21787 articles – 15599 Notices  [english version]
 HAL : hal-00117268, version 1
 arXiv : math.CO/0612003
 The Electronic Journal of Combinatorics 15, 1 (2008) R109
 Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings
 (2008)
 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.
 1 : Centre de Recerca Matematica (CRM) Institut d'Estudis Catalans
 Domaine : Mathématiques/Combinatoire
 Mots Clés : Tutte polynomial – embedding – activities – bijection – subgraphs – orientations – sandpile model
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 hal-00117268, version 1 http://hal.archives-ouvertes.fr/hal-00117268 oai:hal.archives-ouvertes.fr:hal-00117268 Contributeur : Olivier Bernardi <> Soumis le : Jeudi 30 Novembre 2006, 18:29:07 Dernière modification le : Jeudi 18 Juin 2009, 14:00:10