Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings. | Tutte polynomial subgraphs orientations and sandpile model new connections via embeddings Olivier Bernardi CNRS Université Paris-Sud Bat 425 91405 Orsay Cedex France Submitted Jan 23 2007 Accepted Aug 13 2008 Published Aug 25 2008 Mathematics Subject Classification 05C20 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 í j 0 i j 2 of the Tutte polynomial in terms of subgraphs orientations outdegree sequences and sandpile conhgurations. 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 conhgurations. All our proofs are based on a single bijection between the spanning subgraphs and the orientations that we specialize in various ways. The bijection 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. 1 Introduction In 1947 Tutte defined a graph invariant that he named the dichromate because he thought of it as bivariate generalization of the chromatic polynomial 42 . Since then the dichromate now known as the Tutte polynomial has been widely studied see 5 7 . This work was partially supported by the Centre de Recerca Matematica of Barcelona and by the French Agence Nationale de la Recherche project SADA ANR-05-BLAN-0372. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R109 1 There are several alternative definitions of the Tutte polynomial 3 23 32 43 . The most straightforward definition for a connected graph G V E is TG x y X x - 1 c S _1 y - 1 c S S _ y 1 1 S spanning subgraph
