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 $n\ge 3$, let $D_{2n}$ be the dihedral group of order $2n$. Define $\mathbb{D}:=\{(\begin{array}{cc}1& a\\ 0& 1\end{array})\mid a\in \mathbb{Z}\}$. 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.

16 pages. Some changes have been made, based on comments of the referees. Accepted for publication in Journal of Graph Theory