ar X iv : 0 70 7 . 23 06 v 1 [ m at h . C O ] 1 6 Ju l 2 00 7 Parity , eulerian subgraphs and the Tutte polynomial
نویسنده
چکیده
Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G on the hyperbolae H 2 and H 4. These evaluations are expressed in terms of eulerian subgraphs of G and the size of subgraphs modulo 2, 3, 4 or 6. In particular, a graph is found to have a nowhere-zero 4-flow if and only if there is a correlation between the event that three subgraphs A, B, C chosen uniformly at random have pairwise eulerian symmetric differences and the event that ⌊ |A|+|B|+|C| 3 ⌋ is even. Some further evaluations of the Tutte polynomial on H 3 are also given to illustrate the unifying power of the methods used. The connection between results of Matiyasevich [10], Alon and Tarsi [2] and Onn [12] is highlighted by indicating how they may all be derived by the techniques adopted in this paper.
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