Tutte Polynomial, Subgraphs, Orientations and Sandpile Model: New Connections via Embeddings

For any graph G with n edges, the spanning subgraphs and the orientations of G are both counted by the evaluation TG(2, 2) = 2 n of its Tutte polynomial. We define a bijection Φ between spanning subgraphs and orientations and explore its enumerative consequences regarding the Tutte polynomial. The bijection Φ is closely related to a recent characterization of the Tutte polynomial relying on a combinatorial embedding of the graph G, that is, on a choice of cyclic order of the edges around each vertex. Among other results, we obtain a combinatorial interpretation for each of the evaluations TG(i, j), 0 ≤ i, j ≤ 2 of the Tutte polynomial in terms of orientations. The strength of our approach is to derive all these interpretations by specializing the bijection Φ in various ways. For instance, we obtain a bijection between the connected subgraphs of G (counted by TG(1, 2)) and the root-connected orientations. We also obtain a bijection between the forests (counted by TG(2, 1)) and outdegree sequences which specializes into a bijection between spanning trees (counted by TG(1, 1)) and root-connected outdegree sequences. We also define a bijection between spanning trees and recurrent configurations of the sandpile model. Combining our results we obtain a bijection between recurrent configurations and root-connected outdegree sequences which leaves the configurations at level 0 unchanged.