On The Cycle Double Cover Problem

TL;DR

This paper introduces a new condition based on Kuratowski minors that guarantees the existence of a cycle double cover in graphs, ultimately proving the cycle double cover conjecture and Goddyn's conjecture.

Contribution

It presents a novel condition for cycle double cover existence and proves the long-standing cycle double cover conjecture and Goddyn's conjecture.

Findings

Established a condition guaranteeing naive cycle double cover existence

Proved the cycle double cover conjecture

Confirmed Goddyn's conjecture for bridgeless graphs

Abstract

In this paper, for each graph G, a free edge set F is defined. To study the existence of cycle double cover, the naive cycle double cover of G and F have been defined and studied. In the main theorem, the paper, based on the Kuratowski minor properties, presents a condition to guarantee the existence of a naive cycle double cover for couple (G,F). As a result, the cycle double cover conjecture has been concluded. Moreover, Goddyn's conjecture - asserting if C is a cycle in bridgeless graph G, there is a cycle double cover of G containing C - will have been proved.