A permutation snark is a snark which has a 2-factor F 2 consisting of two chordless circuits; F 2 is called the permutation 2-factor of G. We construct an infinite family H of cyclically 5-edge connected permutation snarks. Moreover, we prove for every member G ∈ H that the permutation 2-factor given by the construction of G is not contained in any circuit double cover of G.

The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.

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The strong cycle double cover conjecture states that for every circuit C of a bridgeless cubic graph G, there is a cycle double cover of G which contains C. We conjecture that there is even a 5-cycle double cover S of G which contains C, i.e. C is a subgraph of one of the five 2-regular subgraphs of S. We prove a necessary and sufficient condition for a 2-regular subgraph to be contained in a 5-cycle double cover of G.

Which 2-regular subgraph R of a cubic graph G can be extended to a cycle double cover of G? We provide a condition which ensures that every R satisfying this condition is part of a cycle double cover of G. As one consequence, we prove that every 2-connected cubic graph which has a decomposition into a spanning tree and a 2-regular subgraph C consisting of k circuits with k ≤ 3, has a cycle double cover containing C.