Parity, eulerian subgraphs and the Tutte polynomial

Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G at certain points (a,b) where (a−1)(b−1)∈{2,4}. 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 is even. Some further evaluations of the Tutte polynomial at points (a,b) where (a−1)(b−1)=3 are also given that illustrate the unifying power of the methods used. The connection between results of Matiyasevich, Alon and Tarsi and Onn is highlighted by indicating how they may all be derived by the techniques adopted in this paper.