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, f(1)) and (D, f(2)) is an (h, k)-flow parity-pair-cover of G if the union of their supports covers the entire graph; f(1), is an h-flow and f(2) is a k-flow, and E-f1=odd = E-f2=odd. Then G admits a nowhere-zero 6-flow if and only if G admits a (4, 3)-flow parity-pair-over; and G admits a nowhere-zero 5-flow if G admits a (3, 3)-flow parity-pair-cover. A pair of integer flows (D,f(1)) and (D.f(2)) is an (h, k)-flow even-disjoint-pair-cover of G if the union of their supports covers the entire graph, f(1) is an h-flow and f(2) is a k-flow, and E-fi=even,E-f1 not equal 0 subset of E-fj=0 for each {i, j} = {1, 2}. Then G has a 5-cycle double cover if G admits a (4, 4)-flow even-disjoint-pair-cover; and G admits a (3, 3)-flow parity-pair-cover if G has an orientable 5-cycle double cover.
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