On dihedral flows in embedded graphs

Let [Formula: see text] be a multigraph with for each vertex a cyclic order of the edges incident with it. For [Formula: see text] , let [Formula: see text] be the dihedral group of order [Formula: see text]. Define [Formula: see text]. Goodall et al in 2016 asked whether [Formula: see text] admits a nowhere‐identity [Formula: see text] ‐flow if and only if it admits a nowhere‐identity [Formula: see text] ‐flow with [Formula: see text] (a “nowhere‐identity dihedral [Formula: see text] ‐flow”). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere‐identity [Formula: see text] ‐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.