Abstract
Tutte initiated the study of nowhere-zero flows and proved the following fundamental theorem: For every graph G there is a polynomial f so that for every abelian group Gamma of order n, the number of nowhere-zero Gamma-flows in G is f (n). For signed graphs (which have bidirected orientations), the situation is more subtle.
For a finite group Gamma, let epsilon(2)(Gamma) be the largest integer d so that Gamma has a subgroup isomorphic to Z(2)(d). We prove that for every signed graph G and d >= 0 there is a polynomial f(d) so that f(d) (n) is the number of nowhere-zero Gamma-flows in G for every abelian group Gamma with epsilon(2)(Gamma) = d and vertical bar Gamma vertical bar = 2(d)n.
Beck and Zaslaysky [JCTB 2006] had previously established the special case of this result when d = 0 (i.e., when Gamma has odd order).