A note on counting flows in signed graphs

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 $\mathrm{\Gamma }$ of order $n$, the number of nowhere-zero $\mathrm{\Gamma }$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a finite group $\mathrm{\Gamma }$, let ${ϵ}_2(\mathrm{\Gamma })$ be the largest integer $d$ so that $\mathrm{\Gamma }$ has a subgroup isomorphic to ${\mathbb{Z}}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$thereisapolynomial$f_d$sothat$f_d(n)$ is the number of nowhere-zero $\mathrm{\Gamma }$-flows in $G$ for every abelian group $\mathrm{\Gamma }$ with ${ϵ}_2(\mathrm{\Gamma })=d$ and $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }=2^{d}n$. Beck and Zaslavsky had previously established the special case of this result when $d=0$ (i.e., when $\mathrm{\Gamma }$ has odd order).