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Let G be a cubic graph which has a decomposition into a spanning tree T and a 2-regular subgraph C, i.e. \(E(T) \cup E(C) = E(G)\) and \(E(T) \cap E(C) = \emptyset \). We provide an answer to the following question: which lengths can the cycles of C have if G is a snark? Note that T is a hist (i.e. a spanning tree without a vertex of degree two) and that every cubic graph with a hist has the above decomposition.

Snarks with Special Spanning Trees