Nowhere-zero 5-flows and (1, 2)-factors

A graph G = (V,E) has a nowhere-zero k-flow if there exists an orientation H = (V,A) of G and an integer flow φ : A → Z such that for all a ∈ A, 0 < |φ(a)| < k. A (1, 2)-factor of G is a set F ⊆ E such that the degree of any vertex v in the subgraph induced by F , denoted by dF (v), is 1 or 2. The main result of this work is the following. A bridgeless cubic graph G has a nowhere-zero 5-flow if and only if there is a (1, 2)-factor F such that, the cardinality of the set {uv ∈ E(C) : uv ∈ F or dF (u) = dF (v)} is even, for every cycle C in a basis of cycles associated to an spanning tree.