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 fall 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 matchings as flows whose flow values represent points and outflow patterns represent lines of a configuration of ten points and 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.

Berge的一个推测表明，每个无桥三次方图的边缘最多可以覆盖五个完美匹配。因为只有当所讨论的图形是3边色时，三个完全匹配才足够，所以其余三次方图都可以分为两类：可以被四个完全匹配覆盖的图，以及需要至少五个完全匹配的图。立方图需要四个以上的完美匹配来覆盖其边缘，因此作为一些深奥且长期存在的猜想（包括著名的周期双覆盖猜想）的潜在反例，它特别有趣。但是，到目前为止，它们很难找到。

在本文中，我们建立了一种理论，该理论将具有四个完美匹配的覆盖物描述为流，其流量值表示点，流出模式表示10点配置的线，而3维投影空间的四个点所跨的六条线 ${\mathbb{P}}_3（{\mathbb{F}}_{\mathrm{2个}}）$一般情况。该理论为研究不允许这种覆盖的图形提供了强大的工具，并提供了多种构建方法。作为说明性示例，我们生成了一个丰富的snarks（无平凡的三次图形，没有3边着色），它们无法用四个完美匹配覆盖。该族包含具有此属性的所有先前已知的图。