# A note on counting flows in signed graphs

## TL;DR

This paper extends Tutte's polynomial approach to counting nowhere-zero flows from graphs to signed graphs, establishing a polynomial formula for the number of flows based on the group's 2-torsion subgroup size.

## Contribution

It proves the existence of a polynomial counting function for nowhere-zero flows in signed graphs, parameterized by the 2-torsion subgroup size of the abelian group.

## Key findings

- Polynomial counting functions for flows in signed graphs are established.
- The result generalizes previous work for groups of odd order.
- The polynomial depends on the subgroup structure of the group.

## 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).