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Computational Graph Decompositions I: Oriented Berge-Fulkerson Conjecture

Nikolay Ulyanov

TL;DR

The paper investigates the oriented Berge–Fulkerson conjecture $o6c4c$, proposing that every bridgeless graph admits a collection of 6 cycles orientable so that each edge is covered twice in each direction. It develops a framework to study $o6c4c$ across graph families, proves the conjecture for Isaacs flower snarks, and analyzes how $o6c4c$ solutions can be glued into orientable surfaces or split into multiple cycle double covers. A key contribution is the ribbon-graph interpretation, enabling the construction of $o6cdc$ from $o6c4c$ and yielding explicit realizations for Petersen and related snarks. The work also includes extensive computational verification up to 30-vertex snarks, a detailed study of boundary structures and ordered/disordered vertices, and open-source code to reproduce and extend the results. The results provide new topological and combinatorial perspectives on cycle covers and pave the way for future exploration of unit-vector flows and other graph-decomposition conjectures in the series CGD.

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

Computational Graph Decompositions I: Oriented Berge-Fulkerson Conjecture

TL;DR

The paper investigates the oriented Berge–Fulkerson conjecture , proposing that every bridgeless graph admits a collection of 6 cycles orientable so that each edge is covered twice in each direction. It develops a framework to study across graph families, proves the conjecture for Isaacs flower snarks, and analyzes how solutions can be glued into orientable surfaces or split into multiple cycle double covers. A key contribution is the ribbon-graph interpretation, enabling the construction of from and yielding explicit realizations for Petersen and related snarks. The work also includes extensive computational verification up to 30-vertex snarks, a detailed study of boundary structures and ordered/disordered vertices, and open-source code to reproduce and extend the results. The results provide new topological and combinatorial perspectives on cycle covers and pave the way for future exploration of unit-vector flows and other graph-decomposition conjectures in the series CGD.

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
Paper Structure (21 sections, 5 equations, 14 figures, 1 table)

This paper contains 21 sections, 5 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. tl;dr
  3. Motivation and related conjectures
  4. Main contribution
  5. Note about the paper series title
  6. o6c4c panorama
  7. o6c4c for 3-edge-colourable cubic graphs
  8. o6c4c for Petersen graph
  9. o6c4c for small snarks
  10. o6c4c for Isaacs flower snarks, and almost strong Petersen colouring
  11. Gluing an orientable surface from o6c4c
  12. A split into 2$\times$6cdcs, and another pair of surfaces
  13. o6cdc from o6c4c
  14. Ribbon graphs
  15. Petersen graph, and o5cdc
  16. ...and 6 more sections

Figures (14)

  • Figure 1: Solution for Petersen graph
  • Figure 3: First set of 3 cycles of $J_3$ vs $J_5$
  • Figure 4: Second set of 3 cycles of $J_3$ vs $J_5$
  • Figure 5: Solution for second Blanuša snark
  • Figure 6: 2 types of edges in 6c4c
  • ...and 9 more figures

Theorems & Definitions (8)

  • Conjecture 1.1: o6c4c, oriented 6-cycle 4-cover
  • Conjecture 1.1: o6c4c, oriented 6-cycle 4-cover
  • proof
  • Definition 5.1: ribbon graph
  • proof
  • proof
  • proof
  • proof