A proof of the Erdős--Gallai cycle decomposition conjecture

In the 1960s, Erdős and Gallai conjectured that the edges of every $n$-vertex graph can be decomposed into $O(n)$ cycles and edges. We prove this conjecture. Equivalently, every Eulerian graph on $n$ vertices can be decomposed into $O(n)$ cycles, which confirms Hajós’ conjecture up to a constant factor.