Cubic graphs that cannot be covered with four perfect matchings

TL;DR

This paper develops a theoretical framework to identify and construct cubic graphs that cannot be covered with four perfect matchings, including a new family of such graphs called snarks, which are relevant to longstanding conjectures.

Contribution

It introduces a flow-based theory linking four-perfect-matching covers to configurations in projective space, enabling the construction of new non-coverable cubic graphs.

Findings

Established a flow configuration model in projective space for four-perfect-matching covers.

Constructed a new family of snarks that cannot be covered with four perfect matchings.

Unified previously known examples within this new theoretical framework.

Abstract

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic graphs falls into two classes: those that can be covered with four perfect matchings, and those that need at least five. Cubic graphs that require more than four perfect matchings to cover their edges are particularly interesting as potential counterexamples to several profound and long-standing conjectures including the celebrated cycle double cover conjecture. However, so far they have been extremely difficult to find. In this paper we build a theory that describes coverings with four perfect match\-ings as flows whose flow values and outflow patterns form a configuration of six lines spanned by four points of the 3-dimensional projective space $\mathbb{P}_3(\mathbb{F}_2)$ in general position. This theory provides powerful tools for investigation of graphs that do not admit such a cover and offers a great variety of methods for their construction. As an illustrative example we produce a rich family of snarks (nontrivial cubic graphs with no $3$-edge-colouring) that cannot be covered with four perfect matchings. The family contains all previously known graphs with this property.