On dihedral flows in embedded graphs

Let $\Gamma$ be a multigraph with for each vertex a cyclic order of the edges incident with it. For $n\ge 3$ , let $D_{2n}$ be the dihedral group of order $2n$ . Define $D≔\left\{\left(\begin{array}{cc}\pm 1& a\\ 0& 1\end{array}\right)\mid a\in Z\right\}$ . Goodall et al in 2016 asked whether $\Gamma$ admits a nowhere-identity $D_{2n}$ -flow if and only if it admits a nowhere-identity $D$ -flow with $\mid a\mid <n$ (a "nowhere-identity dihedral $n$ -flow"). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere-identity $2$ -flows is discussed. Lastly, graphs in which the equivalence of the existence of flows as above is true are described. We focus particularly on cubic graphs.