A note on counting flows in signed graphs (Q2420566)

Summary: 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 \(\mathbb{Z}_2^d\). We prove that for every signed graph \(G\) and \(d \ge 0\) 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 \(|\Gamma| = 2^d n\). \textit{M. Beck} and \textit{T. Zaslavsky} [J. Comb. Theory, Ser. B 96, No. 6, 901--918 (2006; Zbl 1119.05105)] had previously established the special case of this result when \(d=0\) (i.e., when \(\Gamma\) has odd order).