A note on 5-cycle double covers

TL;DR

This paper explores the strong cycle double cover conjecture for bridgeless cubic graphs, proposing a conjecture for 5-cycle double covers containing a given circuit, and provides a necessary and sufficient condition for such covers.

Contribution

It introduces a new conjecture about 5-cycle double covers and characterizes when a 2-regular subgraph can be included in such covers.

Findings

Proposes a conjecture that every circuit is contained in a 5-cycle double cover.

Provides a necessary and sufficient condition for a 2-regular subgraph to be in a 5-cycle double cover.

Abstract

The strong cycle double cover conjecture states that for every circuit $C$ of a bridgeless cubic graph $G$, there is a cycle double cover of $G$ which contains $C$. We conjecture that there is even a 5-cycle double cover $S$ of $G$ which contains $C$, i.e. $C$ is a subgraph of one of the five 2-regular subgraphs of $S$. We prove a necessary and sufficient condition for a 2-regular subgraph to be contained in a 5-cycle double cover of $G$.