On dihedral flows in embedded graphs

摘要

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 D≔{(±1a01)∣a∈Z}. 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 ∣a∣<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.