Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can be decomposed into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(n log log n) cycles and edges suffice. We also prove the Erdős–Gallai conjecture for random graphs showing that whp G(n, p) (for most values of p) can be decomposed into a union of n/4 + np/2 + o(n) cycles and edges, which is asymptotically tight.
This is joint work with Conlon and Fox, and also Korandi and Krivelevich.