# Proof of the Caccetta-Haggkvist conjecture for digraphs with small independence number

## TL;DR

This paper proves the Caccetta-H"aggkvist conjecture for digraphs with small independence number, extending previous results to a broader class of graphs with independence number up to (k+1)/2.

## Contribution

It generalizes the proof of the conjecture to digraphs with independence number at most (k+1)/2, beyond the previously known case of independence number two.

## Key findings

- Confirmed the conjecture for digraphs with independence number ≤ (k+1)/2.
- Extended the class of graphs for which the conjecture holds.
- Provided a new bound relating independence number and cycle length.

## Abstract

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge n/k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $k$. In [2], N. Lichiardopol proved that this conjecture is true for digraphs with independence number equal to two. In this paper, we generalize that result, proving that the conjecture is true for digraphs with independence number at most $(k+1)/2$.