Short cycle covers and the cycle double cover conjecture

AbstractThe assertion of the long standing cycle double cover conjecture is shown to be true if either one of the following statements holds: 1.(1) Every graph G=(V,E) with no isthmus has a cycle cover, whose length does not exceed 75|E|.2.(2) There exists an integer k≥9 such that every regular matroid M, which admits a k-nowherer zero flow, has a cycle cover of length at most 2(1−1k)|M|.The values in (1) and (2) have been mentioned in several previous articles as the largest known lengths of shortest cycle covers for the corresponding families of matroids