  In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any
$n$-vertex graph can be decomposed into $O(n)$ cycles and edges. We improve
upon the previous best bound of $O(n\log\log n)$ cycles and edges due to
Conlon, Fox and Sudakov, by showing an $n$-vertex graph can always be
decomposed into $O(n\log^{*}n)$ cycles and edges, where $\log^{*}n$ is the
iterated logarithm function.
