Whitney's Theorem for 2-Regular Planar Digraphs

TL;DR

This paper extends Whitney's theorem to 2-regular planar digraphs, establishing conditions for their embeddings and relating them to Tutte's theorem on peripheral cycles, thus advancing topological graph theory.

Contribution

It introduces a new analogue of Whitney's theorem for 2-regular digraphs and explores the related Tutte's theorem on peripheral cycles in this context.

Findings

Established a Whitney-type theorem for 2-regular planar digraphs.

Identified a natural analogue of Tutte's theorem on peripheral cycles.

Provided conditions for embeddings of 2-regular digraphs in surfaces.

Abstract

A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embedding of the underlying graph in a closed surface with the added property that for every vertex~$v$, the two edges directed away from $v$ are not consecutive in the local rotation around $v$. In other words, at each vertex the incident edges are oriented in-out-in-out. The goal of this article is to provide an analogue of Whitney's theorem on planar embeddings in the setting of 2-regular digraphs. In the course of doing so, we note that Tutte's Theorem on peripheral cycles also has a natural analogue in this setting.