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Annals of Combinatorics 12, 2 (2008) 139-153

A characterization of the Tutte polynomial via combinatorial embeddings

Olivier Bernardi 1

(2008)

We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to activities. Tutte's notion of activity requires to choose a \emph{linear order} on the edge set (though the generating function of the activities is, in fact, independent of this order). We define a new notion of activity, the embedding-activity, which requires to choose a \emph{combinatorial embedding} of the graph, that is, a cyclic order of the edges around each vertex. We prove that the Tutte polynomial equals the generating function of spanning trees counted according to embedding-activities (this generating function being, in fact, independent of the embedding).

1 :	Laboratoire Bordelais de Recherche en Informatique (LaBRI)
	CNRS : UMR5800 – Université Sciences et Technologies - Bordeaux I – Ecole Nationale Supérieure d'Electronique, Informatique et Radiocommunications de Bordeaux – Université Victor Segalen - Bordeaux II

Domaine

:

Mathématiques/Combinatoire

Mots Clés : Tutte polynomial – graph – embedding – internal activity – external activity

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Contributeur : Olivier Bernardi <>

Soumis le : Mercredi 2 Août 2006, 15:21:22

Dernière modification le : Jeudi 18 Juin 2009, 14:07:06