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 , 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 f 1 = odd = E f 2 = odd . Then G admits a nowhere-zero 6-flow if and only if G admits a ( 4 , 3 ) -flow parity-pair-cover; 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 f i = even , f i ≠ 0 ⊆ E f j = 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.