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, DiscreteMath. 244 (2002) 77–82] about edge-disjoint bipartizingmatchings of a cubic graphwith 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) and (D, f2) is an (h, k)-flow parity-pair-cover of G if the union of their supports covers the entire graph; f1 is an h-flow and f2 is a k-flow, and Ef1=odd = Ef2=odd. Then G admits a nowhere-zero 6-flow if and only if G admits a (4, 3)-flow parity-paircover; and G admits a nowhere-zero 5-flow if G admits a (3, 3)-flow parity-pair-cover. A pair of integer flows (D, f1) and (D, f2) is an (h, k)-flow even-disjoint-pair-cover of G if the union of their supports covers the entire graph, f1 is an h-flow and f2 is a k-flow, and Efi=even,fi 6=0 ⊆ Efj=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. © 2008 Elsevier B.V. All rights reserved.