On dihedral flows in embedded graphs

TL;DR

This paper investigates dihedral flows in embedded graphs, providing counterexamples to previous conjectures, analyzing obstructions, and exploring the complexity and conditions for the existence of such flows, especially in cubic graphs.

Contribution

It offers counterexamples to a conjecture about dihedral and affine flows, introduces obstructions, and characterizes graphs where flow equivalences hold, with a focus on cubic graphs.

Findings

Counterexamples to the dihedral flow conjecture

Obstructions to the existence of certain flows

Complexity analysis of $ ext{D}_2$-flows in graphs

Abstract

Let $\Gamma$ be a multigraph with for each vertex a cyclic order of the edges incident with it. For $n \geq 3$, let $D_{2n}$ be the dihedral group of order $2n$. Define $\mathbb{D} := \{(\begin{smallmatrix} 1 & a \\ 0 & 1 \end{smallmatrix}) \mid a \in \mathbb{Z}\}$. 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.