Abstract: | Let O(G) denote the set of odd-degree vertices of a graph G. Let t ? N and let ??t denote the family of graphs G whose edge set has a partition. E(g) = E1 U E2 U … U Etsuch that O(G) = O(GEi]) (1 ? i ? t). This partition is associated with a double cycle cover of G. We show that if a graph G is at most 5 edges short of being 4-edge-connected, then exactly one of these holds: G ? ??3, G has at least one cut-edge, or G is contractible to the Petersen graph. We also improve a sufficient condition of Jaeger for G ? ??2p+1(p ? N). |