Approximate cycle double cover
Babak Ghanbari, Robert Šámal
TL;DR
This work addresses the Cycle double cover conjecture for bridgeless graphs by reframing the problem as the quest for embeddings with no singular face-edges. It introduces the partial CDC concept and develops an embedding-extension framework that leverages local rotations and the perfect matching polytope, enabling polynomial-time constructions that bound the number of singular edges. The main results give nontrivial upper bounds: an embedding with at most $n/2$ singular edges in general, improved to at most $|M_1 \cap M_2| \le \frac{n}{10}$ (hence at most $\frac{n}{10}$ singular edges) using two perfect matchings, and further tightened to at most $\frac{n}{2k}$ for cyclically $k$-edge-connected cubic graphs, with $O(n^{3/2}\log n)$ time algorithms; a faster variant uses Edmonds’ theorem and a single perfect matching. Overall, the paper advances CDC research by providing a concrete, algorithmic route to construct embeddings with bounded singular edges and identifying structural limitations that inform future improvements toward resolving CDC.
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.