Flows, flow-pair covers and cycle double covers

In this paper, some earlier results by Fleischner [H. Fleischner, Bipartizing matchings and Sabidussi’s compatibility conjecture, Discrete Math. 244 (2002) 77–82] about edge-disjoint bipartizing matchings of a cubic graph with a dominating circuit are generalized for graphs without the assumption of the existence of a dominating circuit and 3-regularity. A pair of integer flows (D,f1)(D,f1) and (D,f2)(D,f2) is an (h,k)(h,k)-flow parity-pair-cover of GG if the union of their supports covers the entire graph; f1f1 is an hh-flow and f2f2 is a kk-flow, and Ef1=odd=Ef2=odd. Then GG admits a nowhere-zero 6-flow if and only if GG admits a (4,3)(4,3)-flow parity-pair-cover; and GG admits a nowhere-zero 5-flow if GG admits a (3,3)(3,3)-flow parity-pair-cover. A pair of integer flows (D,f1)(D,f1) and (D,f2)(D,f2) is an (h,k)(h,k)-flow even-disjoint-pair-cover of GG if the union of their supports covers the entire graph, f1f1 is an hh-flow and f2f2 is a kk-flow, and Efi=even,fi≠0⊆Efj=0 for each {i,j}={1,2}{i,j}={1,2}. Then GG has a 5-cycle double cover if GG admits a (4,4)(4,4)-flow even-disjoint-pair-cover; and GG admits a (3,3)(3,3)-flow parity-pair-cover if GG has an orientable 5-cycle double cover.