A proof of the Erd\H{o}s-Gallai cycle decomposition conjecture

Abstract

In the 1960s, Erd\H{o}s 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 Haj\'os'conjecture up to a constant factor.