial edge of the tour.
  • Figure 2: An example for the tour-rotor action. See Example \ref{['ex:tour_rotor_action']} for the details. The ribbon structure is given by the plane embedding.
  • Figure 3: Left panel: A graph $G$ embedded into the torus (blue vertices, red edges), its dual $G^*$ (red vertices, blue edges), and the medial digraph $G^{\bowtie}$ (emerald nodes, black edges). Right panel: A quasitree of $G$ (thick red edges) and the corresponding Eulerian tour of $G^{\bowtie}$ (indicated by a grey curve).
  • Figure 4: The tour-rotor action of $(-1,0,1,0)\in S(G^{\bowtie})$ on the Eulerian tour of the left panel (and the corresponding quasi-tree) gives the Eulerian tour (and quasi-tree) on the right panel. If we choose the dashed medial edge as first edge of the tour, the thick edges are obtained as in-arborescence corresponding to the tour.
  • Figure 5: Left: An embedded graph and its dual. Right: A reference orientation and the induced orientation on the dual.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Example 2.1
  • Definition 2.2: bijection between compatible Eulerian tours and ${\rm Arb}(D,v)$, de1951circuits
  • Definition 2.3: Sandpile group
  • Proposition 2.5
  • proof
  • Definition 2.6: Rotor-router action, Holroyd08
  • Definition 2.7: linear equivalence of chip-and-rotor configurations, alg_rotor
  • proof
  • Proposition 2.9
  • Definition 2.10: Rotor-routing action for undirected graphs Holroyd08
  • ...and 28 more