Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture

TL;DR

This paper advances the understanding of graph cycle decompositions by proving that any n-vertex graph can be decomposed into O(n log* n) cycles and edges, improving previous bounds significantly.

Contribution

It improves the upper bound for the Erd ext{o}s-Gallai Cycle Decomposition Conjecture from O(n log log n) to O(n log* n).

Findings

Achieved a bound of O(n log* n) cycles and edges for graph decomposition.

Enhanced the theoretical understanding of cycle decompositions in graphs.

Progressed towards resolving the Erd ext{o}s-Gallai Cycle Decomposition Conjecture.

Abstract

In the 1960's, Erd\H{o}s 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 $O(n\log\log n)$ 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.