Directed cycle double covers and cut-obstacles

TL;DR

This paper explores the problem of directed cycle double covers in graphs, proposing a new conjecture on avoiding cut-obstacles that, if true, would prove Jaeger's conjecture that bridges are the only obstacles.

Contribution

It introduces a novel conjecture on graph connections and demonstrates that avoiding cut-obstacles would imply Jaeger's directed cycle double cover conjecture.

Findings

Formulation of a new conjecture on graph connections.

Establishment that avoiding cut-obstacles implies Jaeger's conjecture.

Provides a new approach to proving directed cycle double covers.

Abstract

A directed cycle double cover of a graph G is a family of cycles of G, each provided with an orientation, such that every edge of G is covered by exactly two oppositely directed cycles. Explicit obstacles to the existence of a directed cycle double cover in a graph are bridges. Jaeger conjectured that bridges are actually the only obstacles. One of the difficulties in proving the Jaeger's conjecture lies in discovering and avoiding obstructions to partial strategies that, if successful, create directed cycle double covers. In this work, we suggest a way to circumvent this difficulty. We formulate a conjecture on graph connections, whose validity follows by the successful avoidance of one cut-type obstruction that we call cut-obstacles. The main result of this work claims that our 'cut-obstacles avoidance conjecture' already implies Jaeger's directed cycle double cover conjecture.