Cycle double covers and non-separating cycles

TL;DR

This paper investigates conditions under which 2-regular subgraphs of cubic graphs can be extended to cycle double covers, proving that certain decompositions guarantee the existence of such covers.

Contribution

It provides a new condition ensuring 2-regular subgraphs are part of a cycle double cover and proves existence results for specific decompositions.

Findings

Every 2-connected cubic graph with a spanning tree and a 2-regular subgraph of at most 3 circuits has a cycle double cover containing that subgraph.

A sufficient condition is identified for extending 2-regular subgraphs to cycle double covers.

The results contribute to understanding cycle double covers in cubic graphs.

Abstract

Which $2$-regular subgraph $R$ of a cubic graph $G$ can be extended to a cycle double cover of $G$? We provide a condition which ensures that every $R$ satisfying this condition is part of a cycle double cover of $G$. As one consequence, we prove that every $2$-connected cubic graph which has a decomposition into a spanning tree and a $2$-regular subgraph $C$ consisting of $k$ circuits with $k\leq 3$, has a cycle double cover containing $C$.