On dihedral flows in embedded graphs

Abstract

Let $\Gamma$ be a multigraph with for each vertex a cyclic order of the edges incident with it. For $n \geq 3$, let $D_{2n}$ be the dihedral group of order $2n$. Define $\mathbb{D} := \{(\begin{smallmatrix} 1 & a \\ 0 & 1 \end{smallmatrix}) \mid a \in \mathbb{Z}\}$. In [5] it was asked whether $\Gamma$ admits a nowhere-identity $D_{2n}$-flow if and only if it admits a nowhere-identity $\mathbb{D}$-flow with $|a| < n$ (a `nowhere-identity $\mathbb{D}_n$-flow'). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of the existence of nowhere-identity $\mathbb{D}_2$-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.