Facial diagrams and cycle double cover

TL;DR

This paper investigates the cycle double cover conjecture by exploring circular 2-cell embeddings of cubic graphs on surfaces, analyzing edge twisting operations to understand face-graph interactions.

Contribution

It introduces a detailed study of edge twisting operations in embeddings, providing bounds on singular edges to advance understanding of the conjecture.

Findings

Bounds on the number of singular edges in embeddings

Analysis of edge twisting operations in 2-cell embeddings

Method to transform arbitrary embeddings into circular embeddings

Abstract

We approach the cycle double cover conjecture by looking for a circular 2-cell embedding of cubic graphs on an arbitrary surface. It is easy to see that if such an embedding exists, we can get to it from an arbitrary starting 2-cell embedding by repeating ``twists of an edge''. We study this twisting operation in detail and deduce bounds on the number of singular edges (edges where a face meets itself).