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A Characterization of the Tutte Polynomial via Combinatorial Embeddings
Olivier Bernardi
LaBRI, Universit\'{e} Bordeaux 1, 351 cours de la Lib\'{e} ration, 33405 Talence cedex, France
olivier.bernardi@gmail.com
Annals of Combinatorics 12 (2) p.133-147 March, 2008
AMS Subject Classification: 05C99
Abstract:
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 a choice of a 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 a choice of a 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 is, in fact,
independent of the embedding.
Keywords: Tutte polynomial, spanning trees, activities, embedded graphs

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