Tutte polynomial, subgraphs, orientations and sandpile model: New connections via embeddings

Abstract

We define a bijection between spanning subgraphs and orientations of graphs and explore its enumerative consequences regarding the Tutte polynomial. We obtain unifying bijective proofs for all the evaluations TG(i; j); 0 ≤ i; j ≤ 2 of the Tutte polynomial in terms of subgraphs, orientations, outdegree sequences and sandpile configurations. For instance, for any graph G, we obtain a bijection between connected subgraphs (counted by TG(1; 2)) and root-connected orientations, a bijection between forests (counted by TG(2; 1)) and outdegree sequences and bijections between spanning trees (counted by TG(1; 1)), root-connected outdegree sequences and recurrent sandpile configurations. All our proofs are based on a single bijection Φ between the spanning subgraphs and the orientations that we specialize in various ways. The bijection Φ is closely related to a recent characterization of the Tutte polynomial relying on combinatorial embeddings of graphs, that is, on a choice of cyclic order of the edges around each vertex.