Towards the Erdős-Gallai Cycle Decomposition Conjecture

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(n loglogn) 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. Our arguments make use and further develop the theory of robust sublinear expander graphs.