Many flows in the group connectivity setting

Two well-known results in the world of nowhere-zero flows are Jaeger's 4-flow theorem asserting that every 4-edge-connected graph has a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_2$-flow and Seymour's 6-flow theorem asserting that every 2-edge-connected graph has a nowhere-zero $\mathbb{Z}_6$-flow. Dvo\v{r}\'ak and the last two authors of this paper extended these results by proving the existence of exponentially many nowhere-zero flows under the same assumptions. We revisit this setting and provide extensions and simpler proofs of these results. The concept of a nowhere-zero flow was extended in a significant paper of Jaeger, Linial, Payan, and Tarsi to a choosability-type setting. For a fixed abelian group $\Gamma$, an oriented graph $G = (V,E)$ is called $\Gamma$-connected if for every function $f : E \rightarrow \Gamma$ there is a flow $\phi : E \rightarrow \Gamma$ with $\phi(e) \neq f(e)$ for every $e \in E$ (note that taking $f = 0$ forces $\phi$ to be nowhere-zero). Jaeger et al. proved that every oriented 3-edge-connected graph is $\Gamma$-connected whenever $|\Gamma| \ge 6$. We prove that there are exponentially many solutions whenever $|\Gamma| \ge 8$. For the group $\mathbb{Z}_6$ we prove that for every oriented 3-edge-connected $G = (V,E)$ with $\ell = |E| - |V| \ge 11$ and every $f: E \rightarrow \mathbb{Z}_6$, there are at least $2^{ \sqrt{\ell} / \log \ell}$ flows $\phi$ with $\phi(e) \neq f(e)$ for every $e \in E$.