Extensions of Flow Theorems

As an extension of Seymourʹs 6-flow theorem, we prove that every 2-edge-connected graph has a set of vertex-disjoint circuits and a 3-flowfsuch thatf(e)=0 only ifeis an edge in one of the circuits. An extension of Jaegerʹs 8-flow theorem, together with applications to the short cycle cover problem, is also presented. It is shown that the edges of a 2-edge-connected graphGcan be covered by cycles whose total length is at most|E(G)|+rr+1 (|V(G)|−1),whereris the minimum length of an even circuit (ofG) of length at least 10 (r=∞, if there is no such circuit).