Graph Puzzles I.1: Oriented Berge-Fulkerson Conjecture

TL;DR

This paper explores an oriented version of the Berge-Fulkerson conjecture, proves it for Isaacs flower snarks, and discusses topological and combinatorial constructions related to cycle covers and ribbon graphs.

Contribution

It introduces the o6c4c conjecture, proves it for a specific graph family, and connects cycle covers with topological surface constructions and ribbon graphs.

Findings

Proved the o6c4c conjecture for Isaacs flower snarks.

Constructed orientable surfaces from cycle covers.

Linked special cases to oriented ribbon graphs.

Abstract

The Berge-Fulkerson conjecture states that every bridgeless cubic graph can be covered with six perfect matchings such that each edge is covered exactly twice. An equivalent reformulation is that it's possible to find a 6-cycle 4-cover. In this paper we discuss the oriented version (o6c4c) of the latter statement, pose it as a conjecture and prove it for the family of Isaacs flower snarks. Similarly to the case of oriented cycle double cover, we can always construct an orientable surface (possibly with boundary) from an o6c4c solution. If the o6c4c solution itself splits into two (not necessarily oriented) cycle double covers, then it's also possible to build another pair of orientable surfaces (also possibly with boundaries). Finally we show how to build a ribbon graph, and for some special o6c4c cases we show that this ribbon graph corresponds to an oriented 6-cycle double cover. Github: https://github.com/gexahedron/cycle-double-covers