On dihedral flows in embedded graphs

• Published in: arXiv.org
• Main Author: Litjens, Bart
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 02.12.2018
• Subjects: Graph theory; Graphs; Mathematics - Combinatorics; Obstructions
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1709.06469

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.