A note on directed 4-cycles in digraphs

TL;DR

This paper proves that digraphs with a sufficiently large minimum outdegree necessarily contain short directed cycles of length at most 4, using combinatorial methods.

Contribution

It establishes a new threshold for minimum outdegree ensuring the existence of short directed cycles in digraphs.

Findings

Digraphs with outdegree ≥ 0.28866n contain directed 4-cycles.

The proof uses combinatorial techniques.

Provides a new bound for cycle existence in digraphs.

Abstract

Using some combinatorial techniques, in this note, it is proved that if $\alpha\geq 0.28866$, then any digraph on $n$ vertices with minimum outdegree at least $\alpha n$ contains a directed cycle of length at most 4.