Signed Cycle Double Covers

Lingsheng Shi,Zhang Zhang

doi:10.37236/6760

Abstract

The cycle double cover conjecture states that every bridgeless graph has a collection of cycles which together cover every edge of the graph exactly twice. A signed graph is a graph with each edge assigned by a positive or a negative sign. In this article, we prove a weak version of this conjecture that is the existence of a signed cycle double cover for all bridgeless graphs. We also show the relationships of the signed cycle double cover and other famous conjectures such as the Tutte flow conjectures and the shortest cycle cover conjecture etc.

Highlights

The Cycle Double Cover Conjecture is a famous conjecture in graph theory
It states that every bridgeless graph has a collection of cycles which together cover every edge of the graph exactly twice
A k-cycle double cover of G consists of k even subgraphs of G. This conjecture is a folklore and it was independently formulated by Szekeres [23], Itai and Rodeh [9], and Seymour [22]. Szekeres raised this conjecture just for cubic graphs, while Seymour put it in a general case

Summary

Introduction

The Cycle Double Cover Conjecture is a famous conjecture in graph theory. It states that every bridgeless graph has a collection of cycles which together cover every edge of the graph exactly twice. Though Conjectures 1 and 2 are widely open and it is well known that the Petersen graph has no 4-cycle double cover, we establish the following weak version by replacing cycles with signed cycles This is analogous in some sense to a result of Seymour [21] who proved the weak signed versions of the Berge-Fulkerson Conjecture [8, 21] about double covers by perfect matchings and the Tutte Four-Flow Conjecture in its dual coloring form for cubic graphs. There exist infinitely many graphs of size m such that all 4-signed cycle double covers have at least m/15 negative edges. Every bridgeless graph of size m has a 4-signed cycle double cover with at most m/15 negative edges. We prove Theorems 10, 11, and 12 in Section 2 and finish with a discussion in Section 3 on the relationship of the signed cycle double cover and other conjectures, such as the Shortest Cycle Cover Conjecture and the Tarsi Conjecture on the shortest 3-cycle covers etc

Proofs of main results

Related conjectures