On Caccetta-Haggkvist Conjecture

TL;DR

This paper discusses the Caccetta-Haggkvist conjecture, demonstrating that in certain directed graphs with specified out-degree, a short directed cycle necessarily exists, contributing to the understanding of this longstanding problem.

Contribution

The paper provides a proof that in digraphs with minimum out-degree r, a directed cycle of length at most n/r must exist, advancing partial results on the conjecture.

Findings

Existence of short directed cycles in graphs with high out-degree

Proof of a known bound related to the Caccetta-Haggkvist conjecture

Clarification of conditions guaranteeing small cycles

Abstract

We show that we cannot avoid the existence of at least one directed circuit of length less than or equal to (n/r) in a digraph on n vertices with out-degree greater than or equal to r. This is well-known Caccetta-Haggkvist problem.