A note on counting flows in signed graphs

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 ${\mathbb{Z}}_2^d$. 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 $|\Gamma |=2^{d}n$. Beck and Zaslavsky had previously established the special case of this result when $d=0$ (i.e., when $\Gamma$ has odd order).

7 pages