# An improved bound for disjoint directed cycles

## TL;DR

This paper improves the minimum out-degree bound needed to guarantee k vertex-disjoint directed cycles in a graph, reducing it from 64k to 18k, enabling better results for small k.

## Contribution

The authors present a tighter bound on the minimum out-degree for ensuring k disjoint directed cycles, advancing previous results by Alon.

## Key findings

- Minimum out-degree at least 18k guarantees k disjoint cycles
- Improved bound from previous 64k threshold
- Potential for further improvements for small k

## Abstract

We show that every directed graph with minimum out-degree at least $18k$ contains at least $k$ vertex disjoint cycles. This is an improvement over the result of Alon who showed this result for digraphs of minimum out-degree at least $64k$. The main benefit of the argument is that getting better results for small values of $k$ allows for further improvements to the constant.