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

Olivier Bernardi (1)

1 CRM - Centre de Recerca Matematica

For any graph G with n edges, the spanning subgraphs and the orientations of G are both counted by the evaluation T_G(2,2)=2^n of its Tutte polynomial. We define a bijection $\Phi$ between spanning subgraphs and orientations and explore its enumerative consequences regarding the Tutte polynomial. The bijection $\Phi$ 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 T_G(i,j), 0\leq i,j \leq 2 of the Tutte polynomial in terms of orientations. The strength of our approach is to derive all these interpretations by specializing the bijection $\Phi$ in various ways. For instance, we obtain a bijection between the connected subgraphs of G (counted by T_G(1,2)) and the root-connected orientations. We also obtain a bijection between the forests (counted by T_G(2,1)) and outdegree sequences which specializes into a bijection between spanning trees (counted by T_G(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.

Mots clés: Tutte polynomial; embedding; activities; bijection; subgraphs; orientations; sandpile model

Domaines: Combinatoire [math.CO]

Olivier Bernardi. Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings. The Electronic Journal of Combinatorics, 2008, 15 (1), pp.R109. ⟨hal-00117268⟩