A proof of the Cycle Double Cover Conjecture

TL;DR

This paper proves the Cycle Double Cover Conjecture for all bridgeless cubic graphs by analyzing cycles in their line graphs, and extends the result to include any given cycle in such graphs.

Contribution

It provides a complete proof of the conjecture for cubic graphs and introduces a stronger version involving specific cycles.

Findings

Every bridgeless cubic graph has a cycle double cover.

A cycle double cover can be constructed to include any given cycle in a cubic graph.

The proof uses analysis of cycles in the line graph of the original graph.

Abstract

Given a bridgeless graph $G$, the Cycle Double Cover Conjecture posits that there is a list of cycles of $G$, such that every edge appears in exactly two cycles on the list. This conjecture was originally posed independently in 1973 by Szekeres and 1979 by Seymour. In 1985, Jaeger demonstrated that it is sufficient to prove in the case that $G$ is a cubic graph. We here present a proof that every bridgeless cubic graph has a cycle double cover by analyzing certain kinds of cycles in the line graph of $G$. Further, in the case that $G$ is cubic, we prove the stronger conjecture that given a bridgeless graph $G$ and a cycle $C$ in $G$, then there exists a cycle double cover of $G$ containing $C$.