Towards the Erdős-Gallai cycle decomposition conjecture

Abstract

In the 1960's, Erdős 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(nlog⁡log⁡n) cycles and edges due to Conlon, Fox and Sudakov, by showing an n-vertex graph can always be decomposed into O(nlog⋆⁡n) cycles and edges, where log⋆⁡n is the iterated logarithm function.