On dihedral flows in embedded graphs.

Bart Litjens
Published in: Journal of graph theory (2018)
Let Γ be a multigraph with for each vertex a cyclic order of the edges incident with it. For n ≥ 3 , let D 2 n be the dihedral group of order 2 n . Define D ≔ { ( ± 1 a 0 1 ) ∣ a ∈ Z } . Goodall et al in 2016 asked whether Γ admits a nowhere-identity D 2 n -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.