The well-known 5-flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5-flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we summarise some of these methods and we propose new ones based on variations of the known constructions. Afterwards, we prove that all such methods are nothing but particular instances of a more general construction that we introduce into detail. In the second part, we consider many instances of this general method and we determine when our method permits to obtain a snark with circular flow number 5. Finally, by a computer search, we determine all snarks having circular flow number 5 up to 36 vertices. It turns out that all such snarks of order at most 34 can be obtained by using our method, and that the same holds for 96 of the 98 snarks of order 36 with circular flow number 5.

中文翻译：

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用循环流编号 5 构建 snark 的统一方法

著名的 Tutte 5-flow Conjecture 最初是针对整数流提出的，它声称每个无桥图的循环流数最多为 5。 5-flow 猜想的研究可以简化为三次图是一个经典的结果, 尤其是蛇。然而，很少有已知的程序来构建具有循环流编号 5 的 snark。在本文的第一部分，我们总结了其中一些方法，并根据已知结构的变化提出了新的方法。之后，我们证明所有这些方法只不过是我们详细介绍的更一般构造的特定实例。在第二部分中，我们考虑了这种通用方法的许多实例，并确定何时我们的方法允许获得循环流编号为 5 的 snark。最后，通过计算机搜索，我们确定所有具有 5 到 36 个顶点的循环流的 snark。事实证明，使用我们的方法可以获得最多 34 阶的所有此类 snark，并且对于循环流数为 5 的 98 个 36 阶 snark 中的 96 个也是如此。