On dihedral flows in embedded graphs

Abstract

Let Γ be a multigraph with for each vertex a cyclic order of the edges incident with it. For n ≥ 3, let D2n be the dihedral group of order 2n. Define (Formula presented.). Goodall et al in 2016 asked whether Γ admits a nowhere-identity D2n -flow if and only if it admits a nowhere-identity D -flow with (Formula presented.) (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.