Fulkerson′s Conjecture and Circuit Covers

It was conjectured by Fulkerson that the edge-set of any bridgeless graph can be covered by six cycles (union of circuits) such that each edge is in exactly four cycles. We prove that if Fulkerson′s conjecture is true, then the edge-set of every bridgeless graph G can be covered by three cycles whose total length is at most 2215|E(G)|. We also prove that there are infinitely many bridgeless graphs G whose edge-set cannot be covered by three cycles of total length less than 2215|E(G)|.