Approximate cycle double cover

TL;DR

This paper explores the cycle double cover conjecture by analyzing embeddings of cubic graphs, providing bounds on singular edges and algorithms to find embeddings with fewer singular edges.

Contribution

It introduces new upper bounds on singular edges in cubic graph embeddings and offers efficient algorithms to achieve these bounds.

Findings

Established nontrivial upper bounds on singular edges in cubic graph embeddings.

Developed efficient algorithms to find embeddings with minimized singular edges.

Enhanced understanding of the relationship between graph embeddings and the CDC conjecture.

Abstract

The Cycle double cover (CDC) conjecture states that for every bridgeless graph $G$, there exists a family $\mathcal{F}$ of cycles such that each edge of the graph is contained in exactly two members of $\mathcal{F}$. Given an embedding of a graph~$G$, an edge $e$ is called a \emph{singular edge} if it is visited twice by the boundary of one face. The CDC conjecture is equivalent to bridgeless cubic graphs having an embedding with no singular edge. In this work, we introduce nontrivial upper bounds on the minimum number of singular edges in an embedding of a cubic graph. Moreover, we present efficient algorithms to find embeddings satisfying these bounds.