For some time the Petersen graph has been the only known Snark with circular flow number 5 (or more, as long as the assertion of Tutte's 5-flow Conjecture is in doubt). Although infinitely many such snarks were presented eight years ago in [9], the variety of known methods to construct them and the structure of the obtained graphs were still rather limited. We start this article with an analysis of sets of flow values, which can be transferred through flow networks with the flow on each edge restricted to the open interval (1, 4) modulo 5. All these sets are symmetric unions of open integer intervals in the ring R/5Z. We use the results to design an arsenal of methods for constructing snarks S with circular flow number φ c (S) ≥ 5. As one indication to the diversity and density of the obtained family of graphs, we show that it is sufficiently rich so that the corresponding recognition problem is NP-complete.

A k-weak bisection of a cubic graph G is a partition of the vertex-set of G into two parts V 1 and V 2 of equal size, such that each connected component of the subgraph of G induced by V i (i = 1, 2) is a tree of at most k − 2 vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere-zero r-flow has a r -weak bisection. In this paper we study problems related to the existence of k-weak bisections. We believe that every cubic graph which has a perfect matching, other than the Petersen graph, admits a 4-weak bisection and we present a family of cubic graphs with no perfect matching which do not admit such a bisection. The main result of this article is that every cubic graph admits a 5-weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5-flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs which do contain bridges.

Let G be a bridgeless cubic graph. Fulkerson conjectured that there exist six 1-factors of G such that each edge of G is contained in exactly two of them. Berge conjectured that the edge-set of G can be covered with at most five 1-factors. We prove that the two conjectures are equivalent. ᭧

We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hägglund. We construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is computer-assisted. In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number φ C (G) ≥ 5 and admit a 5-cycle double cover.

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