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"... s of every 2-edge-connected graph on n vertices can be covered by n 1 cycles. However, in some instances the eciency of a cover may be more accurately described by the total size of the cover, i.e. the sum of the number of edges of the subgraphs in the cover. Alon and Tarsi [AT] (also [BJJ,T]) prove ..."

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s of every 2-edge-connected graph on n vertices can be covered by n 1 cycles. However, in some instances the eciency of a cover may be more accurately described by the total size of the cover, i.e. the sum of the number of edges of the subgraphs in the cover. Alon and Tarsi [AT] (also [BJJ,T]) proved that every 2-edge-connected graph on e edges can be covered by cycles with total size at most 5 3 e, and they conjecture that this can be sharpened to 7 5 e (achieved by the Petersen graph). Jamshy and Tarsi [JT] showed that this conjecture implies one of the outstanding conjectures in graph theory, the cycle double cover conjecture: the edge-set of every 2-edge-connected graph can be covered by cycles, such that every edge is in exactly 2 cycles. A cut [S; S ] is the set of edges in E(G) between S<